Energy Density

The electric and magnetic fields have energy and hence have an energy density. We can see this for a capacitor:The energy stored in a capacitor is U = 1 2 C V 2 U = 1 2 C V 2 where C C is the capacitance and V V the potential drop (voltage) across the capacitor. For a parallel plate capacitor C = ε 0 A d C = ε 0 A d and V = E d V = E d where A A is the area of the plates d d the distance between them and E E the electric field strength.note that A d A d is the volumeThus U = 1 2 ε 0 A d ( E d ) 2 = 1 2 ε 0 A d E 2 U = 1 2 ε 0 A d ( E d ) 2 = 1 2 ε 0 A d E 2 So we can write the energy density (Energy per Unit volume) of the field as u E = U A d = 1 2 ε 0 E 2 u E = U A d = 1 2 ε 0 E 2 Likewise by calculating the energy stored by a B-field in a current carrying solenoid one can derive: u B = B 2 2 μ 0 u B = B 2 2 μ 0 Since we know E = c B E = c B u E = 1 2 ε 0 E 2 = 1 2 ε 0 c 2 B 2 = 1 2 ε 0 1 ε 0 μ 0 B 2 = 1 2 1 μ 0 B 2 = u B u E = 1 2 ε 0 E 2 = 1 2 ε 0 c 2 B 2 = 1 2 ε 0 1 ε 0 μ 0 B 2 = 1 2 1 μ 0 B 2 = u B In an EM wave u = u E + u B u = u E + u B which is u = ε 0 E 2 u = ε 0 E 2 or equivalently u = B 2 / μ 0 u = B 2 / μ 0