Question

Consider two thin, concentric spherical conductors of radii R1 and R2 where R2 < R1. The conductors are both initially uncharged. If a charge +Q is placed on the outer conductor (of radius R1) the potential on the inner conductor is:
Qu4-Pic3-UP2-2020
2*k_e(Q)/(R_2)
0
k_e(Q)/(R_1 + R_2)
k_e(Q)/(R_2)
k_e(Q)/(R_1)

Answer 1

To solve this problem, let’s analyze the situation step by step. We have two thin, concentric spherical conductors with radii \( R_1 \) (outer sphere) and \( R_2 \) (inner sphere). Initially, both spheres are uncharged, and then a charge \( +Q \) is placed on the outer conductor.

### 1. Distribution of charge
When a charge is placed on the outer conductor, it induces a charge on the inner conductor. Since the inner conductor is also a conductor, it will redistribute charges such that the electric field within the conducting material remains zero. Thus, the inner conductor will have a charge of \( -Q \) while the outer conductor retains a charge of \( +Q \).

### 2. Electric field and potential
The important point about conductors is that the potential inside a conductor is constant and equal to the potential on its surface. The potential at a distance from a charged conductor can be calculated using the formula for the electric potential due to a point charge (or a uniformly charged sphere) which is:

\[
V = k_e \frac{Q}{r}
\]

where \( k_e \) is Coulomb's constant, \( Q \) is the total charge, and \( r \) is the distance from the center of the charge.

For the inner conductor with radius \( R_2 \):

1. The charge on the outer conductor is \( +Q \).
2. We want to calculate the potential at the surface of the inner conductor (at \( R_2 \)) due to the charge \( +Q \) on the outer conductor.

### 3. Calculating the potential on the inner conductor
The potential \( V \) at the surface of the inner conductor (radius \( R_2 \)) is given by:

\[
V = k_e \frac{Q}{R_2}
\]

This is because from the outer conductor, the potential at any point inside it (and thus at \( R_2 \)) is governed solely by the charge on the outer conductor due to symmetry. The inner charge does not contribute to the potential at its own surface.

### Conclusion
The potential on the inner conductor is given by the equation:

\[
\frac{k_e Q}{R_2}
\]

Thus, the potential on the inner conductor when a charge \( +Q \) is placed on the outer conductor is:

\[
\boxed{\frac{k_e Q}{R_2}}
\]