23.3 a charged isolated conductor positive charge ( q ) is placed on a conducting spherical shell with inner radius ( r_1 ) and outer radius ( r_2 ). a particle with charge ( q ) is placed at the center of the cavity. the net charge on the outer surface of the conducting shell is
( -q ).
( q - q ).
( q + q ).
zero.
( q ).
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# Brief Explanations:
1. Recall the property of a conducting shell: In a static situation, the electric field inside a conductor (the material of the shell, between $R_1$ and $R_2$) is zero. By Gauss's law, the net charge enclosed by any Gaussian surface within the conductor must be zero.
2. The charge $q$ at the center induces a charge $-q$ on the inner surface of the shell (to make the net charge enclosed by a Gaussian surface in the conductor zero).
3. The conducting shell initially has a charge $Q$. The total charge on the shell (inner + outer surface) must be $Q$. Let the charge on the outer surface be $Q_{\text{outer}}$. We know that charge on inner surface is $-q$, so $ -q + Q_{\text{outer}}=Q$. Solving for $Q_{\text{outer}}$, we get $Q_{\text{outer}} = Q + q$? Wait, no, wait. Wait, the initial charge on the shell is $Q$? Wait, no, the problem says "Positive charge $Q$ is placed on a conducting spherical shell" – so the shell has charge $Q$ initially. Then a charge $q$ is placed at the center. The inner surface will have $-q$ (by induction, to cancel the field inside the conductor). Then the outer surface charge is $Q + q$? Wait, no, wait: the total charge on the shell is $Q$ (initial). So inner surface charge ($-q$) plus outer surface charge ($Q_{\text{outer}}$) should equal $Q$. So $-q + Q_{\text{outer}} = Q$ → $Q_{\text{outer}} = Q + q$? Wait, no, maybe I misread. Wait, the problem says "Positive charge $Q$ is placed on a conducting spherical shell" – so the shell's total charge is $Q$. Then a charge $q$ is at the center. The inner surface gets $-q$ (because the electric field inside the conductor must be zero, so the Gaussian surface inside the conductor encloses $q + \text{inner surface charge} = 0$ → inner surface charge is $-q$). Then the outer surface charge is total shell charge ($Q$) minus inner surface charge ($-q$)? Wait, no: total charge on shell is inner + outer. So $Q = (\text{inner charge}) + (\text{outer charge})$. Inner charge is $-q$, so outer charge is $Q - (-q)= Q + q$? Wait, but maybe the shell was initially uncharged? Wait, no, the problem says "Positive charge $Q$ is placed on a conducting spherical shell" – so the shell has charge $Q$ initially. Wait, maybe I made a mistake. Wait, let's re-express:

- The conductor is isolated, so total charge on the conductor (shell) is $Q$ (given: "Positive charge $Q$ is placed on a conducting spherical shell").
- A charge $q$ is placed at the center of the cavity.
- By induction, the inner surface (radius $R_1$) will have a charge $-q$ (because in electrostatic equilibrium, the electric field inside the conductor (between $R_1$ and $R_2$) is zero. Using Gauss's law, a Gaussian surface inside the conductor (radius between $R_1$ and $R_2$) encloses $q + \text{inner surface charge} = 0$ → inner surface charge $= -q$.
- The total charge on the shell is the sum of the charge on the inner surface and the outer surface: $Q = (\text{inner charge}) + (\text{outer charge})$.
- Substituting inner charge $= -q$ into the equation: $Q = -q + (\text{outer charge})$ → $\text{outer charge} = Q + q$? Wait, but the options include $Q + q$ as one of the choices (the option " $Q + q$"). Wait, but let's check again. Wait, maybe the shell was initially uncharged? Wait, the problem says "Positive charge $Q$ is placed on a conducting spherical shell" – so the shell has charge $Q$ initially. Then adding a charge $q$ at the center. The inner surface gets $-q$, so the outer surface must have $Q + q$ to keep the total charge on the shell as $Q$? Wait, no: total charge on shell is inner + outer. If inner is $-q$, outer is $Q + q$, then total is $Q$, which matches. So the net charge on the outer surface is $Q + q$. Wait, but let's confirm with the options. The options are: $-Q$, $Q - q$, $Q + q$, Zero, $Q$. So the correct option should be $Q + q$? Wait, no, wait, maybe I messed up the initial charge. Wait, maybe the shell is initially uncharged, and $Q$ is the charge placed on the shell? No, the problem says "Positive charge $Q$ is placed on a conducting spherical shell" – so the shell has charge $Q$. Then a charge $q$ is at the center. So inner surface: $-q$, outer surface: $Q + q$ (since total on shell is $Q = -q + \text{outer}$ → outer = $Q + q$). So the correct option is the one with $Q + q$.
# Answer:
$Q + q$ (the option labeled " $Q + q$")

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