Electromagnetic Induction

Electromagnetic induction is best explained when a conducting wire wound into a coil is placed near a moving bar magnet having a north and a south pole. The magnetic field in the bar magnet is represented by lines of forces that come out from the north pole and terminate into the south pole. The number of lines passing through a given area in space is the magnetic flux.

Magnetic Flux Equation

Suppose the bar magnet a field of strength B. Consider a two-dimensional area A in space such that the lines of force pass through this area and make an angle θ with the area vector. Then, the area projected on a plane perpendicular to the field is A cos θ. The magnetic flux φ is defined as the product of the field and the projected area.

φ = BA cos θ

When the north pole is brought closer to the coil, the flux increases and will continue to increase as long as the magnet is in motion. The change in magnetic flux over time results in an induced current. If an ammeter is connected to the two ends of the coil, it will register a current. The current will continue to flow as long as the magnet is moving. If the magnet is retracted from the coil, the induced current will change direction.

1. Faraday’s Law

As current is induced in the circuit, there will also be an induced voltage or emf (electromotive force) due to the wire’s resistance. According to Faraday’s law, the magnitude of this emf is proportional to the rate of change of the flux. This law establishes a quantitative relationship between the changing magnetic field and the induced emf in an electromagnetic circuit.

Faraday’s Law Equation

Suppose the coil has N turns. It is placed in a magnetic field such that dφ/dt represents the flux rate. The formulas for induced emf ε are given by,

ε = – N dφ/dt (continuous)

ε = – N Δφ/Δt (discrete)

SI Unit of EMF: Volts

The negative sign is because the induced emf opposes the very reason causing it, which is dictated by Lenz’s law.

Factors that Effect Electromagnetic Induction

1. Number of Turns: The induced voltage is directly proportional to the number of turns of the wire. Therefore, the higher the number of turns, the higher is the voltage produced.

2. Changing Magnetic Field: The induced voltage is directly proportional to the flux rate. Changing the magnetic flux affects the induced voltage. This change can be done by moving the magnet closer to or away from the coil and vice-versa.

2. Lenz’s Law

Faraday’s law does not state anything about the conservation of energy. Lenz’s law can explain the negative sign in Faraday’s law equation. Lenz’s law states that “The polarity of the induced emf is such that it opposes the change in magnetic flux that produced it.” In other words, an induced current will always oppose the motion that started it in the first place. Therefore, Lenz’s law can determine the direction of induced current.

Lenz’s law is named after German physicist Heinrich Friedrich Lenz after he deduced it in 1834.

Here are some examples, uses, and applications of electromagnetic in real and daily life.

• Electric generator: It converts kinetic energy into electrical energy and produces electricity in power plants. The electricity is then supplied to homes.

• Electric transformer: It changes the voltage and comes in two types – step up and step down. A step-up transformer increases the voltage, while a step-down transformer decreases the voltage.
• Dynamo: It is a small generator consisting of a permanent magnet that is rotated inside a fixed coil by the movement of a bicycle wheel.
• Inductance: It is a simple coil with a ferromagnetic core and is usually a part of an electrical circuit. It stores energy in the magnetic field of the coil when current passes through it.

Here are some benefits of electromagnetism.

• Cheap, simple, reliable, efficient, and controllable
• Mechanical energy can be converted directly into electrical energy
• Produces electric current and powers homes

P.1. A closed coil of 50 turns and area 150 cm² is rotated in a magnetic field of flux density 2 Wb m^-2. It rotates from a position where its plane makes an angle of 45˚ with the field to a position perpendicular to the field in a time of 0.3 sec. Find the magnitude of the emf induced in the coil due to its rotation.

Soln. Given,

N = 50

B = 2 Wb m^-2

A = 150 cm² = 150 x 10^-4m² = 0.015 m²

θ = 45˚

Δt = 0.3 s

Initial flux: φ_i = BA cos θ_i = 2 Wb m^-2 x 0.015 m² x cos 45˚ = 0.021 Wb

Final flux: φ_f = BA cos θ_f = = 2 Wb m^-2 x 0.015 m² x cos 0˚ = 0.03 Wb

Therefore, Δφ = φ_f – φ_i = 0.03 – 0.021 = 0.009 Wb

Therefore, the magnitude of induced emf is,

|ε| = N Δφ/Δt = 50 x 0.009 Wb/0.3 s = 1.46 V

P.2. A uniform magnetic field is normal to the plane of a circular loop 15 cm in diameter and made of copper wire of diameter 3 mm. (a) Calculate the resistance of the wire (R = ρl/A, the resistivity of copper: ρ = 1.68 x 10 ^-8 Ωm). (b) At what rate must the magnetic field change with time if an induced current of 15 A forms in the loop?

Ans. (a) Given,

d = 3 mm = 3 x 10^-3 m

ρ = 1.68 x 10 ^-8 Ωm

D = 15 cm = 0.15 m

Length of the wire = πD

Cross-sectional area of the wire = πd²/4

Therefore,

R = ρl/A = ρπD/(πd²/4) = 4ρD/d² = 4 x 1.68 x 10 ^-8 Ωm x 0.15 m / (3 x 10^-3 m)² = 0.00112 Ω = 1.1 x 10^-3 Ω

(b) Given,

I = 15 A

R = 1.1 x 10^-3 Ω

N = 1

Therefore,

V = IR = 15 A x 1.1 x 10^-3 Ω = 0.0168 V

From Faraday’s law,

|ε| = Δφ/Δt

Or, Δφ/Δt = 0.0168 V

Or, Δ(BA)/Δt = 0.0168 V

Or, ΔB/Δt = 0.0168 V/A =0.0168 V/(πD²/4) = 0.0168 V/(π(0.15m)²/4) = 0.95 V/m² = 0.95 (V.s/m²).(1/s) = 0.95 T/s