A thin ring of radius equal to 25 cm carries a uniformly distributed charge of 4.7 nC. What is the electric field and potential difference at the center of the ring?
To find the electric field and potential difference at the center of the ring, we can use the formulas for the electric field and potential due to a charged ring.
1. Electric Field:
The electric field at the center of a uniformly charged ring is given by the formula:
E = (k * Q * z) / (2 * π * R^3),
where E is the electric field, k is Coulomb's constant (8.99 × 10^9 Nm^2/C^2), Q is the total charge on the ring, z is the distance from the center of the ring, and R is the radius of the ring.
Using the given values:
Q = 4.7 nC (convert to coulombs by dividing by 10^9) = 4.7 * 10^-9 C
R = 25 cm (convert to meters by multiplying by 0.01) = 0.25 m
z = 0 (as the center of the ring is at z = 0)
Substituting the values into the electric field formula:
E = (8.99 × 10^9 Nm^2/C^2 * 4.7 * 10^-9 C * 0) / (2 * π * (0.25 m)^3),
E = 0 N/C.
Therefore, the electric field at the center of the ring is zero.
2. Potential Difference:
The potential difference at the center of a charged ring can be calculated using the formula:
V = (k * Q) / R,
where V is the potential difference, k is Coulomb's constant, Q is the total charge on the ring, and R is the radius of the ring.
Using the given values:
Q = 4.7 nC (convert to coulombs by dividing by 10^9) = 4.7 * 10^-9 C,
R = 25 cm (convert to meters by multiplying by 0.01) = 0.25 m.
Substituting the values into the potential difference formula:
V = (8.99 × 10^9 Nm^2/C^2 * 4.7 * 10^-9 C) / 0.25 m,
V ≈ 170.67 V.
Therefore, the potential difference at the center of the ring is approximately 170.67 volts.
To find the electric field at the center of the ring, we can use the equation for the electric field due to a charged ring:
E = (k * Q * z) / (2 * π * R^2)
where E is the electric field, k is Coulomb's constant (8.99 × 10^9 N m^2/C^2), Q is the charge, R is the radius of the ring, and z is the distance from the center of the ring to the point where the electric field is being measured.
In this case, the charge Q is 4.7 nC (nanocoulombs) and the radius R is 25 cm. Since we are calculating the electric field at the center of the ring, the distance z is zero.
Plugging in the values, we get:
E = (8.99 × 10^9 N m^2/C^2 * 4.7 × 10^-9 C * 0) / (2 * π * (0.25 m)^2)
Simplifying the equation, we find:
E = 0 N/C
So, the electric field at the center of the ring is zero.
To find the potential difference at the center of the ring, we can use the equation for the potential due to a charged ring:
V = (k * Q) / R
where V is the potential difference.
Plugging in the values, we get:
V = (8.99 × 10^9 N m^2/C^2 * 4.7 × 10^-9 C) / (0.25 m)
Simplifying the equation, we find:
V = 170.36 V
So, the potential difference at the center of the ring is 170.36 volts.