Answer to Question #126010 in Electric Circuits for Meet Darji

Question

A spherical conducting shell of inner radius a and outer radius b carries a total charge +Q
distributed on the surface of a conducting shell as in figure. The quantity Q is taken to be positive.
(a) Find the electric field in the interior of the conducting shell, for r <a, and (b) the electric field
outside the shell, for r >b. (c) If an additional charge of -2Q is placed at the center, find the electric
field for r . b.
(d) What is the distribution of charge on the sphere in part (c)?

Answer 1

Explanations & Calculations

• Refer to the sketch.
• All the charges are concentrated on the outer surface of the shell when conducting shells are considered.
• To find the electric field intensity, apply Gauss''s theorem followed by $\small E = \frac{Q}{A\epsilon_0}$ to a selected region.

a) Circular region ; r<a (figure 1)

• $\small E_1 = 0$ ; Since no charges are found.

b) Circular region ; r>b (figure 1)

• $E_2 = \frac{Q}{(4\pi r^2)\epsilon_0}$ ; outwards

c) Before dealing with the sums, mark the induced charges both inside & outside. Outside surface receives a total charge of Q+(-2Q) = (-Q) (figure 2)

(r < a)

• $E_3 = \frac{-2Q}{(4\pi r^2)\epsilon_0}$ ; a reversed field intensity

(r > b)

• $E_4 = \frac{-Q}{(4\pi r^2) \epsilon_0}$ ; same magnitude as $E_2$ (when the same location is considered) but reversed.

d)

• After the introduction of the -2Q charge, the inner surface of the shell receives an induced charge of +2Q .
• Due to the induction, outer surface get induced a negative charge of 2Q & since it had a +Q already results a net charge of -Q.