Question

A counter consists of a cylindrical cathode of radius 1cm and an anode wire of radius 0.01cm which is placed along the axis of the cathode. A voltage of 2.3kV is applied between the cathode and anode. The electric field on the anode surface must be:
(A) $$2.3\times {10}^5$$ V/m
(B) $$5\times {10}^6$$V/m
(C) $$4.6\times {10}^5$$V/m
(D) $$2.5\times {10}^6$$V/m

Answer 1

Hint:We have a cylindrical shape and given outer radius and inner radius. Geiger counters are used to detect radiation viz beta and alpha particles. The counter consists of a tube filled with an inert gas that becomes conductive of electricity when it is impacted by a high-energy particle.

Complete step by step answer:
The radius of the outer cylinder is $$R_2=1$$cm
The radius of the inner wire cylinder $$R_1=0.01$$cm
For cylindrical capacitor capacitance is given by the formula, $$C={\displaystyle \frac{2\pi {\epsilon }_{0}L}{\ln ({\displaystyle \frac{R_2}{R_1}})}}$$
Now given in the question the value of potential difference across the ends is 2.3kV viz 2300 V
Using the formula to calculate the charge, Q=CV
Q= $$CV={\displaystyle \frac{2\pi {\epsilon }_{0}LV}{\ln ({\displaystyle \frac{R_2}{R_1}})}}$$
Now let us find the charge density that is charge per unit length denoted by $$\lambda$$
$$\lambda$$= $$={\displaystyle \frac{Q}{L}}$$
$$={\displaystyle \frac{2\pi {\epsilon }_{0}V}{\ln ({\displaystyle \frac{R_2}{R_1}})}}$$
We know electric field due to a rod or wire in the shape of cylindrical is given by $$E={\displaystyle \frac{\lambda }{2\pi {\epsilon }_0{R}_1}}$$
Substituting the values,
$$={\displaystyle \frac{V}{R_1\ln ({\displaystyle \frac{R_2}{R_1}})}}$$
$$={\displaystyle \frac{2300}{(0.01\times {10}^{-2})\ln (100)}}$$
= $$5\times {10}^{6}V/m$$
This matches with the option (B), hence, the correct option is (B).

Note: While substituting the values we should keep in mind that all the units must be in standard SI. The voltage was given in kilo Volt, w converted into volt. The counter has a cylindrical capacitor and so formula for cylindrical capacitance is used.