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I had this as one of my homework problems - "A dipole of dipole moment P is kept at a center of a ring of radius R and charge Q (uniformly distributed). It is oriented along the axis of the ring. What is the resultant force on the ring due to the dipole?"

This problem can directly be solved since the electric field due to the dipole will be parallel to the axis at all points on the ring.

However I was wondering, if I did it the other way round (calculate the force by the ring on the dipole) the answer would be the same, and it was. So my question is, will this always be the case for any two systems? If so, is there any elementary way I can go about proving this?

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This is just the result of Newton's third law. Or you can see it in how Coulomb's law is the same when you exchange charges.

$$||\mathbf F||=\frac{kq_1q_2}{r^2}=\frac{kq_2q_1}{r^2}$$

The electrostatic force is an interaction between charged objects. There is a mutual force between them. Therefore, this will always be the case.

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