# $B$ field inside a Magnet

From Lorentz Force law there are experimental proofs for defining the direction of the magnetic field $$B$$ outside the magnet.

But when the interior of magnetic material is concerned how is the Direction of B-Field justified experimentally.

What is the experiment that proved The B field to be a solenoidal field?

But when the interior of magnetic material is concerned how is the direction of B-Field justified experimentally.

You don't need an extra justification, because magnetic material is nothing but a collection of charges, some of them being held in place by nonelectric forces, others moving. Since it's, deep down, just a collection of charges, the same laws (Maxwell's equations) that hold outside the magnetic material also hold inside (as long as they're applied properly).

In addition to that, you can send direct current through the magnetic material by using the magnetic material as a conductor (if it's conductive), observe where the current reappears on the other side, and also observe the magnetic field generated by that current.

The Lorentz force acting on a charged particle at a point,

$$\mathbf{F} = q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$$

(where $$\mathbf{B}$$ is the sum of the magnetic field caused by the current and the magnetic material at that point) means that if (one component of) $$\mathbf{B}$$ had the opposite direction, the resulting force would be different (unless $$\mathbf{v}$$ in the material will always be (anti-)parallel to the $$\mathbf{B}$$-field generated by the material), resulting in the current reappearing at a different place on the other side of the magnetic material.

In addition to measuring the place at which the current exited the material, you can calculate $$\mathbf J$$ and then calculate the magnetic field of the current outside the magnetic material. For that, we can use Biot-Savart law

$${\displaystyle \mathbf {B}_\text{current} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int\limits _{V}\ {\frac {(\mathbf {J} \,\textrm{d}V)\times \mathbf {r} '}{|\mathbf {r} '|^{3}}}}$$

where $$\mu_0$$ is permeability of free space, $$\textrm{d} V$$ is an element of volume of the conductor, $$\mathbf{r'}$$ is the displacement vector between the volume element and the place where we measure the magnetic field outside the magnetic material, and $$V$$ is the volume of the conductor.

If $$\mathbf B$$-field in the magnetic material pointed in the opposite way than what the theory says, the current would flow differently and the magnetic field generated by it would therefore look differently too.

What is the experiment that proved the B field to be a solenoidal field?

The general proof is only empirical - $$\mathbf{B}$$-field being solenoidal is identical to saying

$$\boldsymbol\nabla\cdot\mathbf{B} = 0$$

which is one of Maxwell's equations, and is justified by magnetic monopoles never having been observed (since the nonexistence of magnetic monopoles is equivalent to the equation being true).

If magnetic monopoles are ever observed, we can rewrite Maxwell's equations to account for that (but the direction of the $$\mathbf{B}$$-field in a magnetic material will still be the same).