Question

Question 1.28: (a) A conductor A with a cavity as shown in Fig. 1.36(a) is given a charge Q. Show that the entire charge must appear on the outer surface of the conductor. (b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A is Q + q [Fig. 1.36(b)]. (c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.

Class 12 - Physics - Electric Charges And Fields Electric Charges And Fields Page-49

Answer 1

(a) Let us consider a closed surfacr inside the conductor enclosing the cavity. As electric field inside the conductor is zero.
so, $\phi=\oint E.dS = 0$
from Gaussian theorem, $\phi=\oint E.dS = \frac{q_{enclosed}}{\epsilon_0}=0$
so, $q_{enclosed}$=0

here you see charge enclosed inside the conducting surface becomes zero. this shows that there cannot be any charge on the inner surfcrnof the conductor at the cavity, so if we given any charge Q to the conductor , it will appear only on its outer surface.

(b) Let's consider that charge $-q_1$ is induced on the inner surface of conductor A at cavity , when conductor B of charge q is kept in its cavity . due to this charge of (Q + $q_1$) is induced on the outer surface of conductor.

Lets take a closed Gaussian surface S inside the conductor A enclosing its cavity . As electric field inside conductor is zero.
so, $\phi=\oint E.dS = 0$
But according to Gaussian theorem,
$\phi=\frac{q_{enclosed}}{\epsilon_0}$
so, 0 = $\phi=\frac{q-q_1}{\epsilon_0}$
$q-q_1 = 0$ $\therefore,q_1=q$

e.g., equal and opposite nature charge -q is induced on inner surface of conductor A at the cavity , when conductor B of charge +q is kept in its cavity.
so, charge on outer surface of conductor is (Q + $q_1$) = (Q + q)

(c) we found that electric field inside the cavity of conductor is zero, even on charging of conductor , so true sensitive instrument can be shielded from the strong electrostatic field in its environment, by covering it with a metallic cover.

Answer 2

(a) Let us consider a Gaussian surface that is lying whollywithin a conductor and enclosing the cavity. The electric fieldintensity E inside the charged conductor is zero.

Letq is the charge inside the conductor and isthe permittivity of free space.

Accordingto Gauss’s law,

Flux,

Here,E = 0

Therefore,charge inside the conductor is zero.

Theentire charge Q appears on the outer surface of the conductor.

(b) The outer surface of conductor A has a charge of amount Q.Another conductor B having charge +q is kept inside conductorA and it is insulated from A. Hence, a charge of amount −qwill be induced in the inner surface of conductor A and +qis induced on the outer surface of conductor A. Therefore, totalcharge on the outer surface of conductor A is Q + q.

(c) A sensitive instrument can be shielded from the strongelectrostatic field in its environment by enclosing it fully inside ametallic surface. A closed metallic body acts as an electrostaticshield.