Two 18.0 cm diameter charged rings face each other, 22.0 cm apart. The left ring is charged to + 40.0 nC and the right ring is charged to + 15.0 nC. What is the force on a + 0.50 nC charge placed at the mid point?
To find the force on a +0.50 nC charge placed at the midpoint between the two charged rings, we can make use of Coulomb's Law. Coulomb's Law states that the force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
First, we need to calculate the electric field created by each charged ring at the midpoint. The formula for the electric field due to a charged ring is E = k * Q / r^2, where k is the Coulomb constant (8.99 x 10^9 N m^2 / C^2), Q is the charge of the ring, and r is the distance from the center of the ring to the observation point.
Let's start with the left ring, which is charged to +40.0 nC. The diameter of the ring is 18.0 cm, so the radius is half that, which is 9.0 cm = 0.09 m. The distance from the midpoint to the left ring is 11.0 cm = 0.11 m. Plugging these values into the formula, we get:
E(left) = (8.99 x 10^9 N m^2 / C^2) * (40.0 x 10^-9 C) / (0.11 m)^2
Next, let's calculate the electric field due to the right ring, which is charged to +15.0 nC. The distance from the midpoint to the right ring is the same as the distance to the left ring, which is 0.11 m. Plugging in the values, we have:
E(right) = (8.99 x 10^9 N m^2 / C^2) * (15.0 x 10^-9 C) / (0.11 m)^2
Now, we sum up the electric fields due to both rings, taking into account their directions. Since the rings have the same charge, they repel each other, so the electric fields due to each ring point away from them. Therefore, we subtract the electric field due to the right ring from the electric field due to the left ring:
E(total) = E(left) - E(right)
After finding the electric field at the midpoint, we can calculate the force on the +0.50 nC charge using the formula F = q * E, where q is the charge and E is the electric field. Plugging in the values, we get:
F = (0.50 x 10^-9 C) * E(total)
Solving this equation will give us the force on the +0.50 nC charge at the midpoint between the two charged rings.