Inductance: Meaning, Self Inductance, Mutual Inductance

Inductance: Before going into inductance, let us first understand what induction is. You know about conduction, right? You might have studied it in heat transfer/thermodynamics. Conduction is the transfer of any form of energy (whether it is electricity or heat) from one body to another body via physical contact between them. Now, going for induction is the transfer of any form of energy without any kind of physical contact between the bodies.

If we transfer electrical energy, then that process is known as electromagnetic induction or EMI. It is inducing an electromotive force (emf) across an electrical conductor under the effect of changing magnetic flux. There are a total of three methods of EMI viz. (1) Static EMI, (2) Dynamic EMI (or Motional EMI), and (3) Periodic EMI. Static EMI means inducing emf without motion. Dynamic EMI is also known as Motional EMI is a method in which emf is induced with the help of a moving conductor and periodic EMI means the induced emf continuously with respect to time and repeats itself after a fixed interval of time. The topic inductance comes under the category of Static EMI. Reading this article, you will get a complete overview of inductance, its types, and its applications.

Inductor and Inductance

You might have studied resistors and capacitors in the previous chapters. A resistor is an electrical device that is used to oppose and control the flow of electric current. The property of a conductor to oppose the current is known as resistance. While capacitor is a device that is used to store charge and energy temporarily, and capacitance is the ability of a capacitor to store energy.

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Mainly, when we talk about an inductor, it is simply a coil, as shown in the picture. A conducting wire is wound tightly to form a coil. When a direct current is passed through it, it will induce its constant magnetic field in it. But instead, if we apply a continuously changing supply or alternating current, the current flowing through the coil will keep on changing. Due to changing magnetic flux, it will induce its emf in the opposite direction to the supply. This is known as inductance. So, inductance is the ability of a conductor to oppose changing current, whereas an inductor is an electrical device that opposes changing current.

Self-Inductance

Self-induction does mean inducing an emf in itself. To make this thing clear, let us understand using the given diagram. Here, a battery is connected to a rheostat which is further connected to a coil.

If the resistance of the rheostat is to be kept constant, the battery will supply a constant current in the coil. Due to the constant current supplied to the coil, a constant magnetic field will be induced inside the coil.
As the resistance of the rheostat is changed, the current flowing through the coil will also change. Since the current is changing, they will be a changing magnetic flix inside the coil. Due to the effect of changing magnetic flux, an emf will be induced inside this coil, trying to oppose the magnetic flux. Hence, due to the induced emf, the direction of current induced will be opposite to that of current supplied.

The flux induced in the coil is dependent on the current flowing in the coil:
\(\phi \propto I\)
Hence, the ratio of flux and current supplied must be a constant value which will determine its ability to induce magnetic flux with respect to the current supplied to the coil. This constant is known as self-inductance \((L)\).
\(L = \frac{\phi }{I}\)
The system thus made is known as an inductor. If the coil has \(N\) number of turns, then
\(L = \frac{{N\phi }}{I}\)
\(\therefore \,N\phi = LI\)
Differentiating this equation on both sides with respect to time
\(\frac{d}{{dt}}\left( {N\phi } \right) = \frac{d}{{dt}}\left( {LI} \right)\)
\(N\frac{{d\phi }}{{dt}} = L\frac{{dI}}{{dt}}\)
But, according to Faraday’s law of EMI, emf induced in a coil \(\varepsilon = – N\frac{{d\phi }}{{dt}}\). Here, the negative sign says that the emf induced is in the opposite direction of the rate of change in current
\(\varepsilon = – L\frac{{dI}}{{dt}}\)
\(\therefore \,L = \frac{{ – \varepsilon }}{{\frac{{dI}}{{dt}}}}\)
Hence, self-inductance can also be defined as the negative of emf induced per unit rate of current change inside the coil. This now forms an inductor.
The SI unit of inductance is \(Wb/A = Vs/A = {\text{Henry}}\,{\text{(H)}}\)

Magnetic Energy Stored in an Inductor

From the above part, we can conclude that since an inductor can induce its emf, it stores energy in the form of a magnetic field. Let us derive the equation of magnetic energy stored in an inductor:
For a current \(I\) in the circuit, the rate of work done can be written as:
\(\frac{{dW}}{{dt}} = \varepsilon I\)
Substituting the equation for emf induced in the coil:
\(\frac{{dW}}{{dt}} = – L\frac{{dI}}{{dt}}I\)
\(\therefore \,dW = – LIdI\)
Integrating this equation on both the sides:
\(\int dW = \int – LIdI\)
\(W = – \frac{1}{2}L{I^2}\)
As we know that work done is the negative of energy stored. So, if the initial magnetic energy of the inductor is zero, then the energy stored in the inductor will be:
\(U = \frac{1}{2}L{I^2}\).

Mutual Inductance

Mutual inductance is somewhat like self-inductance. Instead, it just involves two coils. Current is supplied to one of them while emf is induced in the other. Let us understand it using the picture shown.

You can see that coil-1 in the diagram is connected to a supply and a rheostat, and coil-2 is connected to a galvanometer. A changing current will start flowing through coil-1. Because of this, a changing magnetic flux will be induced in it. This changing magnetic flux will be linked to the other coil because they are placed close to each other.
Since the magnetic flux through coil-2 is continuously changing with respect to time, an emf that will oppose this magnetic flux will be induced in coil-2, and hence a current in the direction opposite to that of supplied current will be induced in coil-2. Therefore, we can say that the flux through coil-2 is the result of the current supplied in coil-1
\({\phi _2} \propto {I_1}\)

Hence, the ratio of flux through coil-2 and current supplied in coil-1 will be constant. This constant is known as a mutual inductance in coil-2 due to coil-1
\({M_{21}} = \frac{{{\phi _2}}}{{{I_1}}}\)
From the above equation, we can define mutual inductance between both the coils as the ratio of flux linked with a coil per unit current supplied in the other coil.
\(\therefore \,{\phi_2} = {M_{21}}{I_1}\)
Differentiating the above equation with respect to time
\(\frac{d}{{dt}}\left( {{\phi_2}} \right) = \frac{d}{{dt}}\left( {{M_{21}}{I_1}} \right)\)
Using Faraday’s law of EMI
\({\varepsilon_2} = – {M_{21}}\frac{{d{I_1}}}{{dt}}\)
\(\therefore \,{M_{21}} = \frac{{ – {\varepsilon _2}}}{{\frac{{d{I_1}}}{{dt}}}}\)
Hence, mutual inductance can also be defined as the negative of emf induced per unit rate of change in current in the inducing coil.

Applications

Inductors are found in many of the electrical and electronic appliances these days due to the use of AC in the circuits. Hence, they are found to be very important. Some of the real-life applications of inductors are as given below:

1. Inductors are mainly used in choking circuits (choke coil).
2. The principle of mutual inductance is used in a transformer.
3. They are used to store energy as the formula derived earlier.
4. Used in LC oscillator circuit to generate alternating current.
5. Also used in power converters (ac-ac or dc-dc).

Solved Examples on Inductance

Example 1: When a current of \(2\;{\rm{mA}}\) is supplied to a coil with \(100\) turns, a magnetic flux of magnitude \(0.2\;{\rm{Wb}}\) is linked with it. Find the self-inductance of this coil.
Solution:
Current supplied to the coil \( I = 2\;{\rm{mA}}\)
Number of turns in the coil \((N) = 100\)
Magnetic flux linked with the coil \((\phi) = 0.2\;{\rm{Wb}}\)
Self-inductance of a coil is given by the equation
\(L = \frac{{N\phi }}{I}\)
Substituting the values
\(L = \frac{{100 \times 0.2}}{2}\)
\(\therefore \,L = 10\;{\text{H}}\)
The self-inductance of this coil is \(10\;{\text{H}}\).

Example 2: Determine the energy stored in an inductor of inductance \(100\;{\rm{mH}}\) when a current of \(0.2\;{\rm{A}}\) is passed through it.
Solution:
The inductance of the inductor \((L) = 100\;{\rm{mH}}\)
Current passed through it \((I) = 0.2\;{\rm{A}}\)
Energy stored in an inductor is given by the equation
\(U = \frac{1}{2}L{I^2}\)
Substituting the values
\(U = \frac{1}{2} \times 100 \times {10^{ – 3}} \times 0.2 \times 0.2\)
\(\therefore \,U = 2\,{\text{mJ}}\)
The energy stored in this coil is \(2\,{\text{mJ}}\).

Summary

Reading this article, we came to know about inductor and inductance, some important equations related to it, and its applications.
Self-inductance: \(L = \frac{{N\phi }}{I} = \frac{{ – \varepsilon }}{{\frac{{dI}}{{dt}}}}\)
Mutual inductance: \({M_{21}} = \frac{{{\phi _2}}}{{{I_1}}} = \frac{{ – {\varepsilon _2}}}{{\frac{{d{I_1}}}{{dt}}}}\)
Magnetic energy stored in an inductor: \(U = \frac{1}{2}L{I^2}\).

Frequently Asked Questions on Inductance

Q.1. What are the different units of inductance?
Ans: The different units of inductance are \(Wb/A = Vs/A = {\text{Henry}}\,{\text{(H)}}\).

Q.2. How do you calculate inductance?
Ans: Inductance can be calculated by knowing the ratio of magnetic flux and the current supplied to the coil.

Q.3. Why is symbol \(L\) is used for inductance?
Ans: Symbol \(L\) is used to honor scientist Heinrich Lenz who was among the pioneers of the study of electromagnetic induction.

Q.4. Why do we need inductance?
Ans: Inductance is needed to induce back emf to maintain current flowing even in the switch-off condition.

Q.5. Why is the inductor not used in DC?
Ans: An inductor is not used in DC because there is no change in current and hence magnetic flux in DC, and there won’t be any emf induced in it and hence no inductance.

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