Electromagnetic Induction
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Direct answer
When the magnetic flux threading a circuit changes, an emf appears: ε = -N dΦ/dt, Faraday's law, with Lenz's minus sign arranging that the induced current opposes the very change producing it — which is energy conservation in costume, since you must work against that opposition to generate electricity. The chapter delivers motional emf Blv, eddy currents, self and mutual inductance, and the AC generator, and NEET-UG mines all of them for numericals and graph questions.
What you must remember
- Flux Φ = BA cosθ (unit weber); changing B, A or the angle θ — rotation counts — each induces emf; the minus sign in ε = -N dΦ/dt is Lenz's law.
- Lenz's law is energy conservation: opposing the change means the external agent does positive work, which reappears as electrical energy; no opposition would mean free energy.
- Motional emf of a rod of length l moving at v perpendicular to B: ε = Blv; the induced charge q = N ΔΦ/R is independent of how fast the flux changed.
- Eddy currents circulate in bulk conductors facing changing flux — used in magnetic braking, induction furnaces and speedometers; suppressed in transformer cores by lamination.
- Self-inductance: ε = -L dI/dt; solenoid L = μ0 n^2 A l; stored energy U = (1/2)LI^2, analogous to the capacitor's ½CV^2.
- Mutual inductance M: ε2 = -M dI1/dt; the ideal transformer relation N_s/N_p = V_s/V_p = I_p/I_s grows from it.
- AC generator: rotating a coil of N turns in field B at angular frequency ω gives ε = NBAω sin ωt; the emf peaks when the flux through the coil is zero (plane parallel to B).
- Lenz's direction rule: induced current flows so its own flux opposes the change in the original flux — oppose the change, never merely the flux.
A rod, a rail and an energy ledger
A rod of length l slides at velocity v on rails closing a resistance R in a uniform field B. Motional emf ε = Blv drives current I = Blv/R, and that current feels a retarding force F = BIl = B^2l^2v/R opposing the motion. Now audit: the mechanical power fed in is F × v = B^2l^2v^2/R, and the Joule power dissipated is I^2R = (Blv)^2/R = B^2l^2v^2/R. The two match to the last digit — mechanical work in equals electrical energy out, and Lenz's opposition is the bookkeeping that enforces the balance. Pull with constant force and the rod settles at the terminal speed where drag equals pull; let go and it coasts to rest, its kinetic energy banked as heat in R. This one arrangement is the generator, the eddy-current brake and the induction launcher in miniature.
The generator is the rotating version: flux Φ = NBA cos ωt, so dΦ/dt ∝ sin ωt — the emf is sinusoidal precisely because the derivative of a cosine is a sine, and the emf peaks where the flux is momentarily zero.
Where students slip
Flux can change three ways — B changes, area changes, or the coil rotates so cosθ changes — and questions regularly induce via rotation while students inspect only B. Second, Lenz direction questions: the rule opposes the change in flux, so an increasing flux into the page provokes counterclockwise current while a decreasing flux into the page provokes clockwise; students who memorise one configuration fail the other. Third, the induced charge q = NΔΦ/R divides out the time: slow and fast flips of the same coil push the same charge through the galvanometer, though the emfs differ enormously — a favourite conceptual MCQ. Fourth, a magnet dropped through a ring is retarded entering and leaving alike, the induced current opposing both approach and departure. Finally, the inductor's energy ½LI^2 lives in the magnetic field, and opening the switch collapses that field through a spark across the gap — the physical reason inductive circuits need flyback protection.
Frequently asked questions
What does the minus sign in Faraday's law encode?
Lenz's law: the induced emf drives a current whose flux opposes the change that induced it, ensuring the external agent does work and energy is conserved.
What is motional emf and its formula?
The emf generated in a conductor moving through a magnetic field: ε = Blv for a rod of length l moving perpendicular to B, positive end fixed by Fleming's right-hand rule.
Why are transformer cores laminated?
Thin insulated sheets interrupt the large eddy-current loops a changing flux would otherwise drive through solid iron, cutting the I^2R heating without disturbing the intended flux path.
Where is the induced emf greatest as a coil rotates in a generator?
When the flux through the coil is instantaneously zero — the coil's plane parallel to B — because the emf is the rate of flux change, and that rate is maximal there.
How much energy does an inductor store?
U = (1/2)LI^2, held in the magnetic field; doubling the current quadruples the stored energy, mirroring the capacitor's ½CV^2.