For a steady magnetic fields, which of the following is true : 1. The tangential component of magnetic field is continuous across any boundary except the surface of perfect conductor. Which one of the following is correct ?
2. The tangential component of magnetic flux density is continuous across any boundary.
3. The normal component of magnetic flux density is continuous across any boundary.
4. The normal component of electric field is continuous across the boundary.
1 and 3
The boundary conditions come straight from two of Maxwell's equations, and they pick out \(H_{t}\) and \(B_{n}\) — statements 1 and 3, option 2.
Normal B is continuous because \(\nabla\cdot\mathbf{B}=0\): no magnetic monopoles exist, so no surface can accumulate magnetic charge, and applying Gauss's law to a pillbox straddling the boundary gives
\(B_{n1}=B_{n2}\)
This holds at every boundary without exception — there is nothing that could break it.
Tangential H is continuous because \(\nabla\times\mathbf{H}=\mathbf{J}\): applying Ampère's law to a thin loop straddling the boundary gives
\(H_{t1}-H_{t2}=K\)
where K is the surface current density. For ordinary media K is zero and \(H_{t}\) is continuous — but on a perfect conductor the current is confined to an infinitesimally thin sheet, K is finite, and the tangential field jumps. Statement 1 states the rule together with exactly that exception, which is what makes it the carefully worded true statement.
| Quantity | Continuous? | From |
|---|---|---|
| Bn | Always | \(\nabla\cdot\mathbf{B}=0\) |
| Ht | Unless surface current flows | \(\nabla\times\mathbf{H}=\mathbf{J}\) |
| Bt | No | Jumps by the ratio μ1/μ2 |
| En | No | Jumps unless ε1 = ε2 |
Why statement 2 is wrong. Since \(\mathbf{B}=\mu\mathbf{H}\) and the two media have different permeabilities, continuity of \(H_{t}\) forces \(B_{t}\) to be discontinuous. Only one of the pair can be continuous, and it is H. This is the single most common slip in the topic.
Why statement 4 is wrong. The electric counterparts run the other way round: \(E_{t}\) is continuous and \(D_{n}\) is continuous in the absence of surface charge, so
\(\varepsilon_{1}E_{n1}=\varepsilon_{2}E_{n2}\)
and \(E_{n}\) jumps by the ratio of permittivities.
The pattern is worth memorising in one line : tangential fields (E and H) are continuous, normal flux densities (D and B) are continuous — with surface current breaking the H rule and surface charge breaking the D rule.
Hence, statements 1 and 3 are correct.
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