A spherical conductor of radius a is at the center of a cavity of radius b withi

Question

A spherical conductor of radius a is at the center of a cavity of radius b within a conducting spherical shell, of outer radius c>b>a. Charge Q is placed on the inner conductor.
a) Assume that the outer conductor is grounded and find i) the charge on its inner surface -- that is, on the wall of the cavity, ii) the charge on its outer surface, iii) the potential function (r) between the inner conductor and the cavity wall (a<r<b) iv) Graph the function (r) for 0r.

b) Now assume that the outer conductor is uncharged, and repeat part a) i, ii, and iii. Find also the potential (r) (iv) within the shell -- that is, between the cavity wall and the outer surface of the outer conductor (b<r<c), and (v) entirely outside the outer conductor (c<r). (vi) Graph the function (r) for 0r.

Explanation / Answer

We know from gauss law that flux through a closed surface is=Q/ where Q is charge enclosed by the surface

we also know that potential anywhere inside a conductor is same ----1 this indicates

that there is no electric field present inside any conductor (else there would be a potential gradient which is against to 1)----2

1)now in this particular case lets consider an imaginary gaussian surface between inner and outer walls of the innermost hollow sphere thereis no electric field inside(---2) hence there should be no flux through the surface therefore the charge enclosed by the surface is zero so no charge present inside the innermost hollow sphere

2)now by conservation of charge principle the outer surface of a should have charge Q

3)The inner surface b has a charge of -Q because of a similar xplanation as above (considering a gaussian surface between inner and outer walls of sphere since elctric field there should be zero [it is a conductor right?])

outer wall of c has Q has no charge at all.(note Gauss law still holds total chage enclosed = Q+-Q+Q+-Q=0 )

conservation of charge doenot hold because it is grounded

the potential function is given by (r)=[Q/(4r)]-[Q/(4b)] for a<r<b (summation of potentials of charges on a and b ) and (r)=0 for r>b because no net charge or elecctric field

simlarly when it is not grounded Conservation of charge holds for outer conductor

(r)=[Q/(4r)]-[Q/(4b)] for a<r<b

(r)=0 for b<r<c

(r)=[Q/(4r)] for r>c

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