Two fully charged cylindrical capacitors are connected to two identical batteries. The capacitors are identical except that the radius of the outer plate of capacitor A is four times the radius of the outer plate of capacitor B. Both inner plates have a radius that is half the outer radius of capacitor B.

(a) What is the relative capacitance of each capacitor? Express your answer as a ratio CA/CB.

(b) What is the relative energy stored by each capacitor? Express your answer as a ratio UA/UB.

(c) What is the relative charge stored by each capacitor? Express your answer as a ratio QA/QB.

Part(a):  The relative capacitance is

Part(b): The relative energy stored is

Part(c): The relative charge stored is

Explanation:

We know the capacitance () of a capacitor having charge () and subjected to a potential difference of () is given by

Also, the energy () stored by a capacitor can be written as

Let us assume that the inner radius of the Capacitor B, as shown in the figure, be , the outer radius be , the inner radius of Capacitor A be and the outer radius be .

Given in the problem,

Now, the capacitance () of a cylindrical capacitor is given by,

where is the permittivity of the free space, is the length of the cylindrical capacitor.

Part(a):

The capacitance of capacitor A,

and the capacitance of capacitor B,

giving the relative capacitance of each capacitor to be

Part(b):

Energy stored by capacitor A,

Energy stored by capacitor B,

giving the relative energy stored by each capacitor to be

Part(c):

The charge stored by capacitor A,

The charge stored by capacitor B,

giving the relative charge stored by each capacitor to be

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .