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I understand that the magnetic moment of a current carrying closed loop is given by the cross product of the current I and area vector.

It will be perpendicular to the plane of the loop. But I don't understand where this vector will be located? Will it be perpendicular to the plane and arise from the center of the loop?

I will add an example.

a current carrying loop

In this particular case the textbook calculates the magnetic moment vector to have coordinates in x and -z axis (the axises have been marked in the image)

However is the vector arising from every point of the loop? If it was coming from the center it still would not have a component in y axis.

If we are given a vector with components in x and y directions shouldn't the vector just lie in x and y plane? How can it move up and down the z axis? Even if it is perpendicular to the z axis

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The magnetic moment is $$ {\boldsymbol \mu}= \frac 12 I\oint ({\bf r}-{\bf r}_0)\times d{\bf l}. $$

Here ${\bf r}-{\bf r}_O$ is the position of the bit of wire $d{\bf l}$ measured from some chosen origin ${\bf r}_0$. But note that $$ \oint {\bf r}_0\times d{\bf l} = {\bf r}_0 \times \oint d{\bf l}=0 $$ because $\oint d{\bf l}=0$ (it is a closed loop after all). Thus ${\boldsymbol \mu}$ is independent of the chosen origin ${\bf r}_0$. In other words a magnetic moment is like a couple, which, unlike the usual moment of a force, does not have a specified point of action.

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  • $\begingroup$ Thank you for the explanation. Since the moment does not have a specified point of action is it correct to take it anywhere perpendicular to the current carrying loop in the mentioned example for the sake of solving numericals(example was attached in the post). $\endgroup$ Commented Jun 29, 2025 at 18:59
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    $\begingroup$ @RitvikBansal If the magnetic field is uniform over the loop then yes. If the magnetic field is not uniform then it is not of much use to define a magnetic moment like this. $\endgroup$ Commented Jun 29, 2025 at 19:20
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Your loop can be equivalently thought of as a collection of many smaller loops such that all the currents which are inside the boundary of the main loop cancel out. From this you get a large number of very small vectors each localized on every point of the loop. One other way could be introducing a vector magnetic moment per unit area, which will have the units of current.

Now, that loop can be thought of as the boundary of many wildly different surfaces and you can make many different magnetic moment distribution pictures in your end. But in the end all of them will result in the same dynamics of your loop with some external field.

This can be thought as converging to some value using different sequences (for example: $1+1/n$ and $1+1/n^2$). Sure those sequences can have very different properties but once they converge to some value, all of them are just as good and their properties do not matter.

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