# H-field and Ampére's law

## Homework Statement

Hi all.

I'm trying to understand the H-field. From Ampére's law we have:

$$\oint {{\bf{H}} \cdot {\rm{d}}{\bf{l}}} = I_{free,enclosed}$$

If I look at an object with zero free, enclosed current, the integral equals zero. The integral can be equal to zero even if H is not zero. But if the integral equals zero and if H is constant and parallel to dl, I can take it outside the integral, and hence H=0.

My question is: How do I know if H is in the same direction as dl and constant in magnitude?

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It will depend on the context. What kind of situations can you think of that would have a constant field of H? Generally, they will be infinite (or limits/approximations), right? Whether or not it is in the direction of your infinitesimal line element will depend on how you choose your amperian loop. Also, what kind of medium you are in could come into play if you are talking about H.

If we look at a cylinder, where a current I runs in the axis of the cylinder (z-axis), we can find B by the right-hand rule. Is there any way to find M?

Well yes, depending on what information you know about the current you could find the magnetization.

You know that in general that
$$\mathbf{J_b}=\nabla \times \mathbf{M}$$

where you could use stoke's theorem to get the magnetization. There are a bunch of other relations you could potentially use. Again, it will depend on context.

I hope it's OK if we take an example. Lets say, for instance, that the magnetization of a cylinder is homogeneous so there are no bound volume currents. I am told that

$$\textbf K_b=K\hat{\phi}$$

In this example, the direction of B is along the axis of the cylinder (right-hand rule). How would I be able to find the direction of M, and thereby H?

I know that K and M are perpendicular to eachother (cross-product), but this means that M can either be in the phi-direction or z-direction?