Example 10.2: Energy density of electric and magnetic fields

Question: In a certain region of space, the magnetic field has a value of $1.0\times 10^{-2}$T, and the electric field has a value of $2.0\times 10^6\,{\rm V\,m}^{-1}$. What is the combined energy density of the electric and magnetic fields?
 
Answer: For the electric field, the energy density is

$$w_E = \frac{1}{2}\,\epsilon_0\,E^2 = (0.5)\,(8.85\times 10^{-12}) \,(2.0\times 10^6)^2 = 18\,\,{\rm J\,m}^{-3}.$$

For the magnetic field, the energy density is

$$w_B = \frac{1}{2} \,\frac{B^2}{\mu_0} = \frac{(0.5)\,(1.0\times 10^{-2})^2} {(4\pi\times 10^{-7})} = 40\,\,{\rm J\,m}^{-3}.$$

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:

$$w = w_E + w_B = (18 + 40) = 58\,\,{\rm J\,m}^{-3}.$$