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Question
Do magnetic forces obey Newton’s third law. Verify for two current elements dl1 = dlî located at the origin and dl2 = dlĵ located at (0, R, 0). Both carry current I.
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Solution
In this problem first, we have to find the direction of magnetic field due to one wire at the point on another wire, then the magnetic force on that current carrying wire.
According to Biot-Savart’s law, magnetic field B is parallel to idl × r and idl is the current carrying element having its direction along the direction of flow of current.
Here, for the direction of magnetic field, at dl2, located at (0, R, 0) due to wire dlx is given by B || idl × r or i × j (because point (0, R, 0) lies ony-axis), but i x j = k.
So, the direction of magnetic field at dl2 is along the z-direction.
The direction of magnetic force exerted at dl2 due to the magnetic field of first wire is along the x-axis.
F-i(l × B), i.e., F||(i × k) or along – j direction.
Therefore, force due to dl1 on dl2 is non-zero.
Now, for the direction of magnetic field, at dx, located at (0, 0, 0) due to wire d2 is given by B || idl × r or j × – j (because origin lies on y-direction w.r.t. point (0, R, 0), but j × – j = 0.
So, the magnetic field at dx does not exist.
Force due to dl2 on dl1, is zero.
So, magnetic forces do not obey Newton’s third law. But they obey Newton’s third law if current-carrying elements is placed parallel to each other.
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