12th Notes

Class 12 Physics Notes

Comprehensive Notes for CBSE Class 12 Physics (2025-26 Syllabus)

Chapter 1: Electric Charges and Fields

1. Electric Charges

Definition: Property of matter causing it to experience a force in an electric field.

• Types: Positive (e.g., protons), Negative (e.g., electrons).
• Unit: Coulomb (C).
• Quantization: Charge is discrete, \( q = ne \), where \( e = 1.6 \times 10^{-19} \, \text{C} \), \( n \): Integer.
• Exam Tip: Use quantization to find number of electrons for a given charge.

2. Conservation of Charge

Principle: Total electric charge in an isolated system remains constant.

• Examples: Charge transfer in conductors, pair production.
• Exam Tip: Apply in problems involving charge transfer or redistribution.

3. Coulomb’s Law

Definition: Force between two point charges is proportional to the product of charges and inversely proportional to the square of distance between them.

\( F = k \frac{|q_1 q_2|}{r^2} \)

• \( k \): Coulomb’s constant (\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)), \( q_1, q_2 \): Charges, \( r \): Distance.
• Vector Form:
  \( \vec{F} = k \frac{q_1 q_2}{r^2} \hat{r} \)
  .
• Exam Tip: Use vector form for direction; attractive for unlike charges, repulsive for like charges.

4. Forces Between Multiple Charges

Superposition Principle: Net force on a charge is the vector sum of forces due to all other charges.

\( \vec{F}_{\text{net}} = \sum \vec{F}_i \)

• Exam Tip: Calculate force on one charge by summing individual Coulomb forces; resolve vectors in components.

5. Continuous Charge Distribution

Types: Linear (\( \lambda \): charge per unit length), Surface (\( \sigma \): charge per unit area), Volume (\( \rho \): charge per unit volume).

\( dF = k \frac{dq \, q}{r^2} \)

• Integrate over charge distribution to find total force or field.
• Exam Tip: Use symmetry to simplify integrations for uniform distributions.

6. Electric Field

Definition: Region where a charge experiences a force.

\( \vec{E} = \frac{\vec{F}}{q_0} \)

Due to Point Charge:

\( \vec{E} = k \frac{q}{r^2} \hat{r} \)

• Unit: N/C or V/m.
• Exam Tip: Direction away from positive charge, toward negative charge.

7. Electric Field Lines

Definition: Imaginary lines showing direction of electric field; tangent gives field direction, density indicates strength.

• Originate from positive charges, terminate at negative charges.
• Never intersect; denser near stronger fields.
• Exam Tip: Sketch field lines for point charges or dipoles; use for qualitative field analysis.

8. Electric Dipole

Definition: Pair of equal and opposite charges separated by a small distance.

\( \vec{p} = q \cdot 2a \hat{d} \)

• \( p \): Dipole moment, \( 2a \): Separation, \( \hat{d} \): Direction from negative to positive charge.
• Unit: C·m.

9. Electric Field Due to a Dipole

Axial Point:

\( \vec{E} = \frac{2 k p}{r^3} \hat{p} \)

Equatorial Point:

\( \vec{E} = -\frac{k p}{r^3} \hat{p} \)

• \( r \): Distance from dipole center (\( r \gg a \)).
• Exam Tip: Axial field is stronger than equatorial; use approximations for large \( r \).

10. Torque on a Dipole in Uniform Electric Field

\( \vec{\tau} = \vec{p} \times \vec{E} \)

\( \tau = p E \sin \theta \)

• \( \theta \): Angle between \( \vec{p} \) and \( \vec{E} \).
• Exam Tip: Maximum torque at \( \theta = 90^\circ \); zero at \( \theta = 0^\circ \) or \( 180^\circ \).

11. Electric Flux

Definition: Measure of electric field lines passing through a surface.

\( \phi_E = \vec{E} \cdot \vec{A} = E A \cos \theta \)

• \( \vec{A} \): Area vector, \( \theta \): Angle between \( \vec{E} \) and \( \vec{A} \).
• Unit: N·m²/C.
• Exam Tip: Use for closed surfaces in Gauss’s theorem; \( \cos \theta \) determines sign.

12. Gauss’s Theorem

Statement: Total electric flux through a closed surface is proportional to the charge enclosed.

\( \phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\epsilon_0} \)

• \( \epsilon_0 \): Permittivity of free space (\( 8.85 \times 10^{-12} \, \text{C}^2/(\text{N·m}^2) \)).
• Exam Tip: Use symmetry (spherical, cylindrical, planar) to simplify field calculations.

13. Applications of Gauss’s Theorem

Infinitely Long Straight Wire:

\( E = \frac{\lambda}{2 \pi \epsilon_0 r} \)

• \( \lambda \): Linear charge density, \( r \): Radial distance.

Uniformly Charged Infinite Plane Sheet:

\( E = \frac{\sigma}{2 \epsilon_0} \)

• \( \sigma \): Surface charge density.
• Field independent of distance from plane.

Uniformly Charged Thin Spherical Shell:

• Outside (\( r \geq R \)):
  \( E = \frac{q}{4 \pi \epsilon_0 r^2} \)
  .
• Inside (\( r < R \)):
  \( E = 0 \)
  .
• \( q \): Total charge, \( R \): Radius of shell.
• Exam Tip: Use Gaussian surfaces (cylinder for wire, plane for sheet, sphere for shell); symmetry simplifies calculations.

Class 12 Physics Notes

Comprehensive Notes for CBSE Class 12 Physics (2025-26 Syllabus)

Chapter 2: Electrostatic Potential and Capacitance

1. Electric Potential

Definition: Work done per unit charge to bring a positive test charge from infinity to a point in an electric field.

\( V = \frac{W}{q} \)

• Unit: Volt (V), where \( 1 \, \text{V} = 1 \, \text{J/C} \).
• Exam Tip: Potential is a scalar; use for points in an electric field.

2. Potential Difference

Definition: Work done per unit charge to move a charge between two points.

\( \Delta V = V_B - V_A = \frac{W_{A \to B}}{q} \)

• Exam Tip: Relate to electric field:
  \( E = -\frac{dV}{dx} \)
  (for uniform field).

3. Electric Potential Due to a Point Charge

\( V = \frac{k q}{r} \)

• \( k \): Coulomb’s constant (\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)), \( q \): Charge, \( r \): Distance.
• Positive for positive charge, negative for negative charge.
• Exam Tip: Use for single charge potential calculations; \( V \to 0 \) as \( r \to \infty \).

4. Electric Potential Due to a Dipole

Axial Point:

\( V = \frac{k p}{r^2} \)

Equatorial Point:

\( V = 0 \)

• \( p \): Dipole moment (\( p = q \cdot 2a \)), \( r \): Distance from dipole center (\( r \gg a \)).
• Exam Tip: Zero potential at equatorial due to cancellation; use approximations for large \( r \).

5. Electric Potential Due to a System of Charges

Superposition Principle: Total potential is the scalar sum of potentials due to individual charges.

\( V_{\text{total}} = \sum \frac{k q_i}{r_i} \)

• Exam Tip: Add potentials as scalars; consider signs of charges.

6. Equipotential Surfaces

Definition: Surfaces where electric potential is constant.

• Perpendicular to electric field lines.
• Examples: Spherical surfaces for point charge, planes for uniform field.
• Exam Tip: No work done moving a charge on an equipotential surface (\( W = q \Delta V = 0 \)).

7. Electrical Potential Energy of a System

Two Point Charges:

\( U = \frac{k q_1 q_2}{r} \)

Electric Dipole in Electric Field:

\( U = -\vec{p} \cdot \vec{E} = -p E \cos \theta \)

• \( \theta \): Angle between dipole moment and electric field.
• Exam Tip: Minimum energy at \( \theta = 0^\circ \); use for stability analysis.

8. Conductors and Insulators

Conductors: Materials allowing free movement of charges (e.g., metals).

Insulators: Materials restricting charge movement (e.g., glass, rubber).

• Free Charges: Mobile charges in conductors.
• Bound Charges: Fixed charges in insulators or dielectrics.
• Exam Tip: Inside a conductor, \( E = 0 \), all charge resides on the surface.

9. Dielectrics and Electric Polarization

Dielectrics: Non-conducting materials that polarize in an electric field.

Polarization: Alignment of molecular dipoles, creating an internal field opposing the external field.

• Dielectric Constant (\( \kappa \)): Ratio of permittivity of medium to free space (\( \kappa = \frac{\epsilon}{\epsilon_0} \)).
• Exam Tip: Polarization reduces effective electric field in dielectrics.

10. Capacitors and Capacitance

Definition: Device storing charge, capacitance is charge per unit potential difference.

\( C = \frac{Q}{V} \)

• Unit: Farad (F), where \( 1 \, \text{F} = 1 \, \text{C/V} \).
• Exam Tip: Higher capacitance stores more charge for same voltage.

11. Combination of Capacitors

Series Combination:

\( \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots \)

Parallel Combination:

\( C_{\text{eq}} = C_1 + C_2 + \ldots \)

• Exam Tip: Series reduces capacitance, parallel increases it; use for equivalent capacitance calculations.

12. Capacitance of a Parallel Plate Capacitor

Without Dielectric:

\( C = \frac{\epsilon_0 A}{d} \)

With Dielectric:

\( C = \frac{\kappa \epsilon_0 A}{d} \)

• \( A \): Plate area, \( d \): Separation, \( \epsilon_0 \): Permittivity of free space (\( 8.85 \times 10^{-12} \, \text{C}^2/(\text{N·m}^2) \)).
• Exam Tip: Dielectric increases capacitance by factor \( \kappa \); use for capacitors with partial dielectric.

13. Energy Stored in a Capacitor

\( U = \frac{1}{2} C V^2 \)

\( U = \frac{1}{2} \frac{Q^2}{C} \)

\( U = \frac{1}{2} Q V \)

• Unit: Joule (J).
• Exam Tip: Use any form depending on given quantities; energy stored in electric field between plates.

Chapter 3: Current Electricity

1. Electric Current

Definition: Rate of flow of electric charge through a conductor.

\( I = \frac{Q}{t} \)

• Unit: Ampere (A), \( 1 \, \text{A} = 1 \, \text{C/s} \).
• Direction: Conventional current flows from positive to negative terminal.
• Exam Tip: Use to calculate charge or time in steady current problems.

2. Flow of Electric Charges in a Metallic Conductor

Concept: Free electrons in metals move randomly; under electric field, they drift opposite to field direction.

Drift Velocity (\( v_d \)): Average velocity of electrons due to electric field.

\( v_d = \frac{I}{n e A} \)

• \( n \): Electron density, \( e \): Electron charge (\( 1.6 \times 10^{-19} \, \text{C} \)), \( A \): Cross-sectional area.
• Exam Tip: Relate drift velocity to current for conductor properties.

3. Mobility

Definition: Drift velocity per unit electric field.

\( \mu = \frac{v_d}{E} \)

• Unit: \( \text{m}^2/(\text{V·s}) \).
• Relation with Current:
  \( I = n e A \mu E \)
  .
• Exam Tip: Use mobility to understand charge carrier response to electric field.

4. Ohm’s Law

Statement: Current through a conductor is proportional to potential difference across it, provided physical conditions remain constant.

\( V = I R \)

• \( R \): Resistance, Unit: Ohm (\( \Omega \)).
• Exam Tip: Apply to linear conductors; check conditions for non-ohmic behavior.

5. V-I Characteristics

Linear (Ohmic): Straight line through origin; obeys Ohm’s law (e.g., metals).

Non-linear (Non-ohmic): Non-linear graph (e.g., diodes, filament lamps).

• Exam Tip: Sketch V-I graphs; slope gives resistance for ohmic conductors.

6. Electrical Energy and Power

Electrical Energy:

\( E = V I t \)

Power:

\( P = V I = I^2 R = \frac{V^2}{R} \)

• Unit: Energy in Joule (J), Power in Watt (W).
• Exam Tip: Choose appropriate power formula based on given quantities.

7. Electrical Resistivity and Conductivity

Resistivity (\( \rho \)): Material property opposing current flow.

\( R = \rho \frac{l}{A} \)

Conductivity (\( \sigma \)): Reciprocal of resistivity.

\( \sigma = \frac{1}{\rho} \)

• Units: \( \rho \): \( \Omega \cdot \text{m} \), \( \sigma \): S/m (Siemens per meter).
• Exam Tip: Use resistivity for material comparison; conductors have low \( \rho \).

8. Temperature Dependence of Resistance

Metals: Resistance increases with temperature.

\( R_t = R_0 (1 + \alpha \Delta T) \)

• \( \alpha \): Temperature coefficient of resistance, \( \Delta T \): Temperature change.
• Semiconductors: Resistance decreases with temperature.
• Exam Tip: Use for problems involving heating effects in conductors.

9. Internal Resistance of a Cell

Definition: Resistance within a cell opposing current flow.

\( V = \mathcal{E} - I r \)

• \( \mathcal{E} \): EMF, \( V \): Terminal voltage, \( r \): Internal resistance.
• Exam Tip: Calculate internal resistance from voltage drop under load.

10. Potential Difference and EMF of a Cell

EMF (\( \mathcal{E} \)): Maximum potential difference a cell can provide (open circuit).

Potential Difference (\( V \)): Voltage across external circuit.

\( \mathcal{E} = V + I r \)

• Unit: Volt (V).
• Exam Tip: EMF > terminal voltage when current flows due to internal resistance.

11. Combination of Cells

Series:

\( \mathcal{E}_{\text{eq}} = \mathcal{E}_1 + \mathcal{E}_2 + \ldots \), \( r_{\text{eq}} = r_1 + r_2 + \ldots \)

Parallel (same EMF):

\( \mathcal{E}_{\text{eq}} = \mathcal{E} \), \( \frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2} + \ldots \)

• Exam Tip: Series for higher voltage, parallel for higher current; check EMF equality in parallel.

12. Kirchhoff’s Rules

First Rule (Junction Rule): Sum of currents entering a junction equals sum leaving.

\( \sum I_{\text{in}} = \sum I_{\text{out}} \)

Second Rule (Loop Rule): Sum of potential differences in a closed loop is zero.

\( \sum V = 0 \)

• Exam Tip: Use for complex circuits; assign current directions, apply sign conventions.

13. Wheatstone Bridge

Definition: Circuit to measure unknown resistance.

\( \frac{R_1}{R_2} = \frac{R_3}{R_4} \)

• Balanced when no current flows through galvanometer.
• Exam Tip: Use for finding unknown resistance or checking bridge balance.

Chapter 4: Moving Charges and Magnetism

1. Concept of Magnetic Field

Definition: Region around a magnet or current-carrying conductor where magnetic forces act on moving charges or magnetic materials.

• Unit: Tesla (T), \( 1 \, \text{T} = 1 \, \text{N/(A·m)} \).
• Oersted’s Experiment: Current in a wire deflects a nearby compass needle, showing that current produces a magnetic field.
• Exam Tip: Understand magnetic field as a vector field; direction given by compass or right-hand rule.

2. Biot-Savart Law

Definition: Magnetic field due to a current element.

\( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \)

• \( \mu_0 \): Permeability of free space (\( 4\pi \times 10^{-7} \, \text{T·m/A} \)), \( I \): Current, \( d\vec{l} \): Current element, \( r \): Distance, \( \hat{r} \): Unit vector.
• Application to Circular Loop:
• At center:
  \( B = \frac{\mu_0 I}{2 R} \)
  .
• \( R \): Radius of loop.
• Exam Tip: Use right-hand rule for field direction; integrate for non-central points.

3. Ampere’s Law

Statement: Line integral of magnetic field around a closed loop equals \( \mu_0 \) times the current enclosed.

\( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \)

• Applications:
• Infinitely Long Straight Wire:
  \( B = \frac{\mu_0 I}{2 \pi r} \)
  .
• Straight Solenoid (Qualitative): Uniform field inside, \( B \approx \mu_0 n I \), where \( n \): Turns per unit length.
• Exam Tip: Use symmetry (cylindrical for wire, solenoid) to simplify calculations.

4. Force on a Moving Charge in Uniform Magnetic and Electric Fields

Magnetic Force (Lorentz Force):

\( \vec{F}_B = q (\vec{v} \times \vec{B}) \)

Total Force (with Electric Field):

\( \vec{F} = q \vec{E} + q (\vec{v} \times \vec{B}) \)

• \( q \): Charge, \( \vec{v} \): Velocity, \( \vec{B} \): Magnetic field, \( \vec{E} \): Electric field.
• Magnetic Force Characteristics: Perpendicular to \( \vec{v} \) and \( \vec{B} \), no work done.
• Exam Tip: Use right-hand rule for direction; zero force if \( \vec{v} \) parallel to \( \vec{B} \).

5. Force on a Current-Carrying Conductor

\( \vec{F} = I (\vec{l} \times \vec{B}) \)

• \( I \): Current, \( \vec{l} \): Length vector, \( \vec{B} \): Magnetic field.
• Magnitude:
  \( F = I l B \sin \theta \)
  .
• Exam Tip: Maximum force when conductor perpendicular to field (\( \theta = 90^\circ \)).

6. Force Between Two Parallel Current-Carrying Conductors

\( F = \frac{\mu_0 I_1 I_2}{2 \pi d} l \)

• \( I_1, I_2 \): Currents, \( d \): Separation, \( l \): Length.
• Definition of Ampere: 1 A is the current in two parallel conductors 1 m apart producing \( 2 \times 10^{-7} \, \text{N/m} \) force.
• Direction: Attractive for same-direction currents, repulsive for opposite.
• Exam Tip: Use for calculating force per unit length; check current directions.

7. Torque on a Current Loop in Uniform Magnetic Field

\( \vec{\tau} = \vec{m} \times \vec{B} \)

\( \tau = m B \sin \theta \)

• \( \vec{m} \): Magnetic dipole moment, \( \theta \): Angle between \( \vec{m} \) and \( \vec{B} \).
• Exam Tip: Maximum torque at \( \theta = 90^\circ \); zero at \( \theta = 0^\circ \).

8. Current Loop as a Magnetic Dipole

Magnetic Dipole Moment:

\( \vec{m} = N I \vec{A} \)

• \( N \): Number of turns, \( I \): Current, \( \vec{A} \): Area vector.
• Direction: Right-hand rule (curl fingers in current direction, thumb gives \( \vec{m} \)).
• Exam Tip: Treat loop as dipole for torque and field calculations.

9. Moving Coil Galvanometer

Definition: Device to detect small currents using torque on a coil in a magnetic field.

\( \tau = N I A B \sin \theta \)

Current Sensitivity:

\( S_I = \frac{\theta}{I} = \frac{N A B}{k} \)

• \( k \): Torsional constant of spring, \( \theta \): Deflection.

Conversion to Ammeter: Add shunt resistance in parallel.

\( R_s = \frac{I_g G}{I - I_g} \)

Conversion to Voltmeter: Add high resistance in series.

\( R = \frac{V}{I_g} - G \)

• \( I_g \): Full-scale current, \( G \): Galvanometer resistance, \( I \): Max current, \( V \): Max voltage.
• Exam Tip: Use shunt for low resistance (ammeter), series for high resistance (voltmeter).

Chapter 5: Magnetism and Matter

1. Bar Magnet

Definition: A permanent magnet with distinct north and south poles, producing a magnetic field.

• Poles: North pole (N) attracts south pole (S), repels another north pole.
• Magnetic Moment (\( \vec{m} \)): Product of pole strength and distance between poles.
\( \vec{m} = q_m \cdot 2l \)
• \( q_m \): Pole strength, \( 2l \): Length of magnet.
• Exam Tip: Magnetic moment direction is from south to north pole; use for torque calculations.

2. Bar Magnet as an Equivalent Solenoid (Qualitative)

Concept: A bar magnet produces a magnetic field similar to a solenoid with current loops.

• Similarity: North pole corresponds to solenoid end where current is anticlockwise (viewed from that end).
• Magnetic Moment: For solenoid,
  \( \vec{m} = N I \vec{A} \)
  , similar to bar magnet’s moment.
• Exam Tip: Understand qualitative equivalence; no derivation required, focus on field pattern similarity.

3. Magnetic Field Intensity Due to a Magnetic Dipole (Bar Magnet) – Qualitative

Axial Point (Along Axis):

• Field is along the magnet’s axis, stronger than equatorial.
\( B_{\text{axial}} \propto \frac{\mu_0}{4\pi} \frac{2 m}{r^3} \)

Equatorial Point (Perpendicular to Axis):

• Field is opposite to magnetic moment direction, weaker than axial.
\( B_{\text{equatorial}} \propto \frac{\mu_0}{4\pi} \frac{m}{r^3} \)

• \( m \): Magnetic moment, \( r \): Distance (\( r \gg l \)), \( \mu_0 \): Permeability (\( 4\pi \times 10^{-7} \, \text{T·m/A} \)).
• Exam Tip: Axial field is twice equatorial; focus on qualitative field direction and strength.

4. Torque on a Magnetic Dipole (Bar Magnet) in a Uniform Magnetic Field – Qualitative

Concept: A bar magnet in a uniform magnetic field experiences torque to align with the field.

\( \vec{\tau} = \vec{m} \times \vec{B} \)

\( \tau = m B \sin \theta \)

• \( \theta \): Angle between \( \vec{m} \) and \( \vec{B} \).
• Behavior: Maximum torque at \( \theta = 90^\circ \), zero at \( \theta = 0^\circ \) or \( 180^\circ \).
• Exam Tip: Use right-hand rule for torque direction; no derivation, focus on alignment tendency.

5. Magnetic Field Lines

Definition: Imaginary lines representing magnetic field direction and strength.

• Emerge from north pole, enter south pole; never intersect.
• Denser lines indicate stronger field.
• Exam Tip: Sketch field lines for bar magnet (closed loops); use for qualitative field analysis.

6. Magnetic Properties of Materials

Paramagnetic Substances:

• Weakly attracted to magnetic fields; align with external field.
• Examples: Aluminium, magnesium, platinum.
• Property: Small positive susceptibility (\( \chi > 0 \)).

Diamagnetic Substances:

• Weakly repelled by magnetic fields; induced field opposes external field.
• Examples: Bismuth, copper, water.
• Property: Small negative susceptibility (\( \chi < 0 \)).

Ferromagnetic Substances:

• Strongly attracted to magnetic fields; form permanent magnets.
• Examples: Iron, cobalt, nickel.
• Property: High positive susceptibility (\( \chi \gg 0 \)); exhibit hysteresis.
• Exam Tip: Compare magnetic behavior; ferromagnets retain magnetism after field removal.

7. Magnetization of Materials

Definition: Magnetic moment per unit volume induced in a material.

\( \vec{M} = \frac{\vec{m}}{V} \)

• Relation:
  \( \vec{B} = \mu_0 (\vec{H} + \vec{M}) \)
  , where \( \vec{H} \): Magnetizing field.
• Unit: A/m.
• Exam Tip: Magnetization is significant in ferromagnets; relates to material’s response to external field.

8. Effect of Temperature on Magnetic Properties

Paramagnetic: Magnetization decreases with temperature (Curie’s law).

\( \chi \propto \frac{1}{T} \)

Ferromagnetic: Magnetization decreases with temperature; vanishes above Curie temperature.

Diamagnetic: Weak effect, nearly independent of temperature.

• Exam Tip: Focus on Curie temperature for ferromagnets; higher temperature disrupts magnetic domains.

Chapter 6: Electromagnetic Induction

1. Electromagnetic Induction

Definition: The phenomenon of generating an electromotive force (EMF) in a conductor due to a change in magnetic flux through it.

• Discovery: Michael Faraday (1831).
• Condition: Requires relative motion between a conductor and magnetic field or a change in magnetic field strength.
• Applications: Transformers, generators, inductors.
• Exam Tip: Focus on the concept of changing magnetic flux as the cause of induced EMF.

2. Faraday’s Laws of Electromagnetic Induction

First Law: Whenever the magnetic flux linked with a closed circuit changes, an EMF is induced in the circuit. The induced EMF persists as long as the flux change continues.

Second Law: The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux.

\( \mathcal{E} = -\frac{d\Phi_B}{dt} \)

• \( \Phi_B = \vec{B} \cdot \vec{A} \): Magnetic flux, where \( \vec{B} \) is magnetic field, \( \vec{A} \) is area vector.
• Unit of Flux: Weber (Wb).
• Exam Tip: Understand flux change (due to area, field, or angle); use formula for numerical problems.

3. Induced EMF and Current

Induced EMF: The potential difference generated across a conductor due to changing magnetic flux.

\( \mathcal{E} = -\frac{d\Phi_B}{dt} \)

Induced Current: If the circuit is closed, induced EMF drives a current.

\( I = \frac{\mathcal{E}}{R} \)

• \( R \): Resistance of the circuit.
• Examples: Moving a coil in a magnetic field, rotating a loop in a generator.
• Exam Tip: Calculate induced EMF first, then use Ohm’s law for current if resistance is given.

4. Lenz’s Law

Statement: The direction of induced EMF and current opposes the change in magnetic flux that produces it.

• Conservation: Follows the law of conservation of energy.
• Negative Sign in Faraday’s Law: Indicates opposition to flux change.
• Example: If magnetic flux increases through a loop, induced current creates a magnetic field opposite to the applied field.
• Exam Tip: Use right-hand rule to determine current direction; focus on opposition to flux change.

5. Self-Induction

Definition: The phenomenon where a changing current in a coil induces an EMF in itself due to its own magnetic field.

\( \mathcal{E} = -L \frac{dI}{dt} \)

• \( L \): Self-inductance (or inductance), measured in Henry (H).
• Cause: Change in current alters the coil’s magnetic flux.
• Analogy: Like inertia in mechanics; opposes current change.
• Expression for L (Solenoid):
  \( L = \frac{\mu_0 N^2 A}{l} \)
  • \( N \): Number of turns, \( A \): Cross-sectional area, \( l \): Length, \( \mu_0 \): Permeability of free space.
• Exam Tip: Focus on self-inductance formula and its role in opposing current changes.

6. Mutual Induction

Definition: The phenomenon where a changing current in one coil induces an EMF in a nearby coil due to mutual magnetic flux linkage.

\( \mathcal{E}_2 = -M \frac{dI_1}{dt} \)

• \( M \): Mutual inductance, measured in Henry (H).
• Cause: Changing current in primary coil alters flux through secondary coil.
• Expression for M:
  \( M = \frac{\mu_0 N_1 N_2 A}{l} \)
  • \( N_1, N_2 \): Number of turns in primary and secondary coils.
• Applications: Transformers, inductors in circuits.
• Exam Tip: Compare self and mutual induction; mutual inductance depends on both coils’ properties.

Chapter 7: Alternating Current

1. Alternating Current (AC)

Definition: An electric current that periodically reverses direction, unlike direct current (DC).

• Expression:
  \( i = I_0 \sin(\omega t) \)
  or
  \( i = I_0 \cos(\omega t) \)
• \( I_0 \): Peak current, \( \omega \): Angular frequency, \( t \): Time.
• Frequency: Typically 50 Hz or 60 Hz in household circuits.
• Exam Tip: Understand AC’s sinusoidal nature; focus on time-varying behavior.

2. Peak and RMS Value of Alternating Current/Voltage

Peak Value: Maximum value of current (\( I_0 \)) or voltage (\( V_0 \)) in one cycle.

RMS (Root Mean Square) Value: Effective value equivalent to DC producing the same power.

\( I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0 \)

\( V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0 \)

• Significance: RMS values are used for power calculations in AC circuits.
• Exam Tip: Memorize \( \sqrt{2} \approx 1.414 \); convert between peak and RMS for numerical problems.

3. Reactance and Impedance

Reactance: Opposition to AC by inductors (inductive reactance) or capacitors (capacitive reactance).

• Inductive Reactance (\( X_L \)):
  \( X_L = \omega L \)
• Capacitive Reactance (\( X_C \)):
  \( X_C = \frac{1}{\omega C} \)
• \( L \): Inductance (H), \( C \): Capacitance (F), \( \omega \): Angular frequency (rad/s).

Impedance (\( Z \)): Total opposition to AC in a circuit, combining resistance and reactance.

\( Z = \sqrt{R^2 + (X_L - X_C)^2} \)

• Unit: Ohm (\( \Omega \)).
• Exam Tip: Use impedance for AC circuits; note that \( X_L \) increases and \( X_C \) decreases with frequency.

4. LCR Series Circuit (Phasors)

Concept: An AC circuit with resistor (R), inductor (L), and capacitor (C) in series, analyzed using phasors.

• Phasor Diagram: Voltage across R in phase with current, \( V_L \) leads by 90°, \( V_C \) lags by 90°.
• Total Voltage:
  \( V = \sqrt{V_R^2 + (V_L - V_C)^2} \)
• Impedance:
  \( Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2} \)
• Phase Angle (\( \phi \)):
  \( \tan \phi = \frac{X_L - X_C}{R} \)
• Exam Tip: Draw phasor diagrams to find resultant voltage and phase angle; focus on vector addition.

5. Resonance in LCR Series Circuit

Definition: Condition where inductive reactance equals capacitive reactance, minimizing impedance.

\( X_L = X_C \implies \omega L = \frac{1}{\omega C} \)

\( \omega_0 = \frac{1}{\sqrt{LC}} \)

• \( \omega_0 \): Resonant angular frequency, \( f_0 = \frac{\omega_0}{2\pi} \): Resonant frequency.
• At Resonance: \( Z = R \) (minimum), current is maximum, voltage across L and C cancels out.
• Applications: Radio tuning, filters.
• Exam Tip: Derive resonant frequency; note maximum current and phase angle (\( \phi = 0 \)) at resonance.

6.jasd Power in AC Circuits

Average Power:

\( P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi \)

• \( \cos \phi \): Power factor, \( \phi \): Phase angle between voltage and current.
• Unit: Watt (W).
• Exam Tip: Power is maximum when \( \cos \phi = 1 \) (purely resistive circuit).

7. Power Factor

Definition: Measure of how efficiently AC power is utilized.

\( \cos \phi = \frac{R}{Z} \)

• Range: 0 to 1; \( \cos \phi = 1 \) for resistive circuits, < 1 for L or C circuits.
• Significance: Higher power factor means less wasted power.
• Exam Tip: Calculate power factor using impedance triangle; relate to power efficiency.

8. Wattless Current

Definition: Component of AC current that does not contribute to average power (due to 90° phase difference).

\( I_{\text{wattless}} = I_{\text{rms}} \sin \phi \)

• Occurs in: Pure inductor or capacitor circuits (\( \phi = 90^\circ \), \( \cos \phi = 0 \)).
• Exam Tip: Wattless current is reactive; no power is dissipated in purely reactive components.

9. AC Generator

Definition: Device that converts mechanical energy into electrical energy (AC) using electromagnetic induction.

• Principle: Rotating coil in a magnetic field induces EMF (Faraday’s law).
• EMF:
  \( \mathcal{E} = NBA \omega \sin(\omega t) \)
  • \( N \): Number of turns, \( B \): Magnetic field, \( A \): Area of coil, \( \omega \): Angular velocity.
• Components: Armature, magnets, slip rings, brushes.
• Exam Tip: Understand sinusoidal EMF output; focus on peak EMF (\( NBA \omega \)).

10. Transformer

Definition: Device that transfers electrical energy between circuits via mutual induction, changing voltage levels.

• Types: Step-up (increases voltage), step-down (decreases voltage).
• Voltage Relation:
  \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \)
• Current Relation:
  \( \frac{I_s}{I_p} = \frac{N_p}{N_s} \)
• \( V_p, V_s \): Primary and secondary voltages, \( N_p, N_s \): Primary and secondary turns.
• Efficiency: Nearly 100% in ideal transformers; losses due to resistance, hysteresis.
• Exam Tip: Use turns ratio for calculations; note power conservation (\( V_p I_p = V_s I_s \)).

Chapter 8: Electromagnetic Waves

1. Displacement Current

Definition: A time-varying electric field between the plates of a capacitor produces a current, termed displacement current, which complements conduction current in maintaining continuity.

\( I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \)

• \( \varepsilon_0 \): Permittivity of free space, \( \Phi_E \): Electric flux.
• Significance: Introduced by Maxwell to correct Ampere’s law for time-varying fields.
• Role: Ensures consistency in circuits with capacitors where conduction current is absent.
• Exam Tip: Understand displacement current as a “virtual” current; focus on its role in Maxwell’s equations.

2. Electromagnetic Waves

Definition: Waves consisting of oscillating electric (\( \vec{E} \)) and magnetic (\( \vec{B} \)) fields that propagate through space, mutually perpendicular to each other and to the direction of propagation.

• Generation: Produced by accelerating charges or time-varying currents.
• Speed in Vacuum:
  \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^8 \, \text{m/s} \)
• \( \mu_0 \): Permeability of free space, \( \varepsilon_0 \): Permittivity of free space.
• Exam Tip: Relate EM wave generation to Maxwell’s equations; memorize speed of light formula.

3. Characteristics of Electromagnetic Waves

• Transverse Nature: \( \vec{E} \) and \( \vec{B} \) fields oscillate perpendicular to each other and to the direction of wave propagation.
• Self-Sustaining: No medium required; propagates in vacuum.
• Energy: Carries energy, momentum, and angular momentum.
• Wave Equation:
  \( E = E_0 \sin(\omega t - kx) \), \( B = B_0 \sin(\omega t - kx) \)
• Relation:
  \( E_0 = c B_0 \)
• Exam Tip: Focus on qualitative aspects like transverse nature and no medium requirement.

4. Transverse Nature (Qualitative Idea)

Concept: In EM waves, electric and magnetic field vectors oscillate perpendicular to the wave’s direction of travel, unlike longitudinal waves (e.g., sound).

• Illustration: If wave travels along x-axis, \( \vec{E} \) oscillates in y-direction, \( \vec{B} \) in z-direction.
• Evidence: Polarization of light confirms transverse nature.
• Exam Tip: No mathematical derivation needed; understand perpendicularity of \( \vec{E} \), \( \vec{B} \), and wave direction.

5. Electromagnetic Spectrum

Definition: The range of all possible frequencies of electromagnetic radiation, categorized by wavelength and frequency.

• Order (Increasing Frequency): Radio waves, microwaves, infrared, visible, ultraviolet, X-rays, gamma rays.
• Exam Tip: Memorize the spectrum order and key uses; relate frequency to energy (\( E = h f \)).

5.1 Radio Waves

• Wavelength: \( > 10 \, \text{cm} \), Frequency: \( < 3 \times 10^9 \, \text{Hz} \).
• Uses: Radio and TV broadcasting, wireless communication, radar.
• Exam Tip: Note long wavelengths enable long-distance communication.

5.2 Microwaves

• Wavelength: \( 1 \, \text{mm} - 10 \, \text{cm} \), Frequency: \( 3 \times 10^9 - 3 \times 10^{11} \, \text{Hz} \).
• Uses: Microwave ovens, satellite communication, Wi-Fi.
• Exam Tip: Highlight heating effect due to water molecule resonance.

5.3 Infrared

• Wavelength: \( 700 \, \text{nm} - 1 \, \text{mm} \), Frequency: \( 3 \times 10^{11} - 4 \times 10^{14} \, \text{Hz} \).
• Uses: Remote controls, thermal imaging, physiotherapy.
• Exam Tip: Associate infrared with heat radiation.

5.4 Visible Light

• Wavelength: \( 400 - 700 \, \text{nm} \), Frequency: \( 4 \times 10^{14} - 7.5 \times 10^{14} \, \text{Hz} \).
• Uses: Vision, photography, optical instruments.
• Exam Tip: Memorize wavelength range; relate colors to frequency (red: low, violet: high).

5.5 Ultraviolet (UV)

• Wavelength: \( 10 - 400 \, \text{nm} \), Frequency: \( 7.5 \times 10^{14} - 3 \times 10^{16} \, \text{Hz} \).
• Uses: Sterilization, tanning, vitamin D synthesis.
• Exam Tip: Note harmful effects (e.g., skin damage) and protective measures (e.g., sunscreen).

5.6 X-Rays

• Wavelength: \( 0.01 - 10 \, \text{nm} \), Frequency: \( 3 \times 10^{16} - 3 \times 10^{19} \, \text{Hz} \).
• Uses: Medical imaging, security scanning, material analysis.
• Exam Tip: Emphasize high penetration and ionizing nature.

5.7 Gamma Rays

• Wavelength: \( < 0.01 \, \text{nm} \), Frequency: \( > 3 \times 10^{19} \, \text{Hz} \).
• Uses: Cancer treatment, sterilization, nuclear research.
• Exam Tip: Highlight high energy and dangers (e.g., radiation exposure).

Chapter 9: Ray Optics and Optical Instruments

1. Reflection of Light

Definition: The bouncing back of light when it strikes a surface, following the laws of reflection.

• Laws of Reflection:
  • The angle of incidence (\( \theta_i \)) equals the angle of reflection (\( \theta_r \)).
  • Incident ray, reflected ray, and normal at the point of incidence lie in the same plane.
• Types of Reflection: Regular (smooth surfaces, e.g., mirrors) and diffuse (rough surfaces).
• Exam Tip: Use ray diagrams to show incident and reflected rays; apply laws for plane mirrors.

2. Spherical Mirrors

Definition: Mirrors with a curved surface, either concave (inward) or convex (outward).

• Terminology:
  • Pole (P): Center of the mirror.
  • Center of Curvature (C): Center of the sphere of which the mirror is a part.
  • Radius of Curvature (R): Distance from pole to center of curvature.
  • Principal Axis: Line passing through pole and center of curvature.
  • Focal Length (f):
    \( f = \frac{R}{2} \)
• Sign Convention: New Cartesian sign convention; distances measured from pole, positive for real objects/images, negative for virtual.
• Exam Tip: Memorize focal length relation and sign convention for mirrors.

3. Mirror Formula

Formula:

\( \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \)

• \( u \): Object distance, \( v \): Image distance, \( f \): Focal length.
• Magnification (m):
  \( m = \frac{h_i}{h_o} = -\frac{v}{u} \)
• \( h_i \): Image height, \( h_o \): Object height.
• Concave Mirror: Real or virtual image depending on object position.
• Convex Mirror: Always virtual, erect, and diminished image.
• Exam Tip: Practice numericals using mirror formula; use ray diagrams to confirm image nature.

4. Refraction of Light

Definition: Bending of light as it passes from one medium to another due to change in speed.

• Snell’s Law:
  \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)
• \( n_1, n_2 \): Refractive indices, \( \theta_1 \): Angle of incidence, \( \theta_2 \): Angle of refraction.
• Refractive Index:
  \( n = \frac{c}{v} \)
  , where \( c \): Speed of light in vacuum, \( v \): Speed in medium.
• Exam Tip: Apply Snell’s law for angle calculations; note denser medium bends light toward normal.

5. Total Internal Reflection and Optical Fibers

Total Internal Reflection (TIR): When light traveling from a denser to a rarer medium is reflected back if the angle of incidence exceeds the critical angle.

\( \sin \theta_c = \frac{n_2}{n_1} \), where \( n_1 > n_2 \).

• \( \theta_c \): Critical angle.
• Conditions: Light must travel from denser to rarer medium; angle of incidence > critical angle.
• Optical Fibers: Thin glass fibers that use TIR to transmit light signals with minimal loss.
• Uses: Telecommunications, medical imaging (endoscopy).
• Exam Tip: Calculate critical angle; understand TIR applications in optical fibers.

6. Refraction at Spherical Surfaces

Formula:

\( \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \)

• \( n_1, n_2 \): Refractive indices of media, \( u \): Object distance, \( v \): Image distance, \( R \): Radius of curvature.
• Sign Convention: Same as for mirrors; positive for real, negative for virtual.
• Exam Tip: Use for single spherical surfaces (e.g., glass-air interface); practice numericals.

7. Lenses

Definition: Transparent objects with curved surfaces that refract light to form images.

• Types: Convex (converging) and concave (diverging).
• Terminology: Optical center, principal axis, focal point, focal length.
• Sign Convention: Similar to mirrors; focal length positive for convex, negative for concave.
• Exam Tip: Draw ray diagrams for convex and concave lenses to determine image properties.

8. Thin Lens Formula

Formula:

\( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \)

• \( u \): Object distance, \( v \): Image distance, \( f \): Focal length.
• Magnification:
  \( m = \frac{h_i}{h_o} = \frac{v}{u} \)
• Exam Tip: Use thin lens formula for image position; confirm with ray diagrams.

9. Lens Maker’s Formula

Formula:

\( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)

• \( n \): Refractive index of lens material, \( R_1, R_2 \): Radii of curvature of lens surfaces.
• Sign Convention: \( R_1 \) positive for convex surface toward incident light, \( R_2 \) negative for convex away from light.
• Exam Tip: Use lens maker’s formula to calculate focal length; understand dependence on refractive index and radii.

10. Power of a Lens

Definition: Measure of a lens’s ability to converge or diverge light.

\( P = \frac{1}{f} \)

• \( f \): Focal length (in meters), \( P \): Power (in diopters, D).
• Sign: Positive for convex lenses, negative for concave lenses.
• Exam Tip: Use power for quick calculations in optical systems; relate to focal length.

11. Combination of Thin Lenses in Contact

Formula:

\( \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \)

• \( F \): Equivalent focal length, \( f_1, f_2 \): Focal lengths of individual lenses.
• Power of Combination:
  \( P = P_1 + P_2 \)
• Exam Tip: Add powers for lenses in contact; use sign convention for convex/concave lenses.

12. Refraction of Light Through a Prism

Definition: A prism disperses light due to refraction at its surfaces, splitting white light into colors.

• Angle of Deviation (\( \delta \)):
  \( \delta = i + e - A \)
• \( i \): Angle of incidence, \( e \): Angle of emergence, \( A \): Angle of prism.
• Minimum Deviation:
  \( n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \)
• Exam Tip: Use minimum deviation formula to find refractive index; draw ray diagrams for prism refraction.

13. Optical Instruments

13.1 Microscope

Definition: An instrument that magnifies small objects using two convex lenses (objective and eyepiece).

• Magnifying Power (Compound Microscope):
  \( M = \frac{v_o}{u_o} \cdot \frac{D}{u_e} \)
• \( v_o, u_o \): Image and object distances for objective, \( u_e \): Object distance for eyepiece, \( D \): Least distance of distinct vision (25 cm).
• Exam Tip: Focus on magnifying power formula; note that image is inverted and magnified.

13.2 Astronomical Telescope

Definition: An instrument for observing distant objects (e.g., stars) using two lenses (refracting) or a mirror and lens (reflecting).

• Refracting Telescope (Normal Adjustment):
  • Magnifying Power:
    \( M = \frac{f_o}{f_e} \)
  • \( f_o \): Focal length of objective, \( f_e \): Focal length of eyepiece.
• Reflecting Telescope: Uses a concave mirror as objective; reduces spherical aberration.
• Exam Tip: Compare magnifying powers; note reflecting telescopes are preferred for large apertures.

Chapter 10: Wave Optics

1. Wavefront and Huygen’s Principle

Wavefront: A surface joining all points of a wave vibrating in phase, perpendicular to the direction of wave propagation.

• Types: Spherical (from point source), cylindrical (from linear source), plane (at large distances).
• Huygen’s Principle: Every point on a wavefront acts as a source of secondary spherical wavelets, which spread out in the forward direction at the speed of the wave. The new wavefront is the tangent to these secondary wavelets.
• Exam Tip: Understand wavefront shapes and Huygen’s principle for wave propagation; use for reflection and refraction proofs.

2. Reflection of Plane Wave at a Plane Surface Using Wavefronts

Concept: A plane wavefront incident on a plane mirror produces a reflected wavefront obeying the laws of reflection.

• Process: Incident wavefront strikes the mirror; secondary wavelets from the mirror form a reflected wavefront.
• Law of Reflection: Angle of incidence (\( \theta_i \)) equals angle of reflection (\( \theta_r \)).
• Exam Tip: Use Huygen’s principle to show reflected wavefront maintains the same angle; draw wavefront diagrams.

3. Refraction of Plane Wave at a Plane Surface Using Wavefronts

Concept: A plane wavefront passing from one medium to another bends due to change in wave speed, following Snell’s law.

• Process: Secondary wavelets in the second medium travel at a different speed, forming a refracted wavefront.
• Snell’s Law:
  \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \)
• \( n_1, n_2 \): Refractive indices, \( \theta_1 \): Angle of incidence, \( \theta_2 \): Angle of refraction.
• Exam Tip: Relate refraction to speed change; use wavefront diagrams to visualize bending.

4. Proof of Laws of Reflection Using Huygen’s Principle

Proof (Qualitative): For a plane wavefront incident on a plane mirror, secondary wavelets emitted from the mirror surface form a reflected wavefront such that the angle of incidence equals the angle of reflection.

• Key Points:
  • Wavefront remains plane after reflection.
  • Time taken for secondary wavelets to form new wavefront ensures \( \theta_i = \theta_r \).
• Exam Tip: Focus on geometric construction using Huygen’s principle; no derivation required, understand equality of angles.

5. Proof of Laws of Refraction Using Huygen’s Principle

Proof (Qualitative): A plane wavefront crossing a boundary between two media changes speed, causing the wavefront to bend. Secondary wavelets in the second medium travel slower (or faster), leading to Snell’s law.

• Key Points:
  • Wave speed: \( v = \frac{c}{n} \), where \( c \) is speed of light in vacuum, \( n \) is refractive index.
  • Time consistency of secondary wavelets derives \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \).
• Exam Tip: Understand speed change causes bending; use Huygen’s construction to explain Snell’s law.

6. Interference

Definition: The superposition of two or more waves resulting in a new wave pattern, producing regions of increased or decreased amplitude.

• Types:
  • Constructive Interference: Waves in phase, amplitudes add.
  • Destructive Interference: Waves out of phase, amplitudes subtract.
• Conditions: Coherent sources, same frequency, constant phase difference.
• Exam Tip: Focus on constructive and destructive interference; relate to bright and dark fringes.

7. Young’s Double Slit Experiment

Setup: Two coherent light sources (slits) illuminated by monochromatic light produce interference patterns on a screen.

• Fringe Width (\( \beta \)):
  \( \beta = \frac{\lambda D}{d} \)
• \( \lambda \): Wavelength, \( D \): Distance from slits to screen, \( d \): Slit separation.
• Bright Fringes: At path difference \( n\lambda \), where \( n = 0, 1, 2, \ldots \).
• Dark Fringes: At path difference \( (n + \frac{1}{2})\lambda \).
• Exam Tip: Memorize fringe width formula; understand dependence on \( \lambda \), \( D \), and \( d \).

8. Coherent Sources and Sustained Interference

Coherent Sources: Sources with a constant phase difference and same frequency, producing stable interference patterns.

• Examples: Two slits in Young’s experiment, laser light.
• Sustained Interference: Requires coherent sources, same wavelength, and overlapping waves.
• Exam Tip: Emphasize constant phase difference for sustained interference; incoherent sources produce no stable pattern.

9. Diffraction Due to a Single Slit

Definition: Bending of light around the edges of a narrow slit, producing a pattern of bright and dark bands.

• Central Maximum: Bright central band where light waves constructively interfere.
• Minima: Dark bands at path difference \( n\lambda \), where \( n = 1, 2, \ldots \).
• Condition for Minima:
  \( a \sin \theta = n \lambda \)
• \( a \): Slit width, \( \theta \): Angle of diffraction.
• Exam Tip: Focus on qualitative understanding; diffraction occurs due to wave nature of light.

10. Width of Central Maxima (Qualitative Treatment)

Definition: The angular width of the central bright band in a single slit diffraction pattern.

• Expression: Angular width of central maximum:
  \( 2\theta = \frac{2\lambda}{a} \)
• Dependence: Wider for smaller slit width (\( a \)) or longer wavelength (\( \lambda \)).
• Qualitative Idea: Central maximum is broader than secondary maxima; intensity decreases away from center.
• Exam Tip: Understand inverse relation between slit width and central maximum width; no derivation needed.

Chapter 11: Dual Nature of Radiation and Matter

1. Dual Nature of Radiation

Definition: Electromagnetic radiation exhibits both wave-like and particle-like properties, depending on the experiment.

• Wave Nature: Exhibited in phenomena like interference, diffraction, and polarization.
• Particle Nature: Exhibited in phenomena like the photoelectric effect and Compton scattering.
• Exam Tip: Understand that light behaves as waves (e.g., Young’s double slit) or particles (e.g., photons in photoelectric effect).

2. Photoelectric Effect

Definition: The emission of electrons (photoelectrons) from a metal surface when illuminated by light of a certain frequency.

• Key Features:
  • Electrons are emitted only if light frequency exceeds the threshold frequency (\( f_0 \)).
  • Photoelectric current is proportional to light intensity.
  • Electron emission is instantaneous.
• Exam Tip: Focus on threshold frequency and intensity dependence; relate to particle nature of light.

3. Hertz and Lenard’s Observations

Hertz’s Observation (1887): Discovered that ultraviolet light on electrodes produced sparks, indicating electron emission (photoelectric effect).

Lenard’s Observation (1900): Found that photoelectrons’ kinetic energy depends on light frequency, not intensity, and emission occurs instantly.

• Significance: Challenged wave theory; supported particle nature of light.
• Exam Tip: Emphasize experimental evidence for frequency dependence and instantaneous emission.

4. Einstein’s Photoelectric Equation

Concept: Light consists of photons, each with energy \( E = h f \), where \( h \) is Planck’s constant and \( f \) is frequency.

Equation:

\( h f = \phi + \frac{1}{2} m v_{\text{max}}^2 \)

• \( h \): Planck’s constant (\( 6.626 \times 10^{-34} \, \text{J·s} \)), \( f \): Frequency of light.
• \( \phi \): Work function (minimum energy to eject an electron).
• \( \frac{1}{2} m v_{\text{max}}^2 \): Maximum kinetic energy of photoelectrons.
• Threshold Frequency:
  \( f_0 = \frac{\phi}{h} \)
• Exam Tip: Use equation to calculate kinetic energy, work function, or threshold frequency in numericals.

5. Experimental Study of Photoelectric Effect

Setup: A photocell with a photosensitive cathode and anode in an evacuated tube, connected to a variable voltage source and ammeter.

• Observations:
  • Threshold Frequency: No emission below \( f_0 \), regardless of intensity.
  • Current vs. Intensity: Photoelectric current increases linearly with light intensity.
  • Kinetic Energy vs. Frequency: Maximum kinetic energy of photoelectrons increases linearly with frequency above \( f_0 \).
  • Stopping Potential: Voltage required to stop photoelectrons;
    \( e V_s = \frac{1}{2} m v_{\text{max}}^2 \)
    .
• Exam Tip: Relate experimental results to Einstein’s equation; sketch graphs (current vs. intensity, stopping potential vs. frequency).

6. Matter Waves – Wave Nature of Particles

Concept: Particles like electrons exhibit wave-like properties, such as diffraction and interference, under certain conditions.

• Evidence: Electron diffraction experiments (e.g., Davisson-Germer experiment).
• Significance: Extends wave-particle duality to matter, complementing the dual nature of light.
• Exam Tip: Understand that all matter has wave-like properties, especially at microscopic scales.

7. De-Broglie Relation

Concept: Every moving particle has an associated wave with a wavelength dependent on its momentum.

\( \lambda = \frac{h}{p} \)

• \( \lambda \): De-Broglie wavelength, \( h \): Planck’s constant, \( p \): Momentum (\( p = m v \)).
• For Electrons:
  \( \lambda = \frac{h}{\sqrt{2 m E}} \)
  , where \( E \): Kinetic energy.
• Applications: Electron microscopy, quantum mechanics.
• Exam Tip: Use de-Broglie relation for wavelength calculations; relate to electron diffraction.

Chapter 12: Atoms

1. Alpha-Particle Scattering Experiment

Setup: Conducted by Geiger and Marsden under Rutherford’s guidance (1911); alpha particles from a radioactive source (e.g., radon) were directed at a thin gold foil, with a fluorescent screen to detect scattered particles.

• Observations:
  • Most alpha particles passed through undeflected.
  • Some were deflected at small angles.
  • A few (1 in 20,000) were deflected at large angles (>90°).
• Conclusions: Atom has a small, dense, positively charged nucleus; most of the atom is empty space.
• Exam Tip: Focus on the significance of large-angle scattering; relate to nuclear model of the atom.

2. Rutherford’s Model of Atom

Concept: Proposed based on alpha-particle scattering; atom consists of a tiny, positively charged nucleus surrounded by electrons in orbits.

• Features:
  • Nucleus contains most of the atom’s mass and positive charge (protons).
  • Electrons revolve around the nucleus like planets around the sun.
  • Atom is mostly empty space.
• Limitations: Could not explain electron stability (electrons should radiate energy and spiral into the nucleus).
• Exam Tip: Compare with Thomson’s plum-pudding model; emphasize nucleus discovery.

3. Bohr Model of Hydrogen Atom

Concept: Niels Bohr (1913) proposed a model to explain electron stability and spectral lines in hydrogen.

• Postulates:
  • Electrons revolve in discrete, stable orbits with quantized angular momentum:
    \( L = m v r = n \frac{h}{2\pi} \)
    , where \( n \) is the principal quantum number.
  • Electrons do not radiate energy in stable orbits.
  • Energy is emitted or absorbed as a photon when an electron jumps between orbits:
    \( \Delta E = h f \)
    .
• Exam Tip: Understand quantization and energy transitions; relate to spectral lines.

4. Expression for Radius of nth Possible Orbit

Formula:

\( r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} \)

• \( r_n \): Radius of nth orbit, \( n \): Principal quantum number, \( h \): Planck’s constant, \( \varepsilon_0 \): Permittivity of free space, \( m \): Electron mass, \( e \): Electron charge.
• For Hydrogen (n=1):
  \( r_1 \approx 0.529 \times 10^{-10} \, \text{m} \) (Bohr radius).
• Dependence: Radius increases with \( n^2 \).
• Exam Tip: Memorize radius formula; use for numerical calculations of orbit size.

5. Velocity of Electron in nth Orbit

Formula:

\( v_n = \frac{e^2}{2 \varepsilon_0 n h} \)

• \( v_n \): Velocity of electron in nth orbit.
• For Hydrogen (n=1):
  \( v_1 \approx 2.19 \times 10^6 \, \text{m/s} \).
• Dependence: Velocity decreases with \( 1/n \).
• Exam Tip: Understand inverse relation with \( n \); use in numerical problems.

6. Energy of Electron in nth Orbit

Formula:

\( E_n = -\frac{13.6}{n^2} \, \text{eV} \)

• \( E_n \): Total energy of electron in nth orbit.
• For Hydrogen (n=1):
  \( E_1 = -13.6 \, \text{eV} \).
• Significance: Negative energy indicates bound state; energy becomes less negative as \( n \) increases.
• Exam Tip: Memorize energy formula; use for calculating energy differences in transitions.

7. Hydrogen Line Spectra (Qualitative Treatment)

Concept: When an electron in a hydrogen atom jumps from a higher orbit (\( n_2 \)) to a lower orbit (\( n_1 \)), it emits a photon with energy corresponding to the energy difference.

\( \Delta E = E_{n_2} - E_{n_1} = h f = \frac{h c}{\lambda} \)

• Rydberg Formula:
  \( \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \)
  , where \( R \): Rydberg constant (\( 1.097 \times 10^7 \, \text{m}^{-1} \)).
• Spectral Series:
  • Lyman Series: \( n_1 = 1 \), \( n_2 = 2, 3, \ldots \) (Ultraviolet).
  • Balmer Series: \( n_1 = 2 \), \( n_2 = 3, 4, \ldots \) (Visible).
  • Paschen Series: \( n_1 = 3 \), \( n_2 = 4, 5, \ldots \) (Infrared).
• Exam Tip: Focus on qualitative understanding of spectral lines; identify series by wavelength region.

Chapter 13: Nuclei

1. Composition and Size of Nucleus

Composition: The nucleus is the central core of an atom, composed of positively charged protons and neutral neutrons, collectively known as nucleons. Protons determine the chemical properties of the element, while neutrons contribute to the stability of the nucleus.

• Atomic Number (Z): Represents the number of protons in the nucleus, which defines the element's position in the periodic table and its chemical behavior.
• Mass Number (A): The total number of protons and neutrons (\( A = Z + N \)), where \( N \) is the number of neutrons. Isotopes of an element have the same Z but different A due to varying N.
• Notation: Represented as \( _Z^A\text{X} \), where X is the chemical symbol of the element. For example, \( ^{12}_6\text{C} \) for carbon-12.
• Size of Nucleus: The nucleus is extremely small compared to the atom's size. Its radius is approximated by the empirical formula
  \( R = R_0 A^{1/3} \)
  , where \( R_0 \) is a constant approximately equal to 1.2 fm (1 fm = 10^{-15} m). This shows that the volume of the nucleus is proportional to A, leading to constant nuclear density.
• Density: Nuclear density is about \( 2.3 \times 10^{17} \, \text{kg/m}^3 \), which is immense—equivalent to compressing the mass of a large mountain into a sugar cube. This density is independent of A for most nuclei, indicating uniform packing of nucleons.
• Exam Tip: Remember the radius formula to calculate nuclear size; emphasize that the nucleus occupies only a tiny fraction (about 10^{-5}) of the atom's volume, with the rest being empty space filled by electron cloud.

2. Nuclear Force

Definition: The nuclear force, also known as the strong nuclear force, is the attractive force that holds protons and neutrons together in the nucleus, counteracting the electrostatic repulsion between protons.

• Characteristics:
  • Short Range: Operates effectively over very short distances, typically 1-3 femtometers. Beyond this range, the force drops rapidly to zero, unlike gravitational or electromagnetic forces.
  • Charge Independent: The force is the same between any pair of nucleons—proton-proton, neutron-neutron, or proton-neutron—regardless of charge. This explains why nuclei with many protons can be stable with sufficient neutrons.
  • Strongest Force: It is the strongest of the four fundamental forces (strong, weak, electromagnetic, gravity) at nuclear distances, about 100 times stronger than electromagnetic force.
  • Non-Central and Saturating: Unlike Coulomb force, it does not follow an inverse-square law and saturates, meaning each nucleon interacts only with its nearest neighbors, similar to molecular bonds.
  • Origin: Arises from the exchange of mesons (like pions) between nucleons, according to quantum chromodynamics (QCD), though at introductory levels, it's treated phenomenologically.
• Exam Tip: Highlight how nuclear force overcomes proton repulsion to stabilize the nucleus; compare its strength and range with electromagnetic force, which is repulsive between protons and longer-range.

3. Mass-Energy Relation

Concept: Based on Einstein's special theory of relativity, mass can be converted into energy and vice versa, quantified by the famous equation that relates mass and energy.

\( E = m c^2 \)

• \( m \): Rest mass of the object, \( c \): Speed of light in vacuum (\( 3 \times 10^8 \, \text{m/s} \)), \( E \): Equivalent energy.
• Application in Nuclei: In nuclear reactions, a small amount of mass is converted into a large amount of energy due to the \( c^2 \) factor. For example, in fission or fusion, the total mass of products is less than reactants, releasing energy.
• Atomic Mass Unit Conversion: 1 atomic mass unit (u) = 931.5 MeV/c^2, so mass defect in u can be directly converted to energy in MeV.
• Exam Tip: Use this relation to compute energy released in nuclear processes; remember that even tiny mass differences yield enormous energy, explaining the power of nuclear reactions.

4. Mass Defect

Definition: The mass defect is the difference between the total mass of the isolated nucleons and the actual mass of the nucleus, reflecting the mass converted into binding energy.

\( \Delta m = [Z m_p + (A - Z) m_n + Z m_e] - M \)

(for atomic mass, including electron masses)

• \( m_p \): Proton mass, \( m_n \): Neutron mass, \( m_e \): Electron mass, \( M \): Atomic mass of the element.
• Significance: Positive mass defect indicates a stable nucleus, as energy is required to break it apart. It quantifies the stability provided by nuclear forces.
• Example: For \( ^4_2\text{He} \), mass defect is about 0.0304 u, leading to high stability.
• Exam Tip: Practice calculating mass defect using precise masses; note that atomic masses are used in calculations to account for electron binding, which is negligible compared to nuclear binding.

5. Binding Energy Per Nucleon and Its Variation with Mass Number

Binding Energy (BE): The minimum energy needed to disassemble the nucleus into its constituent nucleons, equivalent to the energy released when nucleons combine to form the nucleus.

\( BE = \Delta m \cdot c^2 = [\Delta m \, \text{(in u)}] \times 931.5 \, \text{MeV} \)

Binding Energy Per Nucleon:

\( \frac{BE}{A} \)

, a measure of nuclear stability per particle.

• Variation with Mass Number (A):
  • Light Nuclei (A < 20): Low binding energy per nucleon (2-8 MeV), increasing with A as more nucleons allow stronger binding. Fusion of light nuclei releases energy.
  • Medium Nuclei (A ≈ 56, e.g., Fe, Ni): Peak binding energy per nucleon (~8.7-8.8 MeV), indicating maximum stability. These elements are the most stable in the universe.
  • Heavy Nuclei (A > 56): Binding energy per nucleon decreases gradually (~8.5 to 7.5 MeV) due to increasing Coulomb repulsion among protons. Fission of heavy nuclei releases energy.
• Significance: The curve explains why fusion occurs in stars for light elements and why fission is used in power plants for heavy elements. Stability decreases for very heavy or very light nuclei.
• Exam Tip: Draw and interpret the binding energy per nucleon vs. A graph; identify fusion/fission regions and the stability peak at iron-56.

6. Nuclear Fission

Definition: A nuclear reaction where a heavy nucleus splits into two or more lighter nuclei upon absorbing a neutron, releasing tremendous energy, neutrons, and radiation.

• Mechanism: The nucleus becomes unstable after neutron absorption, deforming and splitting. Released neutrons can trigger a chain reaction if critical mass is present.
• Example Reaction: \( ^{235}_{92}\text{U} + ^1_0\text{n} \rightarrow ^{236}_{92}\text{U}^* \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3 ^1_0\text{n} + \sim 200 \, \text{MeV} \).
• Energy Release: Arises from the mass defect, as daughter nuclei have higher binding energy per nucleon than the parent.
• Applications: Controlled in nuclear reactors for power generation; uncontrolled in atomic bombs.
• Exam Tip: Explain chain reaction and criticality; calculate energy release using mass defect for specific reactions.

7. Nuclear Fusion

Definition: A nuclear reaction where two light nuclei combine to form a heavier nucleus, releasing energy due to increased binding energy per nucleon.

• Mechanism: Requires overcoming electrostatic repulsion between positively charged nuclei, achieved at high temperatures (~10^7 K) where plasma forms and kinetic energy is sufficient.
• Example Reaction: Proton-proton chain in stars: \( 4 ^1_1\text{H} \rightarrow ^4_2\text{He} + 2 e^+ + 2 \nu_e + 26.7 \, \text{MeV} \).
• Energy Release: Greater per nucleon than fission, but harder to control due to extreme conditions needed.
• Applications: Powers the sun and stars; used in hydrogen bombs; research ongoing for controlled fusion reactors (e.g., ITER project).
• Exam Tip: Contrast with fission: fusion for light nuclei, requires high temperature/pressure; discuss challenges in achieving net energy gain.

Chapter 14: Semiconductor Electronics – Materials, Devices and Simple Circuits

1. Energy Bands in Conductors, Semiconductors and Insulators

Concept of Energy Bands: In an isolated atom, electrons occupy discrete energy levels. In a solid crystal containing a very large number of atoms, these levels overlap and form continuous energy bands. The two most important are the Valence Band (VB) and Conduction Band (CB), separated by an energy gap called the Forbidden Energy Gap (E_g).

• Conductor: In metals, either the valence band overlaps with the conduction band or the conduction band is partially filled. Hence, electrons can move freely under an applied electric field → very high conductivity. Example: Copper (Cu), Aluminium (Al).
• Insulator: In insulators, the forbidden energy gap is very large (> 3 eV). Electrons in the valence band cannot jump to the conduction band under normal conditions → negligible conductivity. Example: Diamond, glass, rubber.
• Semiconductor: Semiconductors have a small band gap (~1 eV). At room temperature, some valence electrons gain sufficient thermal energy to cross into the conduction band, leaving behind holes in the valence band. Conductivity is moderate and strongly temperature-dependent. Example: Silicon (Si, E_g = 1.1 eV), Germanium (Ge, E_g = 0.7 eV).
• Temperature Effect: In metals, resistance increases with temperature (due to collisions). In semiconductors, resistance decreases with temperature (more electrons cross band gap).
• Exam Tip: Remember band gap values and note the opposite temperature behavior of metals vs semiconductors.

2. Intrinsic and Extrinsic Semiconductors

Intrinsic Semiconductor: A pure semiconductor without any impurity. Its conduction is entirely due to thermally generated electron-hole pairs.

• At T = 0 K, it behaves like an insulator (no free carriers).
• At room temperature, a small number of electrons jump to conduction band, creating holes in valence band.
• Both electron concentration (n) and hole concentration (p) are equal in an intrinsic semiconductor → n = p = n_i (intrinsic carrier concentration).
• Conductivity:
  σ = n e μ_e + p e μ_h = n_i e (μ_e + μ_h)
  where μ_e and μ_h are electron and hole mobilities.

Extrinsic Semiconductor: Conductivity of a semiconductor can be improved by adding small amounts of impurity atoms – a process called doping. Two types:

• n-type Semiconductor: Doping with pentavalent impurity (e.g., P, As, Sb) provides an extra electron. These free electrons become majority carriers, while holes are minority carriers. Example: Si doped with phosphorus.
• p-type Semiconductor: Doping with trivalent impurity (e.g., B, Al, Ga) creates deficiency of one electron (a hole). Holes become majority carriers, electrons are minority carriers. Example: Si doped with boron.
• Exam Tip: Always state majority and minority carriers clearly. Remember: n-type → electrons majority; p-type → holes majority.

3. p–n Junction

Formation: A p–n junction is formed when a p-type and an n-type semiconductor are joined together. At the junction:

• Electrons from the n-side diffuse into p-side and recombine with holes.
• Holes from the p-side diffuse into n-side and recombine with electrons.
• This leaves behind charged immobile ions, creating a depletion region (region depleted of free charge carriers).
• An internal electric field (built-in potential barrier) is set up across the junction, preventing further diffusion.

Barrier Potential: For Si ~0.7 V, for Ge ~0.3 V.

• Forward Bias: External voltage applied such that p-side is positive, n-side is negative → depletion width decreases → current flows easily.
• Reverse Bias: p-side negative, n-side positive → depletion width increases → only a very small reverse saturation current flows due to minority carriers.
• Exam Tip: Draw depletion region diagrams for both forward and reverse bias.

4. Semiconductor Diode

Definition: A semiconductor diode is a p–n junction device that allows current to flow only in one direction. It is the basic building block of many electronic circuits.

• Forward Bias Characteristics:
  • No significant current until threshold voltage (~0.3 V for Ge, ~0.7 V for Si).
  • Beyond threshold, current increases exponentially with applied voltage.
• Reverse Bias Characteristics:
  • Only a tiny reverse saturation current (due to minority carriers) flows.
  • At very high reverse voltage, breakdown occurs (Zener or avalanche breakdown) → large current.
• I–V Curve: Shows exponential rise in forward bias and nearly zero current in reverse bias (till breakdown).
• Exam Tip: Clearly mark threshold voltage, reverse saturation current, and breakdown region in the I–V graph.

5. Application of Diode – Rectifier

Rectification: The process of converting alternating current (AC) into direct current (DC) using diodes is called rectification.

• Half-wave Rectifier:
  • Uses a single diode.
  • Only one half (positive or negative) of AC is allowed through → output is pulsating DC.
  • Inefficient, as half the input is wasted.
• Full-wave Rectifier:
  • Uses two diodes (with center-tap transformer) or four diodes (bridge rectifier).
  • Both halves of AC are converted into pulsating DC.
  • More efficient than half-wave rectifier.
• Filtering: A capacitor or inductor is used to smooth out pulsations, providing nearly steady DC output.
• Applications: Power supply circuits for radios, TVs, chargers, etc.
• Exam Tip: Draw neat circuit diagrams of half-wave and full-wave rectifiers, showing input/output waveforms.

Class 11 Notes (Target)

…your Class 11 chapters…

Class 12 Notes (Target)

…your Class 12 chapters…