Question

An x-ray tube is an evacuated glass tube that produces electrons at one end and then accelerates them to very high speeds by the time they reach the other end. The acceleration is accomplished using an electric field. The high-speed electrons hit a metal target at the other end, and the violence of the collision converts their kinetic energy into high-energy light rays, commonly known as x-rays.

Through what potential difference should electrons be accelerated so that their speed is 2.1 % of the speed of light when they hit the target?

Answer 1

To calculate the potential difference needed to accelerate electrons to 2.1% of the speed of light, one can equate the formula for kinetic energy from motion (0.5mv^2) to the formula for kinetic energy from an electric field (eV) and solve for V.

Step-by-step explanation:

To determine through what potential difference electrons should be accelerated so that their speed is 2.1% of the speed of light when they hit the target, we use the formula for kinetic energy in an electric field:

KE = eV

where KE is the kinetic energy of the electron when it reaches the target, e is the charge of an electron (approximately 1.602 x 10-19 coulombs), and V is the potential difference. The kinetic energy can also be written in terms of the mass m of the electron (about 9.109 x 10-31 kg) and its velocity v:

KE = 0.5mv2

To find the potential difference, we set these equations equal to each other and solve for V:

0.5mv2 = eV

v is given as 2.1% of the speed of light, so v = 0.021c, where c is the speed of light (approx. 3 x 108 m/s). Therefore:

V = (0.5 x m x (0.021c)2) / e

Substituting the known values, we calculate the required potential difference (V).