Question

Define Mutual Inductance. Show that Henry = Newton Meter / Ampere².

Hint:

Mutual inductance quantifies the induced EMF in a coil due to a changing current in a nearby coil. Its SI unit, Henry, is derived from the dimensional analysis of voltage, time, and current.

Answer 1

Mutual Inductance: Mutual inductance \( M \) is the proportionality constant that describes the induced EMF in one coil when the current in a nearby coil changes. If the changing current in coil 1 produces a flux linking coil 2, the induced EMF in coil 2 is given by: \[ \mathcal{E}_2 = -M \frac{dI_1}{dt}, \] where \( \frac{dI_1}{dt} \) is the rate of change of current in coil 1. \textit{Dimensional Formula for Henry:} The unit of mutual inductance is the Henry (H), and it can be derived from the relationship between induced EMF and the rate of current change: \[ [M] = \frac{[\text{Induced EMF}] \cdot [\text{Time}]}{[\text{Current}]} \] Since the dimensional formula for EMF is \( [M] = \text{V} \cdot \text{s}/\text{A} \) and voltage (V) has the dimensional formula \( [M L^2 T^{-3} I^{-1}] \), we arrive at the final result for Henry: \[ \text{Henry} = \frac{\text{Newton} \cdot \text{Meter}}{\text{Ampere}^2}. \]