Two electrodes, separated by a distance d, in a vacuum are maintained at a constant potential difference. An electron, accelerated from one electrode to the other, gains kinetic energy Ek. The distance between the electrodes is now changed to d. What is the gain in kinetic energy of an electron that is accelerated from one electrode to the other?

Given that, the distance between the electrode is d.

The electron kinetic energy is Ek when the electrode are at distance "d" apart.

So, we want to find the K. E when that are at d/3 distance apart.

K. E = ½mv²

Note: the mass doesn't change, it is only the velocity that change.

Also,

K. E = Work done by the electron

K. E = F * d

K. E = W = ma * d

Let assume that if is constant acceleration

Then, m and a is constant,

Then,

K. E is directly proportional to d

So, as d increase K. E increase and as d decreases K. E decreases.

So,

K. E_1 / d_1 = K. E_2 / d_2

K. E_1 = E_k

d_1 = d

d_2 = d/3

K. E_2 = K. E_1 / d_1 * d_2

K. E_2 = E_k * ⅓d / d

Then,

K. E_2 = ⅓E_k

So, the new kinetic energy is one third of the E_k