A pair of thin concentric conducting spherical shells, of radii a and b , are ar

Question

A pair of thin concentric conducting spherical shells, of radii a and b, are arranged as shown. The inner shell carries charge +q while the total charge on the outer shell is +Q. Give answers in terms of these quantities.

a. Find the magnitude of the E-field E(r) where r I, r > b ; II, a < r < b ; III, r < a .

Ans: I:E=k(Q+q)/r2; II:E=kq/r2; III:E=0.

b. Write the formula for the potential in those three regions, taking V(?) = 0 .

Ans: I: V = k(Q + q)/r; II: V = kQ/b + kq/r; III: V = kQ/b + kq/a .

c. Suppose small holes are drilled through the spheres along the dotted line in the figure, and an electron (charge –e) is released from rest in the hole at the top of the outer shell (distance r = b from the center). What is the kinetic energy of the electron when it is at the center of the spheres?

Ans:K(r=0)=keq?1?1?.??a b??

A pair of thin concentric conducting spherical shells, of radii a and b, are arranged as shown. The inner shell carries charge +q while the total charge on the outer shell is +Q. Give answers in terms of these quantities. Find the magnitude of the E-field E(r) where r is the distance from the center, in three regions: b. Write the formula for the potential in those three regions, taking V(0 Suppose small holes are drilled through the spheres along the dotted line in the figure, and an electron (charge -e) is released from rest in the hole at the top of the outer shell (distance rb from the center). What is the kinetic energy of the electron when it is at the center of the spheres? Ans: K(r 0)-keq

Explanation / Answer

a] i) By Gauss law, taking concentric spherical gaussian surface with r > b,

E*4pi r^2 = Qenclosed/e0

E = (Q+q)/4pi r^2 e0 = k(Q+q)/r^2

ii) similaraly for b>r>a, taking concentric spherical gaussian surface with b>r>a,

E = k Qenclosed/r^2 = kq/r^2

iii) similaraly for r<a, taking concentric spherical gaussian surface with r<a,

E = k Qenclosed/r^2 = 0

b] i) Potential for r > b,

V = potential due to both spherical surfaces

= kq/r + kQ/r = k(q+Q)/r

ii) similaraly for b>r>a,

V = potential due to both spherical surfaces

= kq/r + kQ/b by applying formula for potential inside hollow sphere.

iii)similaraly for r<a,

V = potential due to both spherical surfaces

= kq/a + kQ/b by applying formula for potential inside hollow sphere.

c] KE = decrease in PE

= -e * decrease in potential

= -e* [ k(q+Q)/b - [kq/a + kQ/b]]

= -[ ke(q+Q)/b - [keq/a + keQ/b]]

= -keq/b + keq/a

= keq*(1/a - 1/b)

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