Electric force between two equal parallel rings of charge

In summary: 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.
  • #1
Flaming Physicist
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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?
 

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  • #2
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.
 
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  • #3
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?
 
  • #4
Flaming Physicist said:
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##.
 
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  • #5
I think the problem is solved then.

Thank you both for the help!
 
  • #6
kuruman said:
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}##.
 
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  • #7
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!
 
  • #8
Flaming Physicist said:
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.
 
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  • #9
Ok. Thank you for the clarification.
 
  • #10
haruspex said:
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.
 
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Related to Electric force between two equal parallel rings of charge

1. What is the formula for calculating the electric force between two equal parallel rings of charge?

The formula for calculating the electric force between two equal parallel rings of charge is F = (kQ1Q2) / d2, where F is the force in Newtons, k is the Coulomb constant (9 x 109 Nm2/C2), Q1 and Q2 are the charges on the two rings in Coulombs, and d is the distance between the two rings in meters.

2. How does the distance between the two rings affect the electric force?

The electric force between two equal parallel rings of charge is inversely proportional to the square of the distance between them. This means that as the distance between the two rings increases, the force between them decreases.

3. What happens to the electric force if the charges on the two rings are doubled?

If the charges on the two rings are doubled, the electric force between them will also double. This is because the force is directly proportional to the product of the two charges.

4. Can the electric force between two equal parallel rings of charge be repulsive?

Yes, the electric force between two equal parallel rings of charge can be repulsive if the charges on the two rings have the same sign. This means that both rings have either positive or negative charges, and they will repel each other.

5. How does the electric force between two equal parallel rings of charge compare to the electric force between two point charges?

The electric force between two equal parallel rings of charge is similar to the electric force between two point charges, except that the rings have a finite size and the charges are spread out along the circumference of the rings. This means that the force between two rings will be slightly less than the force between two point charges with the same charge and distance between them.

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