For your senior project, you are designing a Geiger tube for detecting radiation in the nuclear physics laboratory. This instrument will consist of a long metal cylindrical tube that has a long straight metal wire running down its central axis. The diameter of the wire will be 0.550 mm and the inside diameter of the tube will be 3.72 cm. The tube is to be filled with a dilute gas in which an electrical discharge (breakdown of the gas) occurs when the electric field reaches 5.55 106 N/C. Determine the maximum linear charge density on the wire if breakdown of the gas is not to happen. Assume that the tube and the wire are infinitely long.

Maximum linear charge density = 84.14 nC/m

Explanation:

Looking at this question, The electric field of a line charge of infinite length is given by : Er = (1 / (2πεo)) x (λ/r)

r = the distance from the center of the line of charge

λ = the linear charge density of the wire.

Now looking at the equatiom, due to the fact that Er varies inveresely with r, its maximum value will occur at the surface of the wire where r = R, the radius of the wire:

And so, Emax = (1 / (2πεo)) x (λ/R)

Let's make λ the subject of the equation and we get;

λ = 2πεo (REmax)

From the question, R = 0.55/2 = 0.275cm

Also, Emax = 5.50 * 10^ (6) N/C

Let's take the value of the electric constant to be εo = 8.854 x 10^ (-9) C^ (2) / Nm^2

R = 0.275mm = 0.000275m

Plugging these values into the equation, we get;

λ = 2π x 8.854 x 10^ (-12) x 0.000275 x 5.50 * 10^ (6) = 84.14 nC/m