ISRO EC 2025 Q.51: solution and answer

ISRO 2025 · ECQ.51 · 1 mark · MCQ

A boundary separates two magnetic materials of permeability $μ_{1}$ and $μ_{2}$ . The magnetic field vector in $μ_{1}$ is $H_{1}$ with a normal component $H_{n_{1}}$ and tangential component $H_{t_{1}}$ while that in $μ_{2}$ is $H_{2}$ with a normal component $H_{n_{2}}$ and a tangential component $H_{t_{2}}$ . Then the derived condition would be

(A)
$H_{1} = H_{2}$ and $H_{t_{1}} = H_{t_{2}}$
(B)
$H_{t_{1}} = H_{t_{2}}$ and $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$
(C)
$H_{1} = H_{2}$ and $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$
(D)
$H_{1} = H_{2}$ , $H_{t_{1}} = H_{t_{2}}$ and $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The conditions are $H_{t_{1}} = H_{t_{2}}$ and $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$ , so the answer is option (b).

Official answer: BSource: ISRO's final answer key of 15 May 2026

The boundary conditions at an interface come from Maxwell's equations applied to a thin loop and a thin pillbox straddling the boundary.
Tangential $H$ : take Ampere's law $\oint H \cdot d l = I_{e n c}$ on a thin rectangular loop with its long sides parallel to the boundary, one in each medium. As the loop height shrinks to zero, and with no free surface current, $(H_{t_{1}} - H_{t_{2}}) Δ l = 0$ , so $H_{t_{1}} = H_{t_{2}}$ .
Normal $B$ : take Gauss's law for magnetism $\oint B \cdot d S = 0$ on a thin pillbox with its flat faces on either side of the boundary. As the height shrinks to zero, $(B_{n_{1}} - B_{n_{2}}) Δ S = 0$ , so $B_{n_{1}} = B_{n_{2}}$ .
With $B = μ H$ in each medium, this becomes $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$ .
The two derived conditions are $H_{t_{1}} = H_{t_{2}}$ and $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$ , which is option (b).
Options (a), (c) and (d) state $H_{1} = H_{2}$ as a whole vector. That would need $H_{n_{1}} = H_{n_{2}}$ as well, which contradicts $μ_{1} H_{n_{1}} = μ_{2} H_{n_{2}}$ unless $μ_{1} = μ_{2}$ . Option (a) also leaves out the normal-component condition.

KEY IDEA

Across a boundary with no surface current, tangential $H$ is continuous (Ampere's law) and normal $B$ is continuous (Gauss's law for magnetism), so normal $H$ jumps by the ratio of permeabilities.

WATCH OUT FOR

Assuming the whole field vector $H$ is continuous, or that normal $H$ is continuous. Only the tangential part of $H$ and the normal part of $B$ are continuous.

From the ISRO EC 2025 paper. Every question of this paper · The full paper (PDF, ISRO) · All 2 ISRO Maxwell's equations questions, with notes