Question

The potential at the center of a uniformly charged ring is 44 kV , and 11 cm along the ring axis the potential is 31 kV .

Find the ring's radius.

Find the ring's total charge.

Answer 1

Given that,

Potential at the center of the ring = V(c) = 44 kV = 44000 V

and at d = 11 cm = 0.11 m along the axis V(d) = 31 kV = 31000 V

We need to find the radius and total charge of the ring. Let it be R and Q respectively.

We know that the potential at the center of the uniformly charged ring is given by:

V (c) = k Q / R = 44000 V

kQ = 44000 R (1)

Now the potential at a distance d is given by:

V(d) = k Q / sqrt ( d² + R²) = 31000

Putting the value of kQ from eqn (1) we get

44000 R / sqrt [ (0.11)² + R²] = 31000

44 R = 31 x sqrt  [ (0.11)² + R²]

squaring both the sides we get:

1936 R² = 961 x ( 0.0121 + R²)

1936  R² = 11.63+ 961  R²

(1936 - 961) R² = 11.63

975  R² = 11.63 =>  R² = 11.63/975

R = 0.11 meters

Hence, radius = 0.11 meters

Again, V(c) = k Q / R = 44000

Q = 44000 x R / K = 44000 x 0.11 / 8.99 x 10⁹ = 538.4 x 10^-9 C = 538.4 nC

Hence, charge on the ring = Q = 538.4 x 10^-9 C = 538.4 nC