PLEASE SHOW ALL WORK. THANK YOU [1] Consider a spherical capacitor consisting of

Question

PLEASE SHOW ALL WORK. THANK YOU

[1] Consider a spherical capacitor consisting of two concentric spherical conducting shells. The inner shell of radius a is uniformly charged with total charge -qwhile the outer shell of radius b carries an equal amount of charge of the opposite sign, +q. Assume a coordinate system with the origin coincident with the center of the shells. Hint: See Example 24-3 in the textbook. (A) Argue why the electric field is zero for radius rb. Hint: utilize appropriate Gaussian surface to make your arguments. (B) E 0 for a

Explanation / Answer

A) Imagine a gaussian spehere with radius r < a.

charge enclosed by the gaussian sphere, Qin = 0

now using Gauss' law,

integral E.dS = Qin/epsilon

==> E = 0

so, at r < a, E = 0

Now again, Imagine a gaussian spehere with radius r > b.

charge enclosed by the gaussian sphere, Qin = q - q

= 0

now using Gauss' law,

integral E.dS = Qin/epsilon

==> E = 0

so, at r > b, E = 0

B) at a < r < b.

Now again, Imagine a gaussian spehere with radius r .

charge enclosed by the gaussian sphere, Qin = q

now using Gauss' law,

integral E.dS = Qin/epsilon

==> E*4*pi*r^2 = q/epsilon

E = q/(4*pi*epsilon*r^2)

Vba = -integral E.dr

= -integral q/(4*pi*epsilon*r^2)*dr ( from r = a to r = b)

= -q/(4*pi*epsilon) integral (1/r^2)*dr ( from r = a to r = b)

= -q/(4*pi*epsilon) (-1/r)

= (q/(4*pi*epsilon))*(1/a - 1/b)

= q*(1/a - 1/b)/(4*pi*epsilon)

C) C = q/Vba

= q/(q*(1/a - 1/b)/(4*pi*epsilon))

= 4*pi*epsilon/(1/a - 1/b)

D) Energy stored, U = q^2/(2*C)

= q^2/(2*4*pi*epsilon/(1/a - 1/b) )

= (q^2/(8*pi*epsilon))*(1/a - 1/b)

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