Key Points

Electromagnetic induction is the phenomenon of producing an electromotive force (EMF) or electric current in a conductor when it is subjected to a varying magnetic field. It was discovered by Michael Faraday and Joseph Henry.

Magnetic flux ($\Phi_B$) through a surface measures the total number of magnetic field lines passing through it. For a uniform magnetic field $\mathbf{B}$ and area vector $\mathbf{A}$, it is defined as $\Phi_B = \mathbf{B} \cdot \mathbf{A} = BA \cos \theta$. Its SI unit is the weber (Wb).

The magnitude of the induced EMF ($\varepsilon$) in a circuit is equal to the time rate of change of magnetic flux through the circuit. For a coil with N turns, the law is expressed as $\varepsilon = -N \frac{d\Phi_B}{dt}$.

Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. The negative sign in Faraday's law signifies this opposition and reflects the law of conservation of energy.

An EMF induced by the motion of a conductor through a magnetic field is called motional EMF. For a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$, the induced EMF is $\varepsilon = Blv$.

Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in itself or a nearby circuit. The SI unit of inductance is the henry (H).

Mutual inductance ($M$) occurs when a changing current in one coil induces an EMF in a neighboring coil. The induced EMF in coil 1 due to the current change in coil 2 is $\varepsilon_1 = -M \frac{dI_2}{dt}$.

Self-inductance ($L$) is the property of a coil to induce a 'back EMF' within itself when the current flowing through it changes. This EMF opposes the change in current and is given by $\varepsilon = -L \frac{dI}{dt}$.

The self-inductance of a long solenoid with $n$ turns per unit length, length $l$, and cross-sectional area $A$ is given by the formula $L = \mu_0 n^2 A l$.

An inductor stores energy in its magnetic field when a current flows through it. The magnetic potential energy ($U_B$) stored in an inductor with inductance $L$ carrying a current $I$ is $U_B = \frac{1}{2} L I^2$.

Magnetic energy density ($u_B$) is the energy stored per unit volume in a magnetic field. It is given by the formula $u_B = \frac{B^2}{2\mu_0}$, where B is the magnetic field strength.

An AC generator converts mechanical energy into electrical energy using electromagnetic induction. It operates by rotating a coil in a magnetic field, which causes a periodic change in magnetic flux and induces an alternating EMF.

The instantaneous EMF ($\varepsilon$) produced by an AC generator with a coil of $N$ turns and area $A$, rotating at an angular velocity $\omega$ in a magnetic field $B$, is $\varepsilon = NBA\omega \sin(\omega t)$. The maximum EMF is $\varepsilon_0 = NBA\omega$.