CUET UG Physics Electromagnetic Induction Self and Mutual Inductance Energy in Inductor Transformers

Must‑Know (15–20 detailed bullets)

• Self-inductance (L) of a coil is defined as the ratio of magnetic flux linkage to the current: ( L = \frac{N\Phi_B}{I} ), SI unit is henry (H).
• The induced emf in a coil due to self-induction is given by ( \varepsilon = -L \frac{dI}{dt} ), where the negative sign indicates Lenz’s law.
• For a long solenoid of length ( l ), area ( A ), and turns ( N ), self-inductance is ( L = \frac{\mu_0 N^2 A}{l} ).
• Energy stored in an inductor carrying current ( I ) is ( U = \frac{1}{2} L I^2 ), analogous to ( \frac{1}{2} C V^2 ) for capacitors.
• Mutual inductance (M) between two coils is ( M = \frac{N_2 \Phi_{21}}{I_1} ), where ( \Phi_{21} ) is flux in coil 2 due to current in coil 1.
• The induced emf in secondary coil due to changing current in primary is ( \varepsilon_2 = -M \frac{dI_1}{dt} ).
• Mutual inductance of two coaxial solenoids (one over the other): ( M = \frac{\mu_0 N_1 N_2 A}{l} ), where ( A ) is common cross-sectional area.
• Coefficient of coupling ( k ) between two coils is ( k = \frac{M}{\sqrt{L_1 L_2}} ), with ( 0 \leq k \leq 1 ).
• In an ideal transformer, input power equals output power: ( V_p I_p = V_s I_s ).
• Transformer turns ratio: ( \frac{V_s}{V_p} = \frac{N_s}{N_p} ); for step-up, ( N_s > N_p ), for step-down, ( N_s < N_p ).
• Eddy currents are reduced in transformers by using laminated cores to minimize energy loss.
• Energy density of magnetic field in a solenoid is ( u_B = \frac{B^2}{2\mu_0} ), derived from energy stored per unit volume.
• The self-inductance of a toroidal solenoid is ( L = \frac{\mu_0 N^2 A}{2\pi r} ), where ( r ) is the mean radius.
• In AC circuits, an inductor opposes changes in current, causing current to lag voltage by ( \frac{\pi}{2} ) radians.
• Ideal transformers operate on AC only; they do not work with steady DC because ( \frac{dI}{dt} = 0 ).
• Mutual inductance is maximum when the two coils are coaxial and close together, and zero when perpendicular.
• The phenomenon of self-induction is analogous to inertia in mechanics—inductors resist changes in current.
• In transformers, flux linkage is nearly perfect in ideal case, so ( \Phi_p = \Phi_s ) (same flux passes through both coils).
• Core losses in transformers include hysteresis loss (due to repeated magnetization) and eddy current loss.
• The SI unit of mutual inductance is the same as self-inductance: henry (H).

Difficulty Level

Intermediate — Requires understanding of electromagnetic induction concepts, mathematical relationships, and application to devices like transformers; numericals often involve ratios and energy calculations.

Common CUET Traps

• Trap: Assuming transformers can work with DC because they have coils.
  Avoid: Transformers require changing magnetic flux, which only AC provides; DC gives ( \frac{dI}{dt} = 0 ), so no induced emf.

• Trap: Confusing the formula for energy in inductor (( \frac{1}{2}LI^2 )) with capacitor (( \frac{1}{2}CV^2 )) and applying it to voltage instead of current.
  Avoid: Energy in inductor depends on current, not voltage; remember: "Inductors store energy in magnetic field via current."

• Trap: Thinking mutual inductance depends only on the number of turns, ignoring geometry and orientation.
  Avoid: Mutual inductance depends on relative position, orientation, distance, and core material—maximum when coils are coaxial and close.

Practice MCQs

Q1. The energy stored in a 0.5 H inductor carrying a current of 2 A is:
A. 0.5 J
B. 1 J
C. 2 J
D. 4 J
Answer: B
Explanation: Using ( U = \frac{1}{2} L I^2 = \frac{1}{2} \times 0.5 \times (2)^2 = 1 ) J.
Why others fail: Option C (2 J) comes from incorrectly using ( L I^2 ) without the 1/2 factor.

Q2. Which of the following is true for an ideal step-up transformer?
A. ( N_s < N_p )
B. ( I_s > I_p )
C. ( V_s > V_p )
D. Output power < Input power
Answer: C
Explanation: In step-up transformer, secondary voltage is greater than primary voltage.
Why others fail: Option B is tempting because students confuse current and voltage relationships.

Q3. Two coils have self-inductances 4 H and 9 H. If the coefficient of coupling is 0.5, their mutual inductance is:
A. 3 H
B. 6 H
C. 1.5 H
D. 4.5 H
Answer: A
Explanation: ( M = k \sqrt{L_1 L_2} = 0.5 \times \sqrt{4 \times 9} = 0.5 \times 6 = 3 ) H.
Why others fail: Option B (6 H) comes from ignoring the coupling coefficient and assuming perfect coupling.

Q4. Eddy currents in transformers are minimized by:
A. Using thick copper wires
B. Laminating the core
C. Increasing number of turns
D. Using high resistance wire
Answer: B
Explanation: Laminated cores increase resistance to eddy current paths, reducing energy loss.
Why others fail: Option D is tempting, but wire resistance does not affect core eddy currents.

Q5. A current in a coil changes from 0 to 3 A in 0.1 s, producing an average emf of 6 V. The self-inductance of the coil is:
A. 0.1 H
B. 0.2 H
C. 0.3 H
D. 0.6 H
Answer: B
Explanation: Using ( |\varepsilon| = L \frac{\Delta I}{\Delta t} \Rightarrow 6 = L \times \frac{3}{0.1} = L \times 30 \Rightarrow L = 0.2 ) H.
Why others fail: Option D (0.6 H) results from inverting the time-current ratio.

Last‑Minute Revision (15–20 one‑liners)

• ⚠️ Self-inductance ( L = \frac{\mu_0 N^2 A}{l} ) — same as solenoid inductance formula.
• ⚠️ Energy in inductor: ( U = \frac{1}{2} L I^2 ) — depends on current, not voltage.
• ⚠️ Induced emf: ( \varepsilon = -L \frac{dI}{dt} ) — negative sign shows opposition to change.
• ⚠️ Mutual inductance ( M = \frac{\mu_0 N_1 N_2 A}{l} ) — for two coaxial solenoids.
• ⚠️ Coefficient of coupling ( k = \frac{M}{\sqrt{L_1 L_2}} ), max value 1.
• ⚠️ Transformer: ( \frac{V_s}{V_p} = \frac{N_s}{N_p} ) — voltage ratio = turns ratio.
• ⚠️ Ideal transformer: ( V_p I_p = V_s I_s ) — power conserved.
• ⚠️ Step-up transformer: ( N_s > N_p ), ( V_s > V_p ), ( I_s < I_p ).
• ⚠️ Transformers work only on AC — no DC output.
• ⚠️ Eddy current loss reduced by laminated core — increases resistance.
• ⚠️ Hysteresis loss minimized by soft iron core — low retentivity.
• ⚠️ Magnetic energy density: ( \frac{B^2}{2\mu_0} ) — in vacuum.
• ⚠️ Toroidal inductor: ( L = \frac{\mu_0 N^2 A}{2\pi r} ) — no flux leakage.
• ⚠️ Inductor in AC: current lags voltage by 90° — phase difference ( \pi/2 ).
• ⚠️ Mutual inductance is symmetric: ( M_{12} = M_{21} ).
• ⚠️ No self-induction in straight wire — negligible L compared to coil.
• ⚠️ Flux linkage = ( N\Phi ) — key for Faraday’s law and inductance.
• ⚠️ In ideal transformer, flux per turn is same in primary and secondary.
• ⚠️ Lenz’s law ensures energy conservation in self and mutual induction.
• ⚠️ "EMF induced opposes change" — core idea in all inductance phenomena.