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:

Question

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:

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