Magnetic Induction Formula: Faraday's Law of Induction, Application and Sample Questions

Magnetic Induction Formula Overview

Magnetic induction is a phenomenon that occurs when a magnetic field is applied to a conductor, creating an electric current within the conductor. This process is known as magnetic induction and is governed by the magnetic induction formula. In this article, we will explore the magnetic induction formula in detail, including its definition, application, and mathematical representation.

Magnetic Induction Formula

Magnetic induction refers to the phenomenon of magnetizing materials when exposed to an external magnetic field, resulting in the emergence of certain magnetic properties in the material. The relationship between electricity and magnetism has been understood for nearly 200 years, with scientists discovering that the movement of electric charges (i.e., electric current) generates magnetic fields. Conversely, the movement of magnets can also produce electric currents. Electromagnetic induction, also known as magnetic induction, is the process by which an electrical conductor placed within a varying magnetic field produces an electromotive force (EMF) or voltage.

The magnetic induction formula is given as: ϵ = dϕb/dt

Faraday’s Law of Magnetic Induction

Faraday's experiments led to the discovery that a coil experiences an induced emf when there is a time-varying magnetic flux passing through it. This principle is known as Faraday's Law, which states that the "rate of change of magnetic flux in a circuit produces an emf in that circuit." The magnitude of the emf induced in a circuit is equivalent to the rate of change of the magnetic flux passing through it.

Magnetic Induction Formula

The equation for electromotive force (EMF) in a closed circuit can be derived from Faraday's law and is represented as follows:

ϵ = - (dϕb / dt)

In this context, ϕb represents the magnetic flux, ϵ represents the electromotive force (EMF), and t is the time. The negative sign in the equation indicates that a current I and magnetic field B, opposite in direction to the change in flux, are generated. This concept is known as Lenz's law.

In the case of a closely wound coil comprising N turns, the change in flux associated with each turn is identical, resulting in the total induced EMF or magnetic induction being represented as:

ϵ = N(dϕb / dt)

Magnetic Induction Formula for Moving Conductor

For a moving rod, with N = 1 and the flux given by Φ = BAcosθ, where θ = 0º and cosθ = 1, the area swept out by the rod is ΔA = lΔx. 

ϵ = BΔA/Δt

Here, velocity is perpendicular to the magnetic field.

If the velocity is at an angle θ with B, its component perpendicular to B is v sinθ.

ε = Blv sinθ

Here,

l = length of the conductor, 

v = velocity of the conductor 

θ = the angle between the magnetic field and the direction of motion.

Therefore, the induced current formula denotes a close relation between electric field and magnetic field that is dependent on a specific time variation.

Magnetic Induction Formula: Electricity and Magnetism

Electricity and magnetism are concepts that are closely related. Almost 200 years ago, scientists discovered that moving electric charges (electric current) generated magnetic fields.

1. Electric currents are generated by moving magnets.
2. Experiments conducted by Michael Faraday and Joseph Henry proved conclusively that when closed coils were subjected to changing magnetic fields, electric currents were induced.
3. This is known as electromagnetic induction.

Magnetic Induction Formula: Magnetic Flux

The magnetic flux, denoted by ΦB, is a measurement of the number of magnetic field lines that intersect a surface with cross-sectional area A. The amount of magnetic flux through a small surface is calculated by multiplying the magnetic flux density that is perpendicular to the surface by the area of the surface. Similar to electric flux, magnetic flux is defined as ΦB = B.Acosϕ, 

Here ϕ is the angle between B and A. Magnetic Flux is a scalar quantity. Its SI unit is Weber.

1 Weber = 1 Tesla.meter2.

Magnetic Induction Formula: Lenz’s Law

According to Lenz's law, the direction of the induced emf is such that it generates a current that opposes the change in magnetic flux that caused it.

As shown in the figure below, the north pole of a bar magnet is being moved towards a closed coil, resulting in an increase in the magnetic flux through the coil. Consequently, the induced current in the coil flows in a direction that opposes the increase in flux, which is in a counterclockwise direction.

Magnetic Induction Formula: Motional Electromotive Force

If we consider a straight conductor moving in a magnetic field, we can see in the figure that the rod PQ moves towards the left at a constant velocity v. As long as there is no energy loss, the circuit PQRS forms a closed loop that encloses an area changing as PQ moves. The magnetic flux can be expressed as ΦB = Blx, and since x changes with time, a rate of change of flux induces an emf given by:

ε = -dΦB/dt = -d/dt(Blx) = -Bld/dt = Blv.

The motional emf expression can also be explained by the Lorentz force acting on the free charge carriers of conductor PQ. As the rod moves with speed v in the magnetic field B, the charge also moves with the same speed v. The Lorentz force acting on this charge is qvB in magnitude and it is directed towards Q. The work done in moving the charge from P to Q is given by the product of the force and the distance moved, which is qvBL.

Since emf is defined as the work done per unit charge, it can be expressed as:

ϵ = W/q = BLv

Magnetic Induction for Different Values of Current and Distance

The table below shows the magnetic induction for different values of current and distance:

As can be seen from the table, the magnetic induction increases with the current and decreases with the distance from the wire.

Magnetic Induction Formula: Applications

The magnetic induction formula is used in various applications, including:

1. Electric Motors: Electric motors use the magnetic field generated by a wire carrying a current to produce a rotating force. The magnetic field interacts with the magnetic field of a permanent magnet, resulting in a torque that rotates the motor.
2. Generators: Generators use the principle of electromagnetic induction to convert mechanical energy into electrical energy. The magnetic field generated by a wire carrying a current is used to induce a voltage in a nearby wire, resulting in the production of electrical energy.
3. Transformers: Transformers use the magnetic field to transfer electrical energy from one circuit to another. The magnetic field generated by a wire carrying a current is used to induce a voltage in a nearby wire, resulting in the transfer of electrical energy.

Magnetic Induction Formula: Things to Remember

1. The Magnetic Induction Formula expresses the relationship between the induced electromotive force (EMF) and the rate of change of magnetic flux.
2. The induced EMF is directly proportional to the rate of change of magnetic flux through the conductor.
3. The Magnetic Induction Formula is given by: ϵ = dϕb/dt, where ϵ is the induced EMF, ϕb is the magnetic flux, and dt is the change in time.
4. The unit of measurement for magnetic induction is the volt (V).
5. The direction of the induced EMF is such that it opposes the change in magnetic flux that produces it, in accordance with Lenz's Law.
6. The Magnetic Induction Formula is used to calculate the induced EMF in a conductor placed inside a varying magnetic field, such as in a transformer or an electric generator.
7. The Magnetic Induction Formula is fundamental to the study of electromagnetism and has important practical applications in electrical engineering and physics.

Magnetic Induction Formula: Sample Questions

Q. A transformer has a primary coil with 500 turns and a secondary coil with 1000 turns. If the magnetic flux through the primary coil changes at a rate of 1000 T/s, what is the induced EMF in the secondary coil?

Sol: 

ϵ2/ϵ1 = N2/N1 (From Transformer Equation)

Where ϵ1 is the induced EMF in the primary coil, ϵ2 is the induced EMF in the secondary coil, N1 is the number of turns in the primary coil, and N2 is the number of turns in the secondary coil.

Also, ϵ1 = N1(dϕb/dt) (From Magnetic Induction Formula)

Given:

N1 = 500

N2 = 1000

dϕb/dt = 1000 T/s

Substituting the values, we get:

ϵ1 = 500 x (1000 T/s)

ϵ1 = 500000 V

Using the transformer equation, we get:

ϵ2/500000 = 1000/500

ϵ2 = (1000/500) x 500000

ϵ2 = 1000 V

Therefore, the induced EMF in the secondary coil is 1000 volts.

Q. A metal rod of length 0.5 cm is placed perpendicular to a field of flux density 0.6 Tesla and moves at a right angle to the field with a speed of 2 m/s. Calculate the emf induced in the rod.

Sol: 

l = 0.5cm

B = 0.6T

θ = 90°

v = 2 m/s.

E = Blv

= 0.5 × 0.6 × 2

= 0.6 V

Hence induced emf is 0.6Volt.

Q. The magnetic field of 2*10-2T acts at a right angle to a coil of area 100 cm2 with 50 turns. The average emf induced in the coil is 0.1 V when it is removed from the magnetic field in time t. Calculate the value of t. 

Sol: 

B = 2 × 10-2 T

θ = 90°

A = 100cm2 = 0.01m2

n = 50 turns.

E = 0.1 V

Or, θ = B × A

= 2 × 10-2 × 0.01

= 0.0002

E = N. θt

Or, 0.1 = 50 × 0.002 t

So, t = 0.1 sec.

Q.  A coil of area 50cm2is placed perpendicular to a uniform field of flux density 10-3T.i)What is the flux passing through the coil? ii) If the magnetic field drops down to 0 in 3 sec, what is the value of emf induced? 

Sol: 

i) Area (A) = 50 cm2 = 5 × 10-3 m2

B = 10-3T

Now,

(i) Flux(θ) = B × A

= 5 × 10-2 Tm2

ii) E = dθ/dt 

= 5 × 10-6 − 0

= 1.67 × 10-6V

Q. A wire coil with 200 turns is placed in a magnetic field that changes at a rate of 2 T/s. If the magnetic flux through the coil is 5 x 10^-3 Wb, what is the induced EMF in the coil?

Sol: 

ϵ = N (dϕb/dt) (From Magnetic Induction Formula)

Where N is the number of turns in the coil.

Given:

N = 200

dϕb/dt = 2 T/s

ϕb = 5 x 10^-3 Wb

Substituting the values, we get:

ϵ = 200 x (2 T/s) x (5 x 10^-3 Wb)

ϵ = 2 V

Therefore, the induced EMF in the coil is 2 volts.