Question

(1) An ion engine has 1 Tesla magnetic field and a hydrogen plasma is to be short out at an ExB velocity of 1150 Km/s. How much internal electric field must be present in the plasma?

Answer 1

Energy density of the magnetic field:

$$u_B=\frac{B^2}{2\mu_0}.$$

According to the problem, the electric and magnetic field must provide 1150 km/s per kilogram. Apply conservation of energy per unit mass:

$$u_B+u_E=\frac{v^2}{2},\\ \space\\ u_E=\frac{v^2}{2}-u_B=\frac{v^2}{2}-\frac{B^2}{2\mu_0}.$$

On the other hand, the electric field energy density is

$$u_E=\frac{\epsilon_0E}{2},\\ \space\\E=\frac{2u_E}{\epsilon_0}=\frac{1}{\epsilon_0}\bigg(v^2-\frac{B^2}{\mu_0}\bigg)=1.49\cdot10^{23}\text{ V/m}.$$