To solve the problem, we need to determine the final charges on two spherical shells after they are connected by a conducting wire. Let's break down the solution step by step.
### Step 1: Understand the Problem
We have two spherical shells:
- Shell 1 (radius \( R_1 = 10 \, \text{cm} \)) with charge \( Q_1 = +0.5 \, \mu\text{C} \)
- Shell 2 (radius \( R_2 = 20 \, \text{cm} \)) with charge \( Q_2 = +0.7 \, \mu\text{C} \)
When connected by a conducting wire, the total charge will redistribute between the two shells.
### Step 2: Calculate Total Charge
The total charge \( Q_{\text{total}} \) is the sum of the charges on both shells:
\[
Q_{\text{total}} = Q_1 + Q_2 = 0.5 \, \mu\text{C} + 0.7 \, \mu\text{C} = 1.2 \, \mu\text{C}
\]
### Step 3: Use the Concept of Equal Potential
When the two shells are connected by a wire, they will reach the same electric potential \( V \). The potential \( V \) of a spherical shell is given by:
\[
V = \frac{kQ}{R}
\]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the radius.
### Step 4: Set Up the Equation for Equal Potentials
Since the potentials are equal:
\[
V_1 = V_2 \implies \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2}
\]
Cancelling \( k \) from both sides gives:
\[
\frac{Q_1}{R_1} = \frac{Q_2}{R_2}
\]
Substituting the values of \( R_1 \) and \( R_2 \):
\[
\frac{Q_1}{10} = \frac{Q_2}{20}
\]
### Step 5: Express One Charge in Terms of the Other
Rearranging gives:
\[
Q_2 = 2Q_1
\]
### Step 6: Substitute into Total Charge Equation
We know from Step 2 that:
\[
Q_1 + Q_2 = 1.2 \, \mu\text{C}
\]
Substituting \( Q_2 \) from Step 5:
\[
Q_1 + 2Q_1 = 1.2 \, \mu\text{C}
\]
This simplifies to:
\[
3Q_1 = 1.2 \, \mu\text{C}
\]
So,
\[
Q_1 = \frac{1.2 \, \mu\text{C}}{3} = 0.4 \, \mu\text{C}
\]
### Step 7: Find \( Q_2 \)
Using \( Q_2 = 2Q_1 \):
\[
Q_2 = 2 \times 0.4 \, \mu\text{C} = 0.8 \, \mu\text{C}
\]
### Final Answer
The final charges on the shells after they are connected by a wire are:
- Charge on shell 1: \( Q_1 = 0.4 \, \mu\text{C} \)
- Charge on shell 2: \( Q_2 = 0.8 \, \mu\text{C} \)
