6.7. INDUCTANCE

An electric current can be induced in a coil by flux change produced by another coil in its vicinity or flux change produced by the same coil. These two situations are described separately in the next two sub-sections. However, in both the cases, the flux through a coil is proportional to the current. That is, Φ α I.

Inductance is a property of a coil that quantifies how much voltage can be induced in it due to a change in current (flux change) either from a nearby coil or from its own current variation. The statement 'Φ α I' means that the magnetic flux (Φ) enclosed by the coil is directly proportional to the electric current (I) flowing through it. Therefore, if the current through the coil changes, the magnetic flux also changes, which results in an induced voltage (emf) in the coil.

Imagine you have a water tank (the coil). When you fill the tank with water (the current), the level of water (the magnetic flux) rises. If you then drain the tank (change the current), the level of water drops, which creates a wave of pressure in the pipes connected to it (induced voltage). This is similar to how changing current affects magnetic flux and induces a voltage in the coil.

Further, if the geometry of the coil does not vary with time then, dΦ/dt ∝ dI/dt. For a closely wound coil of N turns, the same magnetic flux is linked with all the turns. When the flux Φ through the coil changes, each turn contributes to the induced emf. Therefore, a term called flux linkage is used which is equal to NΦ for a closely wound coil and in such a case NΦ ∝ I.

The relationship between the change in flux and the change in current indicates that for a constant geometry, the rate at which flux changes over time (dΦ/dt) directly relates to how fast the current changes (dI/dt). For a coil with multiple turns (N), this situation becomes more pronounced since each turn experiences the same change in flux. The total flux linkage is represented as NΦ, and it tells us how the current through the coil correlates with the flux change. The inductance is defined as the proportionality constant relating these two quantities.

Think of a multi-layer cake where each layer represents a turn of the coil. As you remove frosting (change the current) from the top layer and it reaches lower layers, those layers also lose frosting due to the flow of frosting down. Each layer feels a change in frosting equivalently; thus, when one layer loses frosting, all layers (turns of the coil) experience that change.

The constant of proportionality, in this relation, is called inductance. We shall see that inductance depends only on the geometry of the coil and intrinsic material properties. This aspect is akin to capacitance which for a parallel plate capacitor depends on the plate area and plate separation (geometry) and the dielectric constant K of the intervening medium (intrinsic material property).

Inductance is a fundamental property of coils that reflects how effectively they can store magnetic energy and induce voltage in response to changing current. The inductance (measured in henries) is determined not only by the physical dimensions of the coil (geometry) but also by the material properties, such as its magnetic permeability. This is similar to how a capacitor's capacitance depends on its size and the materials between its plates. A larger inductance means the coil can store more energy.

Think of inductance like a giant sponge (the coil) that can soak up water (magnetic energy). The size and material of the sponge - whether it's made of cotton or silicone - determine how much water it can absorb at once. Similarly, a coil's inductance tells you how much magnetic 'energy' it can hold as the current changes.

Inductance is a scalar quantity. It has the dimensions of [M L2 T–2 A–2] given by the dimensions of flux divided by the dimensions of current. The SI unit of inductance is henry and is denoted by H. It is named in honour of Joseph Henry who discovered electromagnetic induction in USA, independently of Faraday in England.

Inductance is measured as a scalar quantity, which implies it has magnitude but no direction. Its dimensions involve mass (M), length (L), time (T), and electric current (A), leading to the derived unit 'henry' (H). This unit represents how much electromotive force (voltage) is induced per unit of current change, showcasing the direct relationship between flux change and current change.

Consider measuring how large a water tank is. We describe the tank's size by the volume unit (liters, for instance) which is similar to how we measure inductance in henries. Just as a larger tank can hold more water, a higher inductance means more voltage generation capability when current changes.

Consider Fig. 6.12 which shows two long co-axial solenoids each of length l. We denote the radius of the inner solenoid S1 by r1 and the number of turns per unit length by n1. The corresponding quantities for the outer solenoid S2 are r2 and n2, respectively. Let N1 and N2 be the total number of turns of coils S1 and S2, respectively. When a current I is set up through S2, it in turn sets up a magnetic flux through S1.

Mutual inductance is the phenomenon where an electrically induced current in one coil (say S2) produces a magnetic field that can affect another coil placed nearby (S1). The induced magnetic flux in S1 depends on the current in S2. This process is mathematically expressed as flux linkage (Φ) in S1 equals mutual inductance (M) times the current in S2. This demonstrates how closely linked systems can influence each other.

Imagine two wireless charging pads placed close to each other. When one charging pad is activated (current flows), it generates a magnetic field that can also induce power flow to a phone sitting on the second pad - even if they are not directly connected. This exemplifies mutual inductance in action.

In the previous sub-section, we considered the flux in one solenoid due to the current in the other. It is also possible that emf is induced in a single isolated coil due to change of flux through the coil by means of varying the current through the same coil. This phenomenon is called self-induction.

Self-inductance occurs when a coil experiences a change in current that leads to a change in its own magnetic field, thus inducing a voltage within itself. The relationship is expressed as flux linkage (NΦ) being proportional to the current (I) through the same coil, resulting in the self-inductance denoted as L.

Think of a swing on a playground. As you push the swing (increase the current), it moves higher (increased magnetic field) and after you stop pushing (change in current), it swings back due to the momentum from previous pushes (induced emf). This shows how the system keeps influencing itself during changes.

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