# Why doesn't the H field form closed loops?

As one learns really early on in electro-magnetism, the field lines of the magnetic flux density B form closed loops. However, the magnetic field intensity H, related to B = $\mu_r\mu_0$ H, doesn't always.

How can we understand the difference?

• the source of H are the poles – hyportnex Jan 8 '18 at 14:45
• I am looking for a bit more insight @hyportnex than "the source of H are the poles", that can be found in every textbook but I would like to understand better why that is the case – Jhonny Jan 8 '18 at 18:36

A vector field forms closed loops is it is divergence free. This is the case for the magnetic field $\mathbf{B}$ $$\nabla \cdot \mathbf{B} =0$$ From your definition of $\mathbf{H}$, taking $\mu_0$ to be constant over space $$\mathbf{H} = \frac {\mathbf{B}} {\mu_0 \mu_r}$$ $$\nabla\cdot\mathbf{H} = \frac{1} {\mu_0}\left(\frac{1} {\mu_r}\nabla \cdot\mathbf{B} + \mathbf{B}\cdot \nabla\frac{1} {\mu_r} \right) \\ = - \frac{\mathbf{B}\cdot\nabla\mu_r} {\mu_0 \mu_r^2}$$ as such it is the varying of $\mu_r$ over space that causes the divergence of $\mathbf{H}$ to be nonzero, and its field lines to not be closed.