Electromagnetic Induction

This chapter explains how a changing magnetic field can produce an electric current. This principle, known as electromagnetic induction, is the foundation for technologies like electric generators, transformers, and inductors. The study begins with Faraday's Laws of Induction, which quantify the electromotive force (EMF) generated, and Lenz's Law, which gives the direction of the induced current.

The main sub-topics include calculating induced EMF in different scenarios, such as for a conductor moving through a magnetic field (motional EMF) or for a stationary loop in a time-varying field. You will then learn about self-inductance, where a changing current in a coil induces an EMF in the coil itself, and mutual inductance, which involves an EMF being induced in one coil due to a changing current in a nearby coil. The chapter often concludes with applications in AC generators and transformers.

Students often find applying Lenz's Law to determine the direction of current to be the most challenging part, as it requires strong conceptual and spatial reasoning. The mathematical aspects require a good command of calculus, particularly differentiation. This chapter builds directly on the concepts of magnetic field and magnetic flux from the preceding chapter, 'Magnetic Effects of Current and Magnetism', so a thorough understanding of that material is essential.

Formulas for Electromagnetic Induction

Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ For a uniform field B and planar area A, this simplifies to \Phi_B = BA \cos\theta.
Faraday's Law of Induction: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ \mathcal{E} is the induced EMF, N is the number of turns, and \Phi_B is the magnetic flux.
Motional EMF: $\mathcal{E} = Blv$ For a conductor of length l moving with velocity v perpendicular to a magnetic field B.
Induced EMF in a rotating coil: $\mathcal{E} = NBA\omega \sin(\omega t)$ For a coil with N turns and area A, rotating in a uniform field B at angular velocity \omega.
Self-Inductance: $L = \frac{N\Phi_B}{I}$ L is the self-inductance, N\Phi_B is the total flux linkage, and I is the current.
EMF due to Self-Induction: $\mathcal{E} = -L \frac{dI}{dt}$ This is the back EMF induced in a circuit due to a change in its own current.
Energy Stored in an Inductor: $U = \frac{1}{2} L I^2$ U is the magnetic potential energy stored in an inductor L carrying current I.
Mutual Inductance: $M = \frac{N_2\Phi_{21}}{I_1}$ \Phi_{21} is the flux in coil 2 due to current I_1 in coil 1.
EMF due to Mutual Induction: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ EMF induced in coil 2 due to a changing current in coil 1.
Inductance of a long solenoid: $L = \mu_0 n^2 A l$ n is the number of turns per unit length, A is the cross-sectional area, and l is the length.