Question

The figure shows an inner metallic sphere (Conductor I) centered inside a spherical cavity located inside another metallic sphere (Conductor II). The radii a, band care defined in the figure. Charges Q, and Qu have been placed on Conductors I and II respectively, so that the total charge is Qtot = Q1 + Qui- (a) Find the surface charge densities oa, Ob, and oc (see figure). Explain why your solution works. (b) What is the field outside conductor II? (c) What is the field in the region between conductors I and II? (d) What is the potential difference V, - V., between the conductors? (e) Which of these answers would change, if at all, if an additional point charge q is brought near the outside of conductor II?

Answer 1

(a) The surface charge densities

σα - Απα?
   ;as  $Q_{I}$ charge is given to the surface of conductor I.

ਫt -= 40
;as$-Q_{I}$  charge is induced in the inner surface of conductor I.

Oc= Q1 + Qu 41c2
; as
Q1 + Q11
charge appears on outer surface of surface of conductor II

The above charge densities are calculated on the basis of

surface charge density = charge /surface area

(b) The field outside the conductor II =
k(Q1 + Q11)
;where r is the distance from the center of conductor II()

(c) The field in the region between the conductor I and conductor II = $\frac{kQ_{I}}{r^{2}}$ ; where r is the distance from the center of conductor I().

a <r <b

(d) as $E = \frac{-dV}{dr}$

or,

dV = -Edr

or, $\oint_{V_{I}}}^{V_{II}} dV =-\oint_{a}^{r}E dr$

or, $\oint_{V_{I}}^{V_{II}}dV = -\oint_{a}^{r}\frac{kQ_{I}}{r^{2}}dr$

or,   $\left [V \right ]_{V_{I}}^{VII} = -kQ_{I}\left [ -\frac{1}{r} \right ]_{a}^{r}$

)04 = 111 – 11

(e) as there is some extra induced charge on outer surface of conductor II,when an additional charge is brought near the outside of conductor II.$\sigma _{c}$ and field outside the conductor II are changed.