Chapter 6: Electromagnetic Induction

Electromagnetic Induction

Magnetic Flux and Faraday's Laws

Electromagnetic induction is the process of generating an electromotive force (emf) or voltage in a conductor due to a change in magnetic flux through it. Magnetic flux (Φ) is the measure of the total magnetic field passing through a given area and is given by:

\[ \Phi = B A \cos \theta \]

where \(B\) is the magnetic field strength, \(A\) is the area, and \(\theta\) is the angle between the magnetic field and the normal to the surface.

Faraday's First Law states that an emf is induced in a conductor when the magnetic flux linked with it changes. Faraday's Second Law quantifies this emf as proportional to the rate of change of magnetic flux:

\[ \varepsilon = -N \frac{d\Phi}{dt} \]

where \(N\) is the number of turns in the coil. The negative sign indicates the direction of induced emf as given by Lenz's Law.

Self and Mutual Induction

Self-induction occurs when a changing current in a coil induces an emf in the same coil. The magnetic flux linked with the coil is proportional to the current:

\[ \Phi = L I \]

where \(L\) is the self-inductance of the coil. The induced emf is:

\[ \varepsilon = -L \frac{dI}{dt} \]

Mutual induction occurs when a changing current in one coil induces an emf in a nearby coil. The mutual inductance \(M\) relates the magnetic flux in the secondary coil to the current in the primary coil:

\[ \Phi = M I \]

The induced emf in the secondary coil is:

\[ \varepsilon = -M \frac{dI}{dt} \]

Solved Examples

Example 1: A square loop of side 0.1 m is placed in a magnetic field of 0.5 T at an angle of 45°. The magnetic field decreases to zero in 0.1 s. Calculate the induced emf and current if the resistance of the loop is 2 Ω.

Solution:

Area of loop, \( A = 0.1 \times 0.1 = 0.01 \text{ m}^2 \)

Initial magnetic flux, \( \Phi_i = B A \cos \theta = 0.5 \times 0.01 \times \cos 45^\circ = 0.5 \times 0.01 \times 0.707 = 0.003535 \text{ Wb} \)

Final magnetic flux, \( \Phi_f = 0 \) (since field decreases to zero)

Change in flux, \( \Delta \Phi = \Phi_f - \Phi_i = -0.003535 \text{ Wb} \)

Time interval, \( \Delta t = 0.1 \text{ s} \)

Induced emf, \( \varepsilon = \left| \frac{\Delta \Phi}{\Delta t} \right| = \frac{0.003535}{0.1} = 0.03535 \text{ V} \)

Induced current, \( I = \frac{\varepsilon}{R} = \frac{0.03535}{2} = 0.0177 \text{ A} \)

Example 2: A circular coil of radius 10 cm with 500 turns and resistance 2 Ω is rotated 180° in Earth's magnetic field of 0.5 × 10⁻⁴ T in 0.1 s. Find the induced emf and current.

Solution:

Radius, \( r = 0.1 \text{ m} \)

Area, \( A = \pi r^2 = 3.1416 \times (0.1)^2 = 0.0314 \text{ m}^2 \)

Number of turns, \( N = 500 \)

Initial flux, \( \Phi_i = N B A \cos 0^\circ = 500 \times 0.5 \times 10^{-4} \times 0.0314 = 0.000785 \text{ Wb} \)

Final flux after 180° rotation, \( \Phi_f = N B A \cos 180^\circ = -0.000785 \text{ Wb} \)

Change in flux, \( \Delta \Phi = \Phi_f - \Phi_i = -0.000785 - 0.000785 = -0.00157 \text{ Wb} \)

Time interval, \( \Delta t = 0.1 \text{ s} \)

Induced emf, \( \varepsilon = \left| \frac{\Delta \Phi}{\Delta t} \right| = \frac{0.00157}{0.1} = 0.0157 \text{ V} = 15.7 \text{ mV} \)

Induced current, \( I = \frac{\varepsilon}{R} = \frac{0.0157}{2} = 0.00785 \text{ A} = 7.85 \text{ mA} \)

Practice Set

• Level 1 (Easy): Define magnetic flux and state Faraday's First Law of electromagnetic induction.
• Level 2 (Moderate): A coil of 100 turns and area 0.02 m² is placed in a magnetic field of 0.3 T. The coil is rotated such that the magnetic flux changes from maximum to zero in 0.05 s. Calculate the induced emf.
• Level 3 (Challenging): Two coils with inductances 2 H and 3 H are coupled with a coupling constant of 0.8. Calculate the mutual inductance. If the current in the first coil changes at a rate of 5 A/s, find the induced emf in the second coil.

Answer Key

• Level 1: Magnetic flux is the total magnetic field passing through a given area. Faraday's First Law states that an emf is induced in a conductor when the magnetic flux linked with it changes.
• Level 2: Area \( A = 0.02 \text{ m}^2 \), Number of turns \( N = 100 \), Magnetic field \( B = 0.3 \text{ T} \), Time \( \Delta t = 0.05 \text{ s} \). Initial flux \( \Phi_i = N B A = 100 \times 0.3 \times 0.02 = 0.6 \text{ Wb} \). Final flux \( \Phi_f = 0 \). Induced emf \( \varepsilon = \frac{\Delta \Phi}{\Delta t} = \frac{0.6}{0.05} = 12 \text{ V} \).
• Level 3: Mutual inductance \( M = k \sqrt{L_1 L_2} = 0.8 \times \sqrt{2 \times 3} = 0.8 \times 2.45 = 1.96 \text{ H} \). Rate of change of current \( \frac{dI}{dt} = 5 \text{ A/s} \). Induced emf \( \varepsilon = M \frac{dI}{dt} = 1.96 \times 5 = 9.8 \text{ V} \).

Quick Reference Table

Magnetic Flux: \( \Phi = B A \cos \theta \)

Faraday's Law: \( \varepsilon = -N \frac{d\Phi}{dt} \)

Self-Inductance: \( \varepsilon = -L \frac{dI}{dt} \)

Mutual Inductance: \( \varepsilon = -M \frac{dI}{dt} \), \( M = k \sqrt{L_1 L_2} \)

Unit of Inductance: Henry (H)

Common Mistakes and Misconceptions

• Confusing magnetic field strength with magnetic flux; only changing magnetic flux induces emf.
• Ignoring the negative sign in Faraday's law, which indicates the direction of induced emf.
• Mixing up self-induction and mutual induction effects.
• Using inconsistent units, leading to calculation errors.
• Assuming a constant magnetic field induces emf; only changing fields do.

Glossary

• Electromagnetic Induction: Generation of emf due to change in magnetic flux.
• Magnetic Flux: Total magnetic field passing through an area.
• Faraday's Laws: Laws describing induced emf due to changing magnetic flux.
• Lenz's Law: Direction of induced emf opposes the change causing it.
• Self-Inductance: Induction of emf in the same coil due to changing current.
• Mutual Inductance: Induction of emf in one coil due to changing current in another coil.
• Inductance: Property of a coil to oppose change in current.