Electromagnetic Induction
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Direct answer
An emf appears whenever the magnetic flux through a circuit changes: Faraday's law gives its magnitude, epsilon = −N dphi/dt with phi = B A cos(theta), and Lenz's law — the minus sign — gives its direction: the induced current opposes the flux change that creates it, a restatement of energy conservation. Flux may change through B, area or orientation; a rod of length l moving at speed v perpendicular to B develops the motional emf epsilon = B l v.
What you must remember
- Faraday's law: emf = −N dphi/dt; a steady flux, however large, induces nothing — only the rate of change counts.
- Lenz's law: induced effects oppose the flux change; the work done against the opposing force becomes electrical energy, which is why the minus sign is energy conservation in disguise.
- Motional emf: rod sliding with speed v gives epsilon = B l v; holding constant speed needs force B^2 l^2 v/R, and the power supplied, B^2 l^2 v^2/R, equals the Joule heat.
- A rod of length l rotating about one end, with B perpendicular to the plane: emf = (1/2) B omega l^2.
- Induced charge q = N times (flux change)/R — independent of how fast the change occurs.
- Coil rotating in a uniform field: emf = N B A omega sin(omega t), the AC generator; maximum emf N B A omega occurs when flux is zero.
- Self-inductance: emf = −L dI/dt; solenoid L = mu_0 n^2 A l; stored energy (1/2) L I^2 with energy density B^2/(2 mu_0); mutual inductance M links coils through emf = −M dI/dt.
Common confusion
The recurring error is expecting an emf from a strong field rather than a changing flux — a huge steady flux induces nothing, a tiny changing one does everything. The second trap is Lenz's direction: ask which flux change is occurring, then pick the current whose own field opposes that change; reasoning from "opposing current" instead of "opposing flux change" leads astray. Note also that an emf exists across an open rod with no current flowing — a closed path is needed for current, not for emf.
Exam-focused takeaway
JEE Main tests motional emf, induced charge, rotating coils and inductor energy — three formulas cover most numerical-value questions. JEE Advanced prefers the dynamic versions: rods on rails reaching terminal velocity, rails that diverge, loops entering or leaving a field region (the induced force always retards the motion), energy accounting among mechanical input, stored magnetic energy and heat, and mutual inductance of coaxial coils. Write phi(t) first and differentiate; the rest is bookkeeping.
Frequently asked questions
What does the minus sign in Faraday's law mean?
It is Lenz's law: the induced current flows so that its own field opposes the flux change, forcing the external agent to do work — energy conservation.
Does an emf require a closed circuit?
No. A rod moving through a field develops a potential difference across its ends even open-circuited; a closed path is needed only for a sustained current.
Why is induced charge independent of speed?
Charge = integral of emf/R over time = net flux change divided by R; doubling the rate halves the time but doubles the current, leaving the total charge fixed.
What is self-inductance, physically?
A coil's magnetic inertia: any change of its current induces an emf −L dI/dt opposing the change, so current through an inductor cannot jump.
Where is an inductor's energy stored?
In its magnetic field — U = (1/2) L I^2 overall, distributable as energy density B^2/(2 mu_0) at every point.