A capacitor consists of a conducting sphere of radius a surrounded by a concentric conducting shell of radius b. Show that its capacitance is C=[ab/k(b-a)].
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You begin by assuming there's some charge Q on the inner sphere and an equal but opposite Q on the inner surface of the outer sphere.
Use Gauss' Law to find the E-field in between the spheres.
This will turn out to be;
E = kQ/r^2
You then integrate this over "r" from a to b to get the potential difference between the spheres;
Vb - Va = - INTEGRAL[(kQ/r^2)dr]
If it comes out negative just use the absolute value. Then apply the definition;
C = IVb - VaI/Q
EDIT___________________________________…
When you do the integration "r" is evaluated at the limits r=a & r=b and no longer exists as "r".
The integration should give you something like;
kQ/b - kQ/a
Just write it as a single fraction with a common denominator and that should do the job.
Oh I just noticed I wrote the definition of capacitance upside down. Should be;
C = Q/IVb - VaI
Use Gauss' Law to find the E-field in between the spheres.
This will turn out to be;
E = kQ/r^2
You then integrate this over "r" from a to b to get the potential difference between the spheres;
Vb - Va = - INTEGRAL[(kQ/r^2)dr]
If it comes out negative just use the absolute value. Then apply the definition;
C = IVb - VaI/Q
EDIT___________________________________…
When you do the integration "r" is evaluated at the limits r=a & r=b and no longer exists as "r".
The integration should give you something like;
kQ/b - kQ/a
Just write it as a single fraction with a common denominator and that should do the job.
Oh I just noticed I wrote the definition of capacitance upside down. Should be;
C = Q/IVb - VaI