Question

A Geiger tube consists of two elements, a long metal cylindrical shell and a long straight metal wire running down its central axis. Model the tube as if both the wire and cylinder are infinitely long. The central wire is positively charged and the outer cylinder is negatively charged. The potential difference between the wire and the cylinder is 1.10 kV. Suppose the cylinder in the Geiger tube has an inside diameter of 3.64 cm and the wire has a diameter of 0.452 mm. The cylinder is grounded so its potential is equal to zero.

(a) What is the radius of the equipotential surface that has a potential equal to 545 V? Is this surface closer to the wire or to the cylinder?

(b) How far apart are the equipotential surfaces that have potentials of 195 and 245 V?

(c) Compare your result in Part (b) to the distance between the two surfaces that have potentials of 685 and 730 V, respectively.

What does this comparison tell you about the electric field strength as a function of the distance from the central wire?

Answer 1

Inner radius of the cylinder is r_c = (3.64/2) = 1.82 cm and the radius of wire is r_w = (0.452/2) = 0.226 mm. Thus the radial distance between cylinder and wire is,

  

The given potential difference between cylinder and wire is V_d = 1.10 kV. Thus the electric field can be calculated as,

Part (a): The radius of the equipotential surface that has a potential equal to 545 V can be calculated as,

  

So this equipotential surface is closer to the wire.

Part (b): The radius of the equipotential surface that has a potential equal to 195 V is,

Now the radius of the equipotential surface that has a potential equal to 245 V is,

Thus the distance between these equipotential surfaces is,

  

Part (c):  The radius of the equipotential surface that has a potential equal to 685 V is,

The radius of the equipotential surface that has a potential equal to 730 V is,

Thus the distance between these equipotential surfaces is,

  

The equipotential surfaces in part (c) are much closer that those given in part (b).

From this comparison, we can see that the electric field strength is constant with respect to the distance from the central wire.