Question

An electron gun on a space probe starts emitting an electron beam into the space. Electrons in the beam have energy $W = 9.0 \times 10^4$ eV and the beam is equivalent to electric current $I = 1.0$ mA. If the space probe is a metal sphere of radius $r = 1.0$ m, find electric field strength at the surface of the space probe 1.0 min after starting the electron gun. Charge on an electron is $e = -1.6 \times 10^{-19}$ C.

Answer 1

5.4 x 10^8 N/C

Answer 2

1. Calculate the total charge accumulated: The current $I$ is the rate of charge leaving the space probe. Since electrons are emitted, the probe loses negative charge and thus acquires a positive charge. Given current $I = 1.0 \text{ mA} = 1.0 \times 10^{-3} \text{ A}$. Given time $t = 1.0 \text{ min} = 60 \text{ s}$. The total charge $Q$ accumulated on the probe is: $Q = I \times t$ $Q = (1.0 \times 10^{-3} \text{ A}) \times (60 \text{ s})$ $Q = 6.0 \times 10^{-2} \text{ C}$

2. Calculate the electric field strength at the surface: The space probe is a metal sphere of radius $r = 1.0 \text{ m}$. For a charged conducting sphere, the electric field strength at its surface is given by: $E = \frac{kQ}{r^2}$ where $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$ (Coulomb's constant). $E = \frac{(9 \times 10^9 \text{ N m}^2/\text{C}^2) \times (6.0 \times 10^{-2} \text{ C})}{(1.0 \text{ m})^2}$ $E = (9 \times 6.0) \times 10^{(9-2)} \text{ N/C}$ $E = 54 \times 10^7 \text{ N/C}$ $E = 5.4 \times 10^8 \text{ N/C}$