Full Physics Short Notes (Part 3) – Electromagnet to Alternating Current | Class 12 & JEE 2026

 

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⭐ Full Physics Short Notes (Part 3) | Class 12 & JEE 2026

Electromagnetic Induction (EMI) is one of the most important topics in Class 12 Physics and JEE/NEET exams. It deals with how changing magnetic fields can produce electric currents. This chapter includes magnetic flux, Faraday’s laws, Lenz’s law, motional emf, self and mutual induction, LR circuits, and energy stored in inductors.

Alternating current (AC) is one of the most important chapters in Class 12 Physics and JEE/NEET exams. It explains the behaviour of current and voltage in AC circuits, reactance, impedance, power, resonance, transformer, and various important formulas.

✅ SECTION 1 - EMI

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🔵 1. Magnetic Flux (Φ)

Magnetic flux measures the total number of magnetic field lines passing through a surface.

⭐ Formula for uniform magnetic field:

\[
\phi = \vec{B}\cdot\vec{A} = BA\cos\theta
\]

Where:

• B = magnetic field

• A = area

• θ = angle between B and normal to area

⭐ For non-uniform field:

\[
\phi = \int \vec{B} \cdot d\vec{A}
\]

Unit: Weber (Wb)

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🔵 2. Faraday’s Laws of Electromagnetic Induction

⭐ First Law

Whenever magnetic flux linked with a circuit changes, an emf is induced in the circuit.

⭐ Second Law

The magnitude of induced emf is proportional to the rate of change of flux:

\[
\epsilon \propto \frac{d\phi}{dt}
\]

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🔵 3. Lenz’s Law

The direction of induced emf is such that it opposes the cause producing it.

This gives rise to the negative sign in Faraday’s law:

\[
e = -\frac{d\phi}{dt}
\]

This law ensures conservation of energy.

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🔵 4. Motional EMF

When a conductor moves across a magnetic field, an emf is induced in it. If it forms a closed loop, a current flows.

⭐ EMF across a moving straight conductor:

\[
E = BLv \sin\theta
\]

Where:

• B = magnetic field

• L = length of conductor

• v = velocity

• θ = angle between v and B

Condition for maximum emf → conductor moves perpendicular to B (θ = 90°).

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🔵 5. EMF of a Rotating Coil (AC Generator Principle)

When a coil rotates in a magnetic field, the flux through it changes periodically.

Instantaneous emf:

\[
E = NAB\omega \sin(\omega t) = E_0 \sin(\omega t)
\]

Where:

• N = number of turns

• A = area

• B = magnetic field

• ω = angular velocity

• E₀ = maximum emf
  

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🔵 6. Self Induction & Self-Inductance (L)

A coil opposes any change in current through it. This property is self-inductance.

⭐ Flux linked:

\[
\phi_s = Li
\]

⭐ Self-induced emf:

\[
e_s = -\frac{d\phi_s}{dt} = -L\frac{di}{dt}
\]

Unit: Henry (H)

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🔵 7. Mutual Induction & Mutual Inductance (M)

When two coils are placed close, a changing current in one induces emf in the other.

⭐ Flux linkage:

\[
\phi_m = MI
\]

⭐ Mutually induced emf:

\[
E_m = -M \frac{dI}{dt}
\]

⭐ M depends on:

1. Geometry of coils

2. Distance & orientation

3. Medium between coils
   

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🔵 8. Solenoid

A long, tightly wound coil behaves like a uniform magnetic field inside it.

⭐ Magnetic field inside:

\[
B = \mu n i
\]

Where:

• μ = permeability

• n = number of turns per unit length

• i = current

⭐ Self-inductance of solenoid:

\[
L = \mu n^2 A l
\]

Where A = cross-sectional area

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🔵 9. Superconducting Loop

For a superconducting loop:

• Resistance R = 0

• Induced emf = 0

Therefore:

\[
\phi_{\text{total}} = \text{constant}
\]

This means flux never changes through a superconducting loop.

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🔵 10. Energy Stored in an Inductor

\[
U = \frac{1}{2} LI^2
\]

⭐ Energy of interaction of two loops:

\[
U = MI_1 I_2
\]

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🔵 11. Growth of Current in L–R Circuit

When a battery is connected to an inductor and resistor in series, current increases gradually.

\[
I = \frac{E}{R}\left(1 - e^{-Rt/L}\right)
\]

Where:

• E/R = maximum current

• L/R = time constant

⭐ Important behaviors:

• At t = 0 → inductor behaves like open circuit

• At t = ∞ → behaves like short circuit
  

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🔵 12. Decay of Current

When supply is removed, current drops to zero:

\[
I = I_0 e^{-Rt/L}
\]

Where \( I_0 \) = initial current.

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🟩 Quick Revision Points

• Magnetic flux: \( \phi = BA\cos\theta \)

• Induced emf: \( e = -\frac{d\phi}{dt} \)

• Motional emf: \( E = BLv\sin\theta \)

• Coil emf: \( E = E_0\sin\omega t \)

• Self induced emf: \( e = -L\frac{di}{dt} \)

• Mutual emf: \( e = -M\frac{dI}{dt} \)

• Solenoid field: \( B = \mu n i \)

• Inductor energy: \( U = \frac{1}{2}LI^2 \)

• LR growth: \( I = I_0(1 - e^{-t/\tau}) \)

• LR decay: \( I = I_0 e^{-t/\tau} \)

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✅ SECTION 2 - AC

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🔵 1. Instantaneous, RMS & Average Values of AC

Alternating current changes its value with time. The instantaneous value of AC at any moment t is:

\[
I = I_0 \sin \omega t
\]
or
\[
I = I_0 \cos \omega t
\]

Similarly, alternating emf:

\[
E = E_0 \sin \omega t
\]

Where:

• \( I_0 \) = peak (maximum) current

• \( E_0 \) = peak emf

• \( \omega \) = angular frequency
  

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⭐ Mean / Average Value of AC

Average value over half cycle:

\[
I_{\text{avg}} = \frac{2I_0}{\pi} = 0.637 I_0
\]

Average value over full cycle = 0
(because positive and negative halves cancel)

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⭐ RMS (Root Mean Square) Value

\[
I_{\text{rms}} = \frac{I_0}{\sqrt{2}} = 0.707 I_0
\]

Similarly:

\[
E_{\text{rms}} = \frac{E_0}{\sqrt{2}}
\]

RMS value is treated as the “effective” value of AC.

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🔵 2. Phase Difference

Phase difference tells how much current lags or leads the voltage.

⭐ When current lags voltage by 90° (π/2):

Circuit is purely inductive.

⭐ When current leads voltage by 90° (π/2):

Circuit is purely capacitive.

⭐ When current & voltage are in phase:

Circuit is purely resistive.

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🔵 3. Sign Convention for Phase Angle φ (in LCR circuit)

Your PDF states:

• φ is positive when \( X_L > X_C \)

• φ is negative when \( X_C > X_L \)

• φ is zero when \( X_L = X_C \) (resonance)
  

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🔵 4. Resonance in LCR Circuit

A series LCR circuit is said to be in resonance when:

\[
X_L = X_C
\]

Since:

\[
X_L = \omega L,\quad X_C = \frac{1}{\omega C}
\]

So resonance frequency is:

\[
\omega_0 = \frac{1}{\sqrt{LC}}
\]

⭐ At resonance:

• Impedance \( Z = R \) (minimum)

• Current \( I_0 = \frac{E_0}{R} \) (maximum)

• Phase angle φ = 0

• Voltage across L and C are equal in magnitude but opposite in direction → cancel out

• Only resistor consumes energy
  

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🔵 5. Energy Stored in Inductor and Capacitor

⭐ In an Inductor:

\[
U_L = \frac{1}{2} L I_0^2
\]

⭐ In a Capacitor:

\[
U_C = \frac{1}{2} C V_0^2
\]

Also:

\[
U_C = \frac{q^2}{2C}
\]

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🔵 6. Power in AC Circuit

Instantaneous power:

\[
P = EI = E_0 I_0 \sin \omega t \cdot \sin(\omega t - \phi)
\]

Power has two components:

⭐ (A) Virtual Power (Non-useful)

\[
P_v = \frac{E_0 I_0}{2} \cos (2\omega t + \phi)
\]

Average over full cycle = 0

⭐ (B) Real Power (Useful Power)

\[
P_{\text{real}} = \frac{E_0 I_0}{2} \cos \phi = E_{\text{rms}} I_{\text{rms}} \cos\phi
\]

Where cos φ is the power factor.

• φ = 0 → purely resistive → maximum power

• φ = 90° → purely L or C → no real power
  

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🔵 7. AC Reactance

⭐ Inductive Reactance:

\[
X_L = \omega L
\]

⭐ Capacitive Reactance:

\[
X_C = \frac{1}{\omega C}
\]

Reactance \( X_L - X_C \) determines phase difference.

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🔵 8. Impedance of LCR Circuit

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

Also:

\[
\tan\phi = \frac{X_L - X_C}{R}
\]

Real power:

\[
P = \frac{E_{\mathrm{rms}} I_{\mathrm{rms}} R}{Z}
\]

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🔵 9. Bandwidth & Sharpness of Resonance

⭐ Bandwidth:

\[
\Delta \omega = \omega_2 - \omega_1
\]

⭐ Sharpness of resonance:

\[
\frac{\omega_0}{\Delta \omega}
\]

Higher the ratio → sharper resonance.

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🔵 10. Q Factor (Quality Factor)

\[
Q = \frac{\text{Voltage across L or C}}{\text{Applied Voltage}}
\]

For series LCR circuit:

\[
Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}
\]

Higher Q = better tuning & sharper resonance.

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🔵 11. Transformer

Transformer changes AC voltage levels without changing frequency.

⭐ Basic transformer equation:

\[
\frac{E_s}{E_p} = \frac{N_s}{N_p} = k
\]

Where:

• \( k \) = transformation ratio

• \( N_p, N_s \) = turns in primary, secondary

• \( E_p, E_s \) = emf in primary, secondary

⭐ Step-up transformer:

• \( k > 1 \)

• \( N_s > N_p \)

• \( E_s > E_p \)

• \( I_s < I_p \)

⭐ Step-down transformer:

• \( k < 1 \)

• \( N_s < N_p \)

• \( E_s < E_p \)

• \( I_s > I_p \)
  

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⭐ Efficiency of transformer:

\[
\eta = \frac{\text{Output power}}{\text{Input power}} = \frac{E_s I_s}{E_p I_p}
\]

For ideal transformer:

\[
\eta = 1,\quad E_s I_s = E_p I_p
\]

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🟩 Quick Revision Notes

• Instantaneous AC: \( I = I_0 \sin \omega t \)

• RMS: \( I_{\text{rms}} = 0.707I_0 \)

• Average (half cycle): \( 0.637I_0 \)

• \( X_L = \omega L \), \( X_C = 1/\omega C \)

• Resonance: \( \omega_0 = 1/\sqrt{LC} \)

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

• Real power: \( P = EI\cos\phi \)

• Transformer: \( E_s/E_p = N_s/N_p \)

• Q factor: \( Q = \omega_0 L/R \)

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