A spherical capacitor consists of two concentric conducting spherical shells of

Question

A spherical capacitor consists of two concentric conducting spherical shells of radii R and 6 R .

Part A

How long would a coaxial cylindrical capacitor made of two concentric cylindrical conductors of radii R and 3 R have to be in order to have the same capacitance as the spherical capacitor?

IT IS NOT 4.3R or 12R.

Part 2

What plate separation distance would be required for a parallel-plate capacitor made of two circular conductors each of radius R in order to have the same capacitance as the spherical capacitor?

It is not 1/4 R.

Explanation / Answer

Here the main formula we have to use:

Area of sphere = 4R2

Area of cylinder = 2R2 + 2Rl

For small radius R,

Area of sphere = 4R2…………….equation 1

Area of cylinder = 2R2 + 2Rl…………….equation 2

Compare equation 1 and equation 2,

4R2 = 2R2 + 2Rl

4R2 - 2R2 = 2Rl

2R2 = 2Rl

So, length of small cylinder is, l= 1 R

For large radius R

Area of sphere = 4(6R)2…………….equation 1

Area of cylinder = 2(3R)2 + 2(3R)l…………….equation 2

Compare equation 1 and equation 2,

4(6R)2 = 2(3R)2 + 2(R)l

4(6R)2 - 2(3R)2 = 2(3R)l

2((72R)2 – 9R2) = 2(3R)l

2(63R)2 = 2(3R)l

63 R2 = 3Rl

So, length of small cylinder is, l= 21 R

C = 4R…………….equation 3

And

Capacitance of parallel plate capacitor is,

C = A/d…………….equation 4

where A is the area, d is the separation between the plates.

Compare equation 3 and equation 4,

4R = A/d

4R = R2/d         (where A = R2)

We can solve this equation for d, we get,

separation between the plates is d = R/4

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