Q.Consider the two idealized systems:

The abrupt-cut-off idealisations cannot be exact. The capacitor field is consistent with Gauss's law, but a loop crossing its boundary violates $\oint\vec E\cdot d\vec l=0$, so (a) and (c) are wrong. For the solenoid, $B=0$ outside makes a Gaussian surface at the end have net magnetic flux (violating $\nabla\cdot\vec B=0$) and an external Amperian loop enclosing the winding current gives $\oint\vec H\cdot d\vec l=0\neq I_{en}$.

Concept understanding.

Case (i) — capacitor: Taking $E$ uniform between the plates and zero outside is consistent with Gauss's law (a pill-box's flux still equals $q_{enc}/\varepsilon_0$), so (a) is wrong. But a loop running from inside (where $E\neq0$) to outside (where $E=0$) would give $\oint \vec{E}\cdot d\vec{l}\neq 0$, contradicting the electrostatic condition $\oint\vec{E}\cdot d\vec{l}=0$. Hence (c) is wrong — case (i) does not agree with that law, it violates it (which is exactly why fringing fields must exist).

Case (ii) — solenoid: If $B$ were exactly zero outside, a closed Gaussian surface straddling the end face would have flux entering (inside, $B\neq0$) with none leaving, giving net $\oint\vec{B}\cdot d\vec{A}\neq0$ and violating Gauss's law for magnetism $\nabla\cdot\vec B=0$ (field lines must close). So (b) is correct. Similarly, an Amperian loop encircling the solenoid from outside encloses the net winding current, yet $B=0$ there would give $\oint\vec{H}\cdot d\vec{l}=0$, contradicting $\oint\vec{H}\cdot d\vec{l}=I_{en}$. So (d) is correct.

Method: Testing an Idealized Field Against Gauss's and Ampere's Laws

Use this method whenever a problem presents an idealized field (sharp cutoff at a boundary: uniform inside a bounded region, exactly zero immediately outside) and asks which fundamental law that idealization actually violates.

Steps

Step 1: State the idealization precisely

Write down exactly what is assumed uniform and where it is assumed to drop to zero — e.g. "field is constant inside a bounded region and exactly zero immediately outside it, with no transition region."

Step 2: Test against Gauss's law for the relevant field

For $\vec{E}$: a Gaussian surface drawn entirely inside, or entirely outside, the region gives flux equal to enclosed charge over $\varepsilon_0$ — a sharp uniform-then-zero field is often still consistent with this, since it correctly reflects the enclosed charge. For $\vec{B}$: a Gaussian surface straddling the boundary (e.g. a pillbox capping the end of a solenoid) has flux entering through one face with none leaving anywhere else if $B=0$ just outside — that gives a nonzero net flux, which violates $\nabla\cdot\vec{B}=0$, since real $\vec{B}$ field lines must always close on themselves.

Step 3: Test against the relevant circulation law

For electrostatics: draw a closed loop running from a region of nonzero $\vec{E}$ into a region of zero $\vec{E}$. The electrostatic condition $\oint\vec{E}\cdot d\vec{l}=0$ must hold for ANY closed loop — if the sharp idealization makes this loop integral come out nonzero, it violates that condition (this is exactly why real fringing fields must exist at the edges). For magnetism: an Amperian loop that encloses the true winding current but passes through the idealized "zero field" region outside gives $\oint\vec{H}\cdot d\vec{l}=0$ from that region's contribution, contradicting $\oint\vec{H}\cdot d\vec{l}=I_{en}$ when the enclosed current is genuinely nonzero.

Step 4 (Applying to this problem): Separate "is consistent with" from "agrees with"

Distinguish a law the idealization happens to satisfy on a bulk Gaussian surface from a circulation-type law it silently violates through the artificial sharp edge. A field can pass one test and fail the other — always check flux-type and circulation-type laws separately rather than assuming one violation implies the other.