Introduction

This chapter explores the phenomenon of electromagnetic induction, where changing magnetic fields induce electric currents. This groundbreaking discovery by Faraday and Henry revealed a fundamental link between electricity and magnetism, paving the way for technologies like electric generators and transformers that power our modern world.

The Experiments Of Faraday And Henry

Faraday and Henry's experiments demonstrated that relative motion between a magnet and a coil, or between two current-carrying coils, induces an electric current in a coil. Crucially, they found that a current is induced only when the magnetic flux through the coil is changing. A changing current in one coil can also induce a current in a nearby coil, even without relative motion.

• Experiment 6.1: Shows that moving a bar magnet towards or away from a coil induces a current, detected by a galvanometer. The direction of the current depends on the pole of the magnet and the direction of motion.
• Experiment 6.2: Demonstrates induction between two coils, where relative motion induces current.
• Experiment 6.3: Shows that a changing current in one coil (when a key is pressed or released) induces a momentary current in a stationary nearby coil.

Magnetic Flux

Magnetic flux ($\Phi_B$) through a surface is a measure of the total magnetic field passing through that surface. For a uniform magnetic field $\vec{B}$ passing through a plane area $\vec{A}$, it is given by:

$\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$

where $\theta$ is the angle between the magnetic field vector and the area vector. For non-uniform fields or curved surfaces, magnetic flux is calculated by integrating the dot product of the magnetic field and the area element over the surface: $\Phi_B = \oint \vec{B} \cdot d\vec{A}$. The SI unit of magnetic flux is the Weber (Wb) or Tesla meter squared (T m$^2$).

Figure 6.4 and Figure 6.5 illustrate magnetic flux calculations.

Faraday’s Law Of Induction

Faraday's Law of Induction states that the magnitude of the induced electromotive force (emf, $\epsilon$) in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit:

$\epsilon = -\frac{d\Phi_B}{dt}$

For a coil with $N$ tightly wound turns, the induced emf is:

$\epsilon = -N \frac{d\Phi_B}{dt}$

The negative sign is crucial and is explained by Lenz's Law (discussed next).

Example 6.1 discusses how to maximize the induced emf and demonstrate its presence.

Example 6.2 calculates the induced emf and current when a rectangular loop is moved out of a magnetic field.

Example 6.3 calculates the induced emf and current when a circular coil is rotated in the Earth's magnetic field.

Lenz’s Law And Conservation Of Energy

Lenz's Law states that the direction of the induced emf (and the resulting induced current in a closed circuit) is always such that it opposes the change in magnetic flux that produced it. This is a consequence of the conservation of energy.

• If the magnetic flux through a coil is increasing, the induced current creates a magnetic field that opposes this increase.
• If the magnetic flux is decreasing, the induced current creates a magnetic field that opposes this decrease.

The negative sign in Faraday's Law ($\epsilon = -d\Phi_B/dt$) mathematically represents Lenz's Law. If Lenz's Law were violated, it would be possible to create a perpetual motion machine, violating energy conservation.

Example 6.4 uses Lenz's Law to determine the direction of induced current in loops moving in magnetic fields.

Example 6.5 explores scenarios related to induced currents, including stationary loops, moving loops in electric fields, and comparing induction in rectangular vs. circular loops.

Motional Electromotive Force

When a conductor moves through a magnetic field, the free charge carriers within the conductor experience a Lorentz force ($q\vec{v} \times \vec{B}$). This force causes charges to separate along the conductor, creating a potential difference across its ends, known as motional emf.

For a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$, the motional emf is:

$\epsilon = Blv$

This emf can also be derived from Faraday's Law by considering the change in magnetic flux due to the changing area of the circuit. For a rotating rod of length $R$ in a uniform field $B$ perpendicular to the axis of rotation, the emf is $\epsilon = \frac{1}{2} B \omega R^2$, where $\omega$ is the angular velocity.

Example 6.6 calculates the motional emf induced in a rotating rod.

Example 6.7 calculates the emf induced in the spokes of a rotating wheel in the Earth's magnetic field.

Figure 6.10 illustrates motional emf in a moving conductor.

Inductance

Inductance ($L$ or $M$) is a measure of a circuit element's ability to oppose changes in current. It's defined by the relationship between flux linkage and current.

Mutual Inductance

Mutual inductance ($M_{12}$) between two coils is the ratio of the flux linkage in coil 1 due to the current in coil 2 to the current in coil 2: $N_1 \Phi_{12} = M_{12} I_2$. When the current in coil 2 changes, it induces an emf in coil 1:

$\epsilon_1 = -M_{12} \frac{dI_2}{dt}$

It is found that $M_{12} = M_{21} = M$. For long co-axial solenoids, $M = \mu_0 n_1 n_2 \pi r_1^2 l$. Mutual inductance depends on the geometry of the coils and the permeability of the medium.

Example 6.8 calculates the mutual inductance between two concentric circular coils.

Self-Inductance

Self-inductance ($L$) of a coil is the ratio of the flux linkage through the coil due to its own current to that current: $N\Phi_B = LI$. When the current in the coil changes, it induces a back emf in the same coil:

$\epsilon = -L \frac{dI}{dt}$

This back emf opposes the change in current. The self-inductance of a long solenoid is $L = \mu_0 n^2 Al$, where $n$ is the number of turns per unit length, $A$ is the cross-sectional area, and $l$ is the length.

The self-inductance $L$ is analogous to mass in mechanics, representing inertia against changes in current. The energy stored in a solenoid due to self-inductance is $U = \frac{1}{2}LI^2$. The magnetic energy density is $u = \frac{1}{2\mu_0} B^2$, analogous to electrostatic energy density $u_E = \frac{1}{2}\epsilon_0 E^2$.

Example 6.9 derives the expression for magnetic energy stored in a solenoid and compares it with electrostatic energy density.

Ac Generator

An AC generator converts mechanical energy into electrical energy using the principle of electromagnetic induction. It consists of a coil rotated in a uniform magnetic field. As the coil rotates, the magnetic flux through it changes, inducing an alternating emf and current.

If a coil of $N$ turns, area $A$, carrying current $I$ rotates with angular frequency $\omega$ in a magnetic field $B$, the induced emf at time $t$ is:

$\epsilon = NBA\omega \sin(\omega t)$

where $e_0 = NBA\omega$ is the maximum emf. The frequency of the induced emf is the same as the frequency of rotation of the coil.

Figure 6.13 illustrates the basic components of an AC generator, and Figure 6.14 shows the induced emf variation with time. Example 6.10 calculates the maximum voltage generated by a bicycle dynamo.

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