Question

Derive an expression for energy stored in the magnetic field in terms of induced current.

Hint:

The energy stored in a magnetic field depends on the square of the current and the inductance of the coil. The larger the current and inductance, the more energy is stored.

Answer 1

Step 1: Induced EMF and Current.
The energy stored in the magnetic field of an inductor can be derived using Faraday’s law of induction. The induced emf \( \mathcal{E} \) in an inductor is related to the rate of change of current \( I \) by: \[ \mathcal{E} = -L \frac{dI}{dt} \] where \( L \) is the inductance and \( \frac{dI}{dt} \) is the rate of change of current.
Step 2: Power Supplied to the Inductor.
The power \( P \) supplied to the inductor is the product of the current and the induced emf: \[ P = I \cdot \mathcal{E} = I \cdot (-L \frac{dI}{dt}) \] \[ P = -L I \frac{dI}{dt} \]
Step 3: Energy Stored in the Magnetic Field.
The energy stored in the magnetic field is the integral of the power over time: \[ U = \int_0^t P \, dt = \int_0^t -L I \frac{dI}{dt} \, dt \] By integrating, we get: \[ U = \frac{1}{2} L I^2 \] This is the expression for the energy stored in the magnetic field in terms of the current.