Two rings of radius 3 cm are 23 cm apart and concentric with a common horizontal x-axis. What is the magnitude of the electric field midway between the rings if both rings carry a charge of +35 nC?
To find the magnitude of the electric field midway between the rings, we can use the principle of superposition. The electric field due to each ring individually can be determined using the equation for the electric field produced by a uniformly charged ring.
The electric field produced by a uniformly charged ring along its axis at a point on the axis is given by:
E = (k * Q * z) / (2π * (R^2 + z^2)^(3/2))
Where:
• E is the electric field
• k is the electrostatic constant (9 * 10^9 N m^2/C^2)
• Q is the charge on the ring
• z is the distance between the center of the ring and the point on the axis
• R is the radius of the ring
For our scenario, both rings have the same radius (R = 3 cm = 0.03 m) and charge (Q = +35 nC = +35 * 10^(-9) C). The rings are concentric, so the distance between their centers is given as 23 cm = 0.23 m.
To find the electric field at the midpoint between the rings, we need to calculate the electric field produced by each ring separately at that point, and then add them together.
Let's calculate the electric field produced by each ring at the midpoint:
1. Electric field produced by the first ring:
For the first ring, the distance from the center of the ring to the midpoint is half the distance between the rings, which is 0.23/2 = 0.115 m.
Using the formula, we can calculate the electric field produced by the first ring at the midpoint:
E1 = (k * Q * z) / (2π * (R^2 + z^2)^(3/2))
E1 = (9 * 10^9 N m^2/C^2) * (35 * 10^(-9) C) * (0.115 m) / (2π * (0.03^2 + 0.115^2)^(3/2))
2. Electric field produced by the second ring:
For the second ring, the distance from the center of the ring to the midpoint is also 0.115 m.
Using the formula, we can calculate the electric field produced by the second ring at the midpoint:
E2 = (k * Q * z) / (2π * (R^2 + z^2)^(3/2))
E2 = (9 * 10^9 N m^2/C^2) * (35 * 10^(-9) C) * (0.115 m) / (2π * (0.03^2 + 0.115^2)^(3/2))
Now, we can find the total electric field at the midpoint by summing up the electric fields from both rings:
E_total = E1 + E2
Substituting the values and calculating E_total will give us the magnitude of the electric field at the midpoint between the rings.