What is the escape speed for an electron initially at rest on the surface of a sphere with a radius of 1.3 cm and a uniformly distributed charge of 2.3 ✕ 10-15 C. That is, what initial speed must the electron have to reach an infinite distance from the sphere and have zero kinetic energy when it gets there?

2.37 * 10^4 m/s

Explanation:

Constants:

Mass of electron = 9.11 * 10^ (-31) kg

Electric charge of an electron = 1.602 * 10^ (-19) C

Parameters given:

Radius of sphere = 1.3cm = 0.013m

Charge of sphere = 2.3 * 10^ (-15) C

Using the law of conservation of energy, we have that:

K. E. (initial) + P. E. (initial) = K. E. (final) + P. E. (final)

K. E. (final) = 0, since final velocity is zero and P. E. (final) = 0 since the electron reaches a final distance of infinity.

Hence,

K. E. (initial) = P. E. (initial)

0.5mv^2 = (kqQ) / r

Where k = Coulumbs constant

Q = charge of the sphere.

r = radius of the sphere.

=> 0.5*m*v^2 = (kqQ) / r

0.5 * 9.11 * 10^ (-31) * v^2 = (9 * 10^9 * 1.602 * 10^ (-19) * 2.3 * 10^ (-15)) / 0.013

4.555 * 10^ (-31) * v^2 = 2550.88 * 10^ (-25)

=> v^2 = 2550.88 * 10^ (-25) / 4.555 * 10^ (-31)

v^2 = 560 * 10^6 = 5.60 * 10^8

=> v = 2.37 * 10^4 m/s