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Q. 82

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

a. Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii (inner shell) and (outer shell).

b. A spherical capacitor with a gap between the shells has a capacitance of . What are the diameters of the two spheres?

a. The expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii andis

b. The diameters of the two spheres are and

See the step by step solution

Step by Step Solution

Step 1: Concept Introduction (Part a)

Concentric Spherical Capacitor:

In a spherical capacitor, either a concentric or a concentric hollow conductor surrounds a solid or hollow spherical conductor with a dissimilar radius.

Step 2: Explanation (Part a) 

Using the formula below, we can calculate the electric field outside a charged conducting sphere,

Here is the charge and is the electric field.

Integrate the electric field along the radial direction for the two concentric spheres to determine the potential difference between them.

Therefore

Step 3: Explanation (Part a)

The formula to find capacitance is as follows,

Substitute for

Therefore, the capacitance of the capacitor is

Step 4: Final Answer (Part a) 

Hence, the expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii and is

Step 5: Concept Introduction (Part b) 

Concentric Spherical Capacitor:

In a spherical capacitor, either a concentric or a concentric hollow conductor surrounds a solid or hollow spherical conductor with a dissimilar radius.

Step 6: Explanation (Part b) 

Find the inner and outer radii of the spherical shells in the capacitor.

Substitute for and rearrange the above equation in the form of .

Substitute for for for , and for in the above equation.

Step 7: Explanation (Part b) 

Now, calculate the value of

Substitute for for .

Therefore,

Also,

Solve equation (1) and (2),

Therefore,

Calculate the diameter of the inner spherical shell.

Calculate the diameter of the outer spherical shell.

Step 8: Final Answer (Part b)

Hence, the diameter of the inner and the outer spherical shells are and

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