To find the potential of the conducting sphere due to the charged ring, we can follow these steps:
### Step 1: Understand the Configuration
We have a nonuniformly charged ring with a total charge \( Q = 1 \, \mu C = 1 \times 10^{-6} \, C \) and a conducting solid sphere. The distance between the centers of the ring and the sphere is \( d = 3 \, m \), and the radius of the ring is \( R = 4 \, m \).
### Step 2: Determine the Distance from the Ring to the Sphere
The potential \( V \) at a point due to a ring of charge is calculated at a distance \( z \) from the center of the ring along the axis perpendicular to the plane of the ring. In this case, the distance from the center of the ring to the center of the sphere is \( d = 3 \, m \).
### Step 3: Calculate the Potential at the Center of the Sphere
The formula for the potential \( V \) at a distance \( z \) from the center of a charged ring is given by:
\[
V = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{\sqrt{R^2 + z^2}}
\]
Where:
- \( Q \) is the total charge on the ring,
- \( R \) is the radius of the ring,
- \( z \) is the distance from the center of the ring to the point where we are calculating the potential,
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, C^2/(N \cdot m^2) \).
In our case:
- \( R = 4 \, m \)
- \( z = 3 \, m \)
### Step 4: Substitute Values into the Formula
Substituting the values into the potential formula:
\[
V = \frac{1}{4\pi (8.85 \times 10^{-12})} \cdot \frac{1 \times 10^{-6}}{\sqrt{4^2 + 3^2}}
\]
Calculating \( \sqrt{4^2 + 3^2} \):
\[
\sqrt{16 + 9} = \sqrt{25} = 5
\]
Now substituting this back into the potential formula:
\[
V = \frac{1}{4\pi (8.85 \times 10^{-12})} \cdot \frac{1 \times 10^{-6}}{5}
\]
### Step 5: Calculate the Numerical Value
Calculating \( \frac{1}{4\pi (8.85 \times 10^{-12})} \):
\[
\frac{1}{4\pi (8.85 \times 10^{-12})} \approx 9 \times 10^9 \, N \cdot m^2/C^2
\]
Now substituting back into the equation:
\[
V \approx 9 \times 10^9 \cdot \frac{1 \times 10^{-6}}{5}
\]
Calculating this gives:
\[
V \approx 9 \times 10^9 \cdot 2 \times 10^{-7} = 1.8 \times 10^3 \, V = 1800 \, V
\]
### Final Answer
The potential of the conducting sphere is approximately \( 1800 \, V \).
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