Self Inductance of a Solenoid - Important Concept and Derivation

Self-inductance is a form of electromagnetic inductance. It can be defined as the property of a current-carrying coil that resists or opposes the change of the current flowing through it. It can also be defined as the induction of a voltage in any current-carrying wire if the current in the wire changes. This happens due to the magnetic field created by the changing current. It induces a voltage in the same circuit, so it can be said that the voltage is self-induced. 

The self-induced voltage or emf always resists the change in current. Hence, if the current is increasing, it will resist the rise of the current and similarly, when the current is decreasing, it will resist the fall of the current. This means that the direction of induced emf is opposite to the applied voltage if the current is increasing. Similarly, the direction of the induced emf would be the same as that of the applied voltage if the current is falling. 

It should be noted that this property of the coil exists only for changing currents, i.e alternating current or AC. This property does not exist for direct or steady current. The self-inductance is measured in Henry which is a SI unit having dimensions ML²T^-2I^-2.

What is an Inductance Coil?

The inductor is a term that is used to describe a circuit that possesses the property of inductance. A coil of wire is one of the most common inductors and so in circuit diagrams, a coil of a wire is used as a symbol for an inductive component.  

The alternating current that runs through any coil creates a magnetic field in and around the coil. This is because the current increases or decreases. The magnetic field created due to the alternating current forms concentric loops around the wire and then they join together to form larger loops. When the current increases in one loop, the surrounding magnetic field expands, and it will cut across some or all the neighboring loops of wires. This induces a voltage in the loops. In this way, a voltage is induced in the coil when the current is changing. 

Given below is a diagram depicting the fields in an inductor coil. This is the principle behind the classic self-inductance of a coil experiment.

Self-inductance Derivation

It is evident from the diagram that the number of turns in the coil will affect the amount of voltage that is induced. Thus, the rate of change of magnetic flux will also affect the induced emf. This is captured precisely in Faraday’s law. Faraday’s law states that the induced emf is directly proportional to the rate of the change of magnetic flux.

Moreover, Lenz’s law states that the induced current has a direction such that its magnetic field opposes the change in the magnetic field which had previously induced the current. 

This means that the induced emf will be given as,

$V_{L}=-N\left(\dfrac{\mathrm{d} \varphi}{\mathrm{d} t}\right)$

Here, VL is the induced voltage, N is the number of turns in the coil and $\dfrac{\mathrm{d} \varphi}{\mathrm{d} t}$ is the rate of change of magnetic flux. The negative sign is due to Lenz’s law.

Since the magnetic field in a current-carrying wire is directly proportional to the current, the flux created by this particular field will also be proportional to the current. So,

$\varphi\propto I$

Here, I is the current and $\phi$ is the magnetic flux. The above expression can also be written as,

$\varphi=LI$

L is the constant of proportionality and is known as ‘self-inductance’. For a coil having N turns, the flux can be written as,

$N\varphi=LI$

Substituting this expression for flux in Faraday’s law expression will give an alternate expression for the induced emf which is,

$V_L=-L\dfrac{dI}{dt}$

To calculate the magnitude of self-inductance, the negative sign can be ignored. Hence, the formula of self-inductance will be,

$L=\dfrac{|\varphi|}{\left|\dfrac{dI}{dt}\right|}$

This is also known as the ‘coefficient of self-inductance formula’. If one asks to state the expression for the self-inductance of a coil, this is the expression. 

Self-inductance of a Solenoid

Let us take a solenoid having N turns with a length l and a cross-section area A and let current I flow through it. There will be a magnetic field at any given point in the solenoid, so let us represent it by B. The magnetic flux per turn will then be equal to the product of B and the area of each turn. 

We know that for a solenoid,

$B=\dfrac{\mu_{0} N I}{l}$

$\mu_{0}$ is the permeability of free space.

So, the magnetic flux per turn will be given by the product of B and the area of each turn which will be, $\dfrac{\mu_{0} N I}{l}A$.

The total magnetic flux will be given by the product of flux present in each turn and the number of turns. 

$\begin{align} \varphi=\dfrac{\mu_{0} N I}{l} \cdot N \\ \varphi=\dfrac{\mu_{0}{N^{2} I}}{l}\dots(1) \end{align}$ 

Now we know the relation between the self-inductance L of a coil and the flux which is given as, $\varphi=LI$.

Combining this equation and (1), we get

$L=\dfrac{\mu N^{2} A}{l}$

This is the self-inductance of a solenoid.

Conclusion

Self-inductance is a kind of electromagnetic induction and it is a property by which a current-carrying conductor opposes any change in the current that flows through it. This property gives rise to an induced emf which opposes the change in the current through the conductor. A coil of wire is usually used as an inductor and so the symbol of inductive components in a circuit diagram is a coil of wire. By using Faraday’s law and Lenz’s law, the self-inductance of a coil is calculated to be, $L=\dfrac{|\varphi|}{\left|\dfrac{\mathrm{d} I }{ \mathrm{d} t}\right|}$. 

Thus, the self-inductance of a solenoid is, $L=\dfrac{\mu_{0} N^{2} A}{l}$.