Question

1. a conducting sphere of radius r initially has charge q, and a conducting sphere of radius 2r has no charge. find the charges on the spheres after they are connected. hint: use the conservation of charge and that the potentials of the spheres are equal after connecting.

Explanation:

Step1: Apply charge - conservation principle

Let the charge on the sphere of radius $R$ after connection be $q_1$ and the charge on the sphere of radius $2R$ be $q_2$. According to the conservation of charge, $q_1 + q_2=Q$ since the total initial charge is $Q$ and no charge is lost during the connection.

Step2: Use the fact that potentials are equal

The electric potential of a conducting sphere of radius $r$ with charge $q$ is given by $V = \frac{kq}{r}$ (for points on or outside the sphere). After connection, the potentials of the two spheres are equal, so $\frac{kq_1}{R}=\frac{kq_2}{2R}$. Cross - multiplying gives $2q_1=q_2$.

Step3: Solve the system of equations

Substitute $q_2 = 2q_1$ into $q_1 + q_2=Q$. We get $q_1+2q_1=Q$, which simplifies to $3q_1 = Q$. So, $q_1=\frac{Q}{3}$.

Step4: Find $q_2$

Since $q_2 = 2q_1$, substituting $q_1=\frac{Q}{3}$ gives $q_2=\frac{2Q}{3}$.

Answer:

The charge on the sphere of radius $R$ is $\frac{Q}{3}$ and the charge on the sphere of radius $2R$ is $\frac{2Q}{3}$.