A charge `q=10muC` is distributed uniformly over the circumference of a ring of radius 3m placed on x-y placed with its centre art origin. Find the electric potential at a point `P(0,0,4m)`

Explanation

To find the electric potential at point P(0,0,4m) due to a uniformly charged ring, we can follow these steps:

### Step 1: Understand the configuration
The charge \( q = 10 \, \mu C \) is uniformly distributed over the circumference of a ring of radius \( R = 3 \, m \) centered at the origin in the x-y plane. The point P is located at (0, 0, 4m).

### Step 2: Calculate the distance from the ring to point P
To find the electric potential at point P, we need to determine the distance \( r \) from any point on the ring to point P. The distance can be calculated using the Pythagorean theorem.

The distance from the center of the ring (which is at the origin) to point P is 4 m (along the z-axis). The radius of the ring is 3 m (in the x-y plane).

Using the Pythagorean theorem:
\[ r = \sqrt{(3^2) + (4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \, m \]

### Step 3: Write the formula for electric potential
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = k \frac{q}{r} \]
where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, N \cdot m^2/C^2 \).

### Step 4: Calculate the total electric potential at point P
Since the charge is uniformly distributed over the ring, the potential at point P due to the entire charge can be calculated as:
\[ V = k \frac{Q}{r} \]
where \( Q = 10 \, \mu C = 10 \times 10^{-6} \, C \) and \( r = 5 \, m \).

Substituting the values:
\[ V = \left(8.99 \times 10^9 \, N \cdot m^2/C^2\right) \frac{10 \times 10^{-6} \, C}{5 \, m} \]

Calculating this gives:
\[ V = \left(8.99 \times 10^9\right) \times \left(2 \times 10^{-6}\right) = 17.98 \times 10^3 \, V \]
or
\[ V \approx 1.798 \times 10^4 \, V \]

### Final Answer
The electric potential at point P(0,0,4m) is approximately \( 1.8 \times 10^4 \, V \).

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