Physics Lesson 16.7.2 - Faraday's Law of Induction

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Welcome to our Physics lesson on Faraday's Law of Induction, this is the second lesson of our suite of physics lessons covering the topic of Faraday's Law of Induction, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Faraday's Law of Induction

Based on the above two experiments, Faraday realized that:

An emf and current can be induced in the loop by changing the amount of magnetic field around the loop;
The amount of magnetic field can be represented through the amount of magnetic field lines passing through the loop.

Combining the above statements, we obtain the Faraday's Law of Induction:

"An induced emf (and current) are induced in a loop only if the number of magnetic field lines passing through the loop is changing."

From this law, we can conclude that the number of actual field lines flowing through the loop is not important; the only thing that matters in this process is the rate of change in the number of magnetic field lines passing through the loop.

Now, let's calculate the amount of magnetic field lines passing through the loop. For this, we have to consider the magnetic flux Φ_M, which is analogue to the electric flux Φ_E we have discussed in the tutorial 14.6 "Electric Flux and the Gauss Law." The integral expression of the electric flux flowing through a loop is

Φ_E = ∫E⃗ dA⃗

where E⃗ is the vector of electric field and A⃗ the area vector of the loop.

Similarly, the magnetic flux Φ_M flowing through a similar loop is

Φ_M = ∫B⃗ dA⃗

The result of this integral is the dot product of the magnetic field vector B⃗ and the area vector A⃗, which is perpendicular to the surface pierced by the magnetic field lines and is numerically equal to the area enclosed by the loop.

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The magnetic field B⃗ between the poles produced by the bar magnet shown above is uniform, i.e. the magnetic field lines are parallel to each other (vertical in direction). The loop is deflected by an angle θ to the horizontal direction; this means the area vector A⃗ forms the same angle θ to the magnetic field lines. Therefore, the value of the integral if the magnetic field is uniform, is

Φ_M = B⃗ ∙ A⃗

and the magnitude of magnetic flux Φ_M, is

Φ_M = B ∙ A ∙ cos⁡θ

In addition, if the loop is placed perpendicular to the direction of magnetic field lines, i.e. the area vector is parallel to the magnetic field lines, we obtain

Φ_M = B ∙ A ∙ cos⁡0⁰
= B ∙ A ∙ 1
= B ∙ A

The unit of magnetic flux (Tesla × square metre) is known as Webber, [Wb]. Thus,

1 Wb = 1 T ∙ m²

The concept of magnetic flux helps us state the Faraday' Law of induction in a more comprehensive and practical way:

"The magnitude of the electromotive force induced in a loop is equal to the rate of change of the magnetic flux in this loop."

We have seen in tutorial 16.2 that the induced current in a coil containing N loops when we move a magnet towards to or away from it, is in the opposite direction to the moving magnet. The same thing is true for the induced emf produced in the coil as well. Thus, based on the Faraday's Law of Induction, we can say that the induced emf tends to oppose the flux change, i.e.

ε_i = -∆Φ_M/Δt

for not-so-small intervals of time and

ε_i = -dΦ_M/dt

for infinitely small intervals of time. Obviously, the last formula gives a more precise result for the emf induced in a loop or coil due to a magnetic field. Therefore, it is known as the mathematical expression of the Faraday's Law of induction.

Example 1

A circular loop of radius equal to 3 cm is placed between the poles of a horseshoe magnet at 300 to the horizontal direction, as shown in the figure.

What is the induced emf produced in the loop if we rotate the coil from the actual position to the horizontal position in 2 seconds? The magnetic field between the poles of magnet is 20 mT.

Solution 1

First, we must find the magnetic flux flowing through the loop in both positions. In the actual position, the angle between magnetic field lines (down-up) and the area vector (normal to the loop) is still 300 (from geometry it is known that two angles that have perpendicular sides to each other are equal).: Physics Tutorials: This image provides visual information for the physics tutorial Faraday's Law of Induction

We have the following clues:

r = 3 cm = 3 × 10^-2 m
B = 20 mT = 2 × 10^-2 T
θ₁ = 30°
θ₂ = 0°
Δt = 2 s

Thus, for the angle θ₁ = 30°

Φ_M(1) = B ∙ A ∙ cos⁡θ₁
= B ∙ π ∙ r² ∙ cos⁡θ₁
= (2 × 10^-2 T) ∙ (3.14) ∙ (3 × 10^-2 m)² ∙ cos⁡30⁰
= (2 × 10^-2 T) ∙ (3.14) ∙ (3 × 10^-2 m)² ∙ (0.866)
= 4.89 × 10^-5 Wb

When rotating the loop at θ₂ = 0° we obtain

Φ_M(2) = B ∙ A ∙ cos⁡θ₂
= B ∙ π ∙ r² ∙ cos⁡θ₂
= (2 × 10^-2 T) ∙ (3.14) ∙ (3 × 10^-2 m)² ∙ cos⁡0⁰
= (2 × 10^-2 T) ∙ (3.14) ∙ (3 × 10^-2 m)² ∙ 1
= 5.65 × 10^-5 Wb

Therefore, the induced electromotive force ε_i produced in the loop when rotating it inside the uniform magnetic field, is

ε_i = -∆Φ_M/Δt
= -(Φ_M(2) -Φ_M(1) )/Δt
= -(5.65 × 10^-5 Wb-4.89 × 10^-5 Wb)/2s
= -0.38 V

Hence, the induced emf during this process is 0.38V (the negative sign is because induced emf is in the opposite direction to the cause of its generation (moving direction of loop).

Note that here we have used the symbol Δ instead of d to represent the time interval as it is too large to consider as infinitely small.

If the coil has N-turns, we obtain for the induced emf:

ε_i = -N ∙ ∆Φ_M/Δt

The magnetic flux through a coil can be changed in four ways:

By changing the value of magnetic field in the space containing the coil,
By changing the area of coil,
By changing the angle of coil to the magnetic field lines, and
By changing the number of turns (loops) in the coil

You have reached the end of Physics lesson 16.7.2 Faraday's Law of Induction. There are 4 lessons in this physics tutorial covering Faraday's Law of Induction, you can access all the lessons from this tutorial below.

More Faraday's Law of Induction Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.7	Faraday's Law of Induction
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.7.1	What did Faraday Observe during His Experiments?
16.7.2	Faraday's Law of Induction
16.7.3	Induced Electromotive Force as Motional Emf
16.7.4	Motional Emf and Electrical Energy

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