Energy Density Formula

Energy Density Formula - Electric Energy Density Formulas with Examples

Energy Density Formula
Energy Density refers to the total amount of energy in a system per unit volume. (Eventhough generally energy per unit mass is also mentioned as energy density, the proper term for the same is specific energy. The term density usually measures the amount per unit spatial extension).
Energy density is denoted by letter U.
Magnetic and electric fields can also store the energy.
In the case of electric field or capacitor, the energy density is given by
\[U = \frac{1}{2}{\varepsilon _0}{E^2}\]
The energy density in the case of magnetic field or inductor is given by,
\[U = \frac{1}{{2{\mu _0}}}{B^2}\]
For electromagnetic waves, both magnetic and electric fields are equally involved in contributing to energy density. Therefore, the energy density is the sum of the energy density of electric and magnetic fields.
i.e., \[U = \frac{1}{2}{\varepsilon _0}{E^2} + \frac{1}{{2{\mu _0}}}{B^2}\]
Example 1
In a certain region of space, the magnetic field has a value of 1.0 × 10–2 T, and the electric field has a value of 2.0 ×106 Vm–1. Find the combined energy density of the electric and magnetic fields.   Solution:

     E = 2.0 ×106 Vm–1; B = 1.0 × 10–2 T

For the electric field, the energy density is
\[{U_E} = \frac{1}{2}{\varepsilon _0}{E^2} = \frac{1}{2} \times 8.85 \times {10^{--12}}{(2.0 \times {10^6})^2} = 18\,J{m^{--3}}\]
For the magnetic field, the energy density is

\[{U_B} = \frac{1}{2}\frac{{{B^2}}}{{{\mu _0}}} = \frac{1}{2} \times \frac{{{{(1.0 \times {{10}^{--2}})}^2}}}{{4\pi \times {{10}^{--7}}}} = 40\,J{m^{--3}}\]
The net energy density is the sum of the energy density due to the electric field and the energy density due to the magnetic field:
     U = UE + UB = 18 + 40 = 58 Jm–3

Practice question

The total energy density associated with an electromagnetic wave is (assuming the same to be equally shared across the electric and magnetic fields):
(a) \[\frac{1}{2}{\varepsilon _0}{E^2}\] (b) \[\frac{1}{{2{\mu _0}}}{B^2}\] (c) \[{\varepsilon _0}{E^2}\] (d) \[\frac{{2{B^2}}}{{{\mu _0}}}\]
Ans (c).