Animated Solution for Physics - Electromagnetic Induction: The SI unit of the inductance, the henry can by written as

Inductance ($L$) is a fascinating property of electrical circuits. It is the electrical equivalent of inertia, opposing any change in current. The SI unit of inductance is the Henry (H), named after the American scientist Joseph Henry.

However, because inductance is deeply interconnected with various physical phenomena—from magnetic fields to energy storage—the Henry can be expressed in several equivalent ways. Let's embark on a journey to uncover these different "faces" of inductance by looking at the fundamental equations of electromagnetism.

The Flux Perspective

The most fundamental definition of inductance comes from its relationship with magnetic flux. When a current $i$ flows through a coil, it generates a magnetic flux $\varphi$ that is directly proportional to the current:

Rearranging this equation to solve for $L$, we get:

The SI unit for magnetic flux is the Weber (Wb), and for current, it is the Ampere (A). Therefore, dimensionally, one Henry is exactly equal to one Weber per Ampere: This confirms that Option (a) is perfectly correct.

The Dynamic EMF Perspective

Inductance truly shines when things change. According to Faraday's Law of Induction, a changing current induces an electromotive force (EMF) that opposes the change. The magnitude of this induced EMF is given by:

If we isolate $L$, we find:

Here, EMF is measured in Volts (V), and the rate of change of current is measured in Amperes per second (A/s). Substituting these units gives: Thus, a Henry can also be written as a Volt-second per Ampere. Option (b) is correct!

The Energy Perspective

Just as a moving mass stores kinetic energy, an inductor carrying a current stores magnetic potential energy. The energy $U$ stored in an inductor is:

Rearranging to solve for $L$:

Energy is measured in Joules (J), and current squared is in Ampere squared (${\text{A}}^2$). This leads us to another beautiful equivalent unit: This proves that Option (c) is also correct.

The Circuit Time Perspective

Finally, let's look at how an inductor behaves in a real circuit with resistance. In an $L-R$ circuit, the current doesn't change instantly; it grows or decays exponentially. The rate of this change is governed by the time constant $\tau$, defined as:

Solving for $L$, we get:

Resistance is measured in Ohms ($\Omega$), and the time constant is simply a time, measured in seconds (s). Therefore, the unit of inductance can be elegantly expressed as: This confirms that Option (d) is correct as well.

This problem is a beautiful reminder of the consistency of physics. Whether you look at it through the lens of magnetic flux, induced voltage, stored energy, or circuit timing, the dimensions always align perfectly. All four options are correct!