# Inductance and Self-Induction

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16.9Inductance and Self-Induction

In this Physics tutorial, you will learn:

• What is inductance?
• What is an inductor?
• What is the self-induced emf?
• How do we measure the self-inductance?
• What are the factors affecting the self-inductance?
• What are some applications of inductance in technology?

## Introduction

What methods do you know that can be used induce emf in a coil?

Do you think the number of turs in the coil has any effect in the induced emf?

Think about changing the value of resistor connected to the coil. What happens to the emf induced in the coil?

In this tutorial, we will discuss about a phenomenon called "self-induction". We encounter it in many electronic devices and it is used to protect them from damage caused by sudden current changes.

## The Meaning of Inductance

So far, we have seen many similarities between electric and magnetic phenomena and we have used the analogy between the corresponding quantities to have a better understanding on the new quantities (usually magnetic ones, because electricity was explained earlier). For instance, we have described the magnetic field through the help of electric field, magnetic flux using the analogy with electric flux and so on.

Let's use the same method to explain a new magnetic-related concept. In tutorial 14.7 "Capacitance and Capacitors", we have seen that capacitors are circuit components used to store electric charges in their plates. In this way, we can produce a desired electric field between the plates of a capacitor because they are charged by opposite signs. We considered the basic arrangement of capacitors as the parallel plate capacitor, having the symbol (-| |-).

Similarly, we call an "inductor" a device used to produce a desired magnetic field. The symbol of inductor is ( ). A solenoid is the most typical example of conductor.

Let's consider as solenoid connected to an electric circuit that contains a battery and a rheostat (variable resistor) as shown in the figure. This is a kind of electromagnet because the current passing around the coil produces a magnetic field similar to that of a bar magnet. The solenoid has N turns and its length is l. The rheostat is initially in the position 1 and therefore, its resistance is R0 and the current produced in the circuit is I0. As a result, the initial magnetic field produced by this electromagnet is B0.

We can change the magnetic field produced in the coil by changing the value of resistance of the circuit. This can be achieved by moving the sliding contact of rheostat in another position. As a result, we will obtain the new values R, I and B for the corresponding quantities, as shown in the figure below. The initial current flowing through the solenoid is

I0 = ε/R0

When the sliding contact of rheostat is moved in the new position, the current flowing through the solenoid becomes

I = ε/R

We have explained in the tutorial 16.2 that the magnetic field produced by a solenoid is

B = μ0N ∙ I/L

where μ0 is the vacuum permeability, N is the number of turns in the solenoid, I is the current flowing through the circuit and l is the solenoid's length. The initial magnetic flux through the solenoid therefore is

Φ0 = B0 ∙ A
= μ0 ∙ N ∙ I0/l ∙ A

where A is the area of solenoid loops, and the final flux through the solenoid is

Φ = B ∙ A
= μ0N ∙ I/L ∙ A

Therefore, the induced emf produced by the solenoid (known as self-induced emf because it is not generated by the flux change due to any motion in respect to an external magnetic field), is

ε' = - N ∙ ∆Φ/Δt
= -N ∙ Φ-Φ0/Δt
= -N ∙ μ0N ∙ I/L ∙ A-μ0N ∙ I0/l ∙ A/Δt
= -N ∙ (μ0N ∙ A/l) ∙ (I-I00/Δt)

Thus, we obtain for the self-induced emf in the coil:

ε' = -μ0 ∙ N2 ∙ A/I∆I/Δt

We denote by L the expression inside the brackets. This quantity is known as self-inductance (or simply inductance). It depends only on the physical features of the solenoid (number of turns, area, length) and not on the electric properties of the circuit. The unit of self-inductance is known as Henry (H).

Thus, having

L = μ0 ∙ N2 ∙ A/I

we obtain for the self-induced emf in the coil in terms of inductance:

ε' = -L ∙ ∆I/Δt

As stated at the beginning of this tutorial, coils are known as "inductors" just because of their property of self-inductance.

### Example 1

A 20 cm long solenoid having a cross sectional area of 4 cm2 contains 500 turns per metre. The solenoid is connected to a 12 V battery through a rheostat.

1. Calculate the self-inductance of the solenoid
2. Find the self-induced emf of solenoid if the resistance in the rheostat decreases from 24 Ω to 6 Ω in 0.4 seconds.
3. (Take the figure shown in theory section as a reference)

### Solution 1

Clues:

l = 20 cm = 0.20 m
A = 4 cm2 = 0.0004 m2 = 4 × 10-4 m2
n = 500 turns/metre = 5 × 102 turns/metre
ε = 12 V
0 = 4π × 10-7 N/A2)
R0 = 24 Ω
R = 6 Ω
Δt = 0.4 s
a) L = ?
b) ε' = ?

1. First, we calculate the number of turns in the solenoid. We have
N = N ∙ l
= 500 turns/m ∙ 0.20 m
= 100 turns
= 102 turns
2. The self-inductance of the solenoid is
L = μ0 ∙ N2 ∙ A/I
= (4 ∙ 3.14 ∙ 10-7 N/A2 ) ∙ (102)2 ∙ (4 × 10-4 m2 )/(0.20 m)
= 2.512 × 10-5 H

## Another Approach on Inductance

If a current I is flowing through the turns of a solenoid (called henceforth an "inductor"), it produces a magnetic flux Φm through the central region of the inductor. As a result, we obtain for the inductance of inductor in terms of magnetic flux:

L = N ∙ ΦM/I

where N is the number of turns in the inductor. Indeed, since

L = μ0 ∙ N2 ∙ A/I

and

ΦM = N ∙ B ∙ A

We obtain by combining the two above formulae (considering also the fact that the magnetic field of a solenoid is B = μ0N ∙ I/L)

L = N ∙ ΦM/I
= N ∙ B ∙ A/I
= N ∙ μ0N ∙ I/L ∙ A/I
= μ0 ∙ N2 ∙ A/I

Therefore, the two formulae of inductance given above are equivalent.

### Example 2

When a 10 cm long solenoid containing 200 turns and having the cross-sectional area of each turn equal to 4 cm2 is connected to a 24 V power source. The resistance of the circuit is 48 Ω. What is the magnetic field the solenoid generates in such conditions? Clues:

l = 10 cm = 10-1 m
N = 200 turns = 2 × 102 turns
A = 4 cm2 = 4 × 10-4 m2
ε = 24V
R = 48 Ω
0 = 4π × 10-7 N/A2)
B = ?

### Solution 2

First, we calculate the current flowing through the circuit through the Ohm's law. We have

I = ε/R
= 24 V/48 Ω
= 0.5 A

Now, let's calculate the inductance in the solenoid. We have

L = μ0 ∙ N2 ∙ A/I
= (4 ∙ 3.14 × 10-7 N/A2 ) ∙ (2 × 102 )2 ∙ (4 × 10-4 m2 )/(10-1 m)
= 2 × 10-4 H

Now, let's use the other formula of inductance to find the magnetic field through the solenoid. Thus, giving that

L = N ∙ ΦM/I = N ∙ B ∙ A/I

we obtain

B = I ∙ L/N ∙ A
= (0.5 A) ∙ (2 × 10-4 H)/(2 × 102 ) ∙ (4 × 10-4 m2 )
= 1.25 × 10-3 T
= 1.25 mT

## Applications of Inductance in Technology

There is a wide range of inductance application in modern electronic devices. Some of them include:

1. Filters. Inductors are used together with capacitors and resistors (as we will see in the upcoming tutorials) to create filters that prevent undesired frequencies from mixing with the signal and allowing only certain bands of signal frequency (the desired ones) to pass through.
2. Sensors. Inductors are very good in detecting magnetic fields or the presence of magnetic materials from a certain distance. Therefore, they are used in cars, traffic lights, measuring devices etc.
However, inductive sensors are limited in two major ways. Either the object to be sensed must be magnetic and induce a current in the sensor, or the sensor must be connected to a power source to detect the presence of materials that interact with a magnetic field. These parameters limit the applications of inductive sensors and influence the designs that use them.
3. Transformers. They are devices that are used to change the value of potential difference in a power transmission system in order to save energy or to make the values fit the electric appliances. Transformers are devices that are built by combining inductors that have a shared magnetic path.
4. Electric Motors. We have seen that electric motors are operated by making a coil turn between two magnets. This action bring a rotation of the circuit that powers the coil as well. This is not practical and consumes a lot of energy. To prevent this, an inductor is installed in the place where the circuit and the coil meet. The inductor does not require any direct contact between the parts it connects, so the coil can rotate while the supplying circuit remains stationary.
5. Tape recorder. Although an outdated device, tape recorder is one of the most important inventions of the last century. It converts sound into electric signal through a microphone and an amplifier. This current then passes through the coil of an electromagnet. After a few processes, the tape is magnetised, causing a varying flux through the playback coil. As a result, the original voice is reproduced in the original frequencies giving an identical pattern when replayed.

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