- #1

Flaming Physicist

- 6

- 2

- Homework Statement
- Two rings of same radius R and same charge Q, are arranged in parallel, both around the z axis, separated by a distance d. Determine the electric force between the rings.

- Relevant Equations
- $$\vec{F} = \frac{Q}{4 \pi \epsilon_0} \int{\frac{dq}{r^2}\hat{r}} $$

The problem is symmetric around the z axis, thus the force must be in the z direction only.

I tried dividing both rings into differential elements, then integrating through the upper ring to get the z component of the total force on the upper ring due to a differential element of the lower ring, getting the following integral.

$$ dF_z = dQ \frac{\lambda R d}{4 \pi \epsilon_0} \int_{0}^{2 \pi} \frac{d\theta}{[d^2+2R^2 (1-\cos{\theta})]^{3/2}}. $$

Where the linear charge density of both rings is $$\lambda = \frac{Q}{2 \pi R}.$$

I am right until this point?

Also, I could not solve that integral. Any hints?

I tried dividing both rings into differential elements, then integrating through the upper ring to get the z component of the total force on the upper ring due to a differential element of the lower ring, getting the following integral.

$$ dF_z = dQ \frac{\lambda R d}{4 \pi \epsilon_0} \int_{0}^{2 \pi} \frac{d\theta}{[d^2+2R^2 (1-\cos{\theta})]^{3/2}}. $$

Where the linear charge density of both rings is $$\lambda = \frac{Q}{2 \pi R}.$$

I am right until this point?

Also, I could not solve that integral. Any hints?