Q.Use

Concept: Magnetic Poles — The distinction between $B$ and $H$ inside a magnet follows from Ampere's law and the continuity of $B$.

1. Ampere's law for $H$: $\oint \vec{H} \cdot d\vec{l} = I_{\text{free}}$. Inside a bar magnet, there are no free currents, so $\oint \vec{H} \cdot d\vec{l} = 0$ along any closed loop. For a loop that goes from the N pole to the S pole inside the magnet and returns outside, the line integral inside must be opposite in sign to that outside. Outside, $H$ lines run from N to S (by definition of pole convention). Hence, inside, $H$ must run from N to S to make the net integral zero. …

The key to this problem is to stop thinking of $\mathbf{H}$ as "just $\mathbf{B}$ in a material" and instead treat it as a separate field defined by $\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$. Inside a permanent magnet, the magnetization $\mathbf{M}$ is fixed and points from the S pole to the N pole (i.e., from south to north inside the magnet). That single fact, combined with Ampere's law for $\mathbf{H}$ and the continuity of $\mathbf{B}$, forces the two fields to point in opposite directions inside the bar.

Let’s walk through it.

1. Ampere’s law for $\mathbf{H}$ tells us the direction of $\mathbf{H}$ outside the magnet. In free space (outside the magnet), $\mathbf{M} = 0$, so $\mathbf{B} = \mu_0 \mathbf{H}$. The familiar picture of a bar magnet’s external field shows lines emerging from the N pole and entering the S pole. Since $\mathbf{B}$ and $\mathbf{H}$ are parallel in free space, the external $\mathbf{H}$ lines also go from N to S. Now apply Ampere’s law for $\mathbf{H}$:

Inside the bar magnet there are no free currents, so the circulation of $\mathbf{H}$ around any closed loop is zero. Take a loop that goes partly outside (where $\mathbf{H}$ points N→S) and partly inside. For the line integral to vanish, the inside contribution must oppose the outside contribution. The only way is for $\mathbf{H}$ inside to point from N to S as well — same direction as outside.

2. Continuity of $\mathbf{B}$ forces the inside direction of $\mathbf{B}$ to be opposite to $\mathbf{H}$. The magnetic flux density $\mathbf{B}$ obeys $\nabla \cdot \mathbf{B} = 0$, meaning its field lines form closed loops — they never start or end. Outside the magnet, $\mathbf{B}$ goes from N to S (same as $\mathbf{H}$). To close the loop, the lines that enter the S pole from outside must continue through the interior and emerge from the N pole. That means inside the magnet, $\mathbf{B}$ must point from S to N. …

Method: Determining Field-Line Direction Inside a Magnetised Material Using Ampere's Law for $H$ and Continuity of $B$

Use this whenever you must work out which way $\vec H$ and $\vec B$ point inside a magnetised body, given how they behave outside it.

Steps

Step 1: Establish the outside direction as your reference

Outside any magnet (in free space, where $\vec M=0$), $\vec B=\mu_0\vec H$, so $\vec B$ and $\vec H$ point the same way there. Use the conventional external field-line picture (lines run from the North pole to the South pole outside the magnet) as your known starting direction for both fields.

Step 2: Apply Ampere's Law for $\vec H$ to find its direction inside

If there is no free current anywhere on your loop (true inside an ordinary permanent magnet — the "currents" here are bound, not free), the circulation of $\vec H$ around any closed loop must be exactly zero. Take a loop that runs partly outside the material (where $\vec H$'s direction is known from Step 1) and partly inside; for the total circulation to cancel to zero with no free current crossing the loop, $\vec H$ inside must continue in the same direction as it does outside — it cannot reverse without a free surface current to justify the jump.

Step 3: Apply continuity of $\vec B$ (no magnetic monopoles) to find its direction inside …