Electric force between two equal parallel rings of charge

[Flaming Physicist]

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?

 

[kuruman]

I checked your integral and you have set it up correctly. You cannot find an analytic expression for it and neither can anybody because it is an elliptic integral. If you have numbers for the ring radius and separation, you can find a numerical answer for it using a high-powered calculational tool such as Mathematica.

 

[Flaming Physicist]

Thanks for the help!
No numbers were given unfortunatelly.

Can I conclude, by symmetry, that the total force on the upper ring is just

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

because the force due to each of the infinitesimal segments would be equal?

 

[etotheipi]

Thanks for the help!
No numbers were given unfortunatelly.

Can I conclude, by symmetry, that the total force on the upper ring is just

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

because the force due to each of the infinitesimal segments would be equal?

You can, yes; since all charge elements $dQ$ are identical by symmetry as you say, it's just a case of integrating from $0$ to $Q$.

 

[Flaming Physicist]

I think the problem is solved then.

Thank you both for the help!

 

[haruspex]

I checked your integral and you have set it up correctly. You cannot find an analytic expression for it and neither can anybody because it is an elliptic integral. If you have numbers for the ring radius and separation, you can find a numerical answer for it using a high-powered calculational tool such as Mathematica.

But the range of integration is given and endows the problem with symmetry. Sometimes this means there are cute ways to solve it in closed form. E.g. the integral of $e^{-x^2}$.

 

[Flaming Physicist]

Hi, Haruspex. Do we have such a situation that allow for the calculation of the integral in this problem in closed form?

Thanks for participating in the discussion!

 

[haruspex]

Hi, Haruspex. Do we have such a situation that allow for the calculation of the integral in this problem in closed form?

Thanks for participating in the discussion!

I am not aware of such, just pointing out that the absence of a closed form for the indefinite integral does not rule out one for some definite cases.
One trick that can help is differentiation with respect to an unknown "constant" that appears in the integrand. E.g. in the present case the distance between the rings. Longshot, though.

 

[Flaming Physicist]

Ok. Thank you for the clarification.

 

[kuruman]

One trick that can help is differentiation with respect to an unknown "constant" that appears in the integrand. E.g. in the present case the distance between the rings. Longshot, though.

Longshot indeed. I spent a couple of hours trying the scaling parameter $\beta =2R/d$ as the constant parameter. Much to my chagrin (but not surprise), differentiating w.r.t. $\beta$ under the integral sign yielded another elliptic integral.