- #1

zenterix

- 524

- 74

- Homework Statement
- Consider the figure below, which shows an infinite wire carrying a current ##I_1## and a semicircular conducting loop with current ##I_2## placed near the wire.

- Relevant Equations
- What happens to to the semicircular loop?

Please see the image further below for the definition of the coordinate system.

The magnetic dipole moment of the loop is

$$\vec{\mu}=\mu\hat{k}\tag{1}$$

The magnetic field for an infinite wire depends only on the distance from the wire.

For the specific configuration above, we have

$$\vec{B}(y)=B(y)\hat{k}=\frac{\mu_0I}{2\pi y}\hat{k}\tag{2}$$

Note that I calculated this field considering an infinite wire and a point ##P## above it. That was a 2d problem (more on why I am saying this further below in my question).

Then

$$\vec{\tau}=\vec{\mu}\times \vec{B}=0$$

That is, there is no torque on the loop.

$$\vec{F}=\nabla(\vec{\mu}\cdot \vec{B})$$

$$=\nabla(\mu B(y))$$

$$=\mu\frac{\partial B}{\partial y}\hat{j}$$

Since the partial derivative above is negative, the loop translates towards the wire.

The problem I just solved online entailed reaching the two conclusions above, namely that the loop experiences no torque and translates towards the wire.

However, I am a little bit confused about some things.

I simply applied the formula I just learned: ##\vec{F}=\nabla(\vec{\mu}\cdot \vec{B})##, which comes from the force being the negative of the gradient of potential energy (the latter being ##-\vec{\mu}\cdot \vec{B}##).

However,

The way I solved the problem, I considered ##\vec{B}## to be a function of ##y## which is position in the ##\hat{j}## direction.

But doesn't ##\vec{B}## have a non-zero derivative relative to ##z## as well, ie in the ##\hat{k}## direction?

After all, ##y## is a function of ##z##.

There is something wrong with my interpretation of the equation ##\vec{F}=\nabla(\vec{\mu}\cdot \vec{B})## I believe.