Induced EMF and Induced Current

Faraday's law says an emf is induced whenever the flux linkage $N\Phi_B$ changes, but since $\Phi_B = BA\cos\theta$ depends on THREE separate quantities -- the field $B$, the area $A$, and the angle $\theta$ -- flux can change because any one (or more) of these three changes. Each case is physically a little different, though all obey the same underlying law.

(a) Changing $B$ -- a stationary coil in a varying field. If a coil is held completely still but the magnetic field passing through it varies with time (for instance, because a nearby electromagnet's current is being switched or varied), the flux $\Phi_B(t)=B(t)A$ changes purely because $B$ changes, and an emf $\mathcal{E}=-N A\,dB/dt$ is induced, exactly as though the coil itself had moved. This is the mechanism behind a transformer (studied in the next sub-topic) and behind induction cooktops and furnaces (Section 6.6).

(b) Changing $A$ -- altering the circuit's effective area. If the field $B$ is steady but the AREA of the circuit exposed to it changes -- for example, a loop being stretched, shrunk, or partly pulled out of the field region -- the flux changes purely through the $A(t)$ term, and $\mathcal{E}=-NB\,dA/dt$. A closed loop being pulled steadily out of a field region (so its effective overlap area with the field shrinks to zero) is the standard example of this case, and is worked out numerically in Numerical 4 of this chapter.

(c) Changing $\theta$ or motional emf -- a conductor physically moving through the field. The most important special case, called MOTIONAL emf, is a straight conducting rod of length $l$ moved with velocity $v$ perpendicular to both its own length and to a uniform field $B$. Every free charge $q$ inside the moving rod experiences a magnetic force $\vec{F}=q\vec{v}\times\vec{B}$, pushing positive charge toward one end of the rod and leaving the other end negatively charged, exactly like a tiny battery being charged up; charge keeps accumulating at the ends until the resulting electric field inside the rod exactly balances the magnetic force on further charges, at which point equilibrium is reached and a steady potential difference

$\mathcal{E} = Blv$ …