(i) a parallel plate capacitor with large plates and small separation and
(ii) a long solenoid of length L≫R, radius of cross-section. In
(i) E is ideally treated as a constant between plates and zero outside. In
(ii) magnetic field is constant inside the solenoid and zero outside. These idealised assumptions, however, contradict fundamental laws as below:
(a) case
(i) contradicts Gauss’s law for electrostatic fields.
(b) case
(ii) contradicts Gauss’s law for magnetic fields.
(c) case
(i) agrees with ∮E.dl = 0.
(d) case
(ii) contradicts ∮H.dl = Ien
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Concept understanding — Magnetic Field Lines
Magnetic Field Lines
A magnet or a current-carrying wire fills the space around it with a magnetic field. We cannot see this field, so we picture it using magnetic field lines — continuous curves that map both the direction and the strength of the field at every point.
What a field line represents
The tangent to a field line at any point gives the direction of the magnetic field B there. If you place a tiny compass needle at that point, it aligns along the tangent, its north pole pointing the way the line runs. The density of the lines (how closely packed they are) represents the magnitude of B: crowded lines mean a strong field, widely spaced lines mean a weak field.
Key properties (exam essentials)
Important
Outside a magnet the lines run from the north pole to the south pole, but they are continuous closed loops — inside the magnet they run south to north, so every line closes on itself.
Two field lines never intersect. If they did, a compass at the crossing point would have to point in two directions at once, which is impossible.
Lines are crowded where the field is strong (near the poles) and spread out where it is weak.
They form smooth, continuous curves with no free ends.
The fact that magnetic field lines always close on themselves is deep: it means there are no isolated magnetic poles (monopoles). This is Gauss's law for magnetism:
∮B⋅dA=0
The net magnetic flux through any closed surface is zero — every line that enters the surface also leaves it.
Contrast with electric field lines
Note
Electric field lines start on positive charges and end on negative charges — they are open curves. Magnetic field lines have no such start or end; they are always closed loops. This single difference reflects that isolated electric charges exist, but isolated magnetic poles do not.
Uniform field
When field lines are parallel, equally spaced, and straight, the field is uniform — the same magnitude and direction everywhere. The region deep inside a long solenoid, or between the flat poles of a large magnet, is very nearly uniform.
Why this matters
Field-line diagrams let you read a field at a glance: where it is strong, which way it points, and whether it is uniform. They underpin magnetic flux, ΦB=B⋅A, Gauss's law for magnetism, and the study of electromagnetic induction. Sketching the correct pattern for a bar magnet, a straight wire (concentric circles around it), and a solenoid (uniform inside, bar-magnet-like outside) is a standard exam skill.
The properties of magnetic field lines, including why they always form closed loops, are introduced in the NCERT Class 12 Physics chapter on magnetism and matter, and "properties of magnetic field lines class 12 physics important questions" is a common CBSE board short-answer topic. This closed-loop explanation, linked to the absence of magnetic monopoles, matches the NCERT-prescribed reasoning.
Why this formula?
Magnetic Field Lines
A magnetic field line is an imaginary curve we draw to picture an invisible field. Its purpose is to encode two things at once: the direction of the field B (the tangent to the line at any point) and the strength of the field (how densely the lines are packed). They are a map, not physical objects.
The four defining rules
1. The tangent gives the field direction. At every point, B points along the tangent to the field line through that point. A compass needle placed on the line aligns with it.
2. Field lines form closed loops. Unlike electric field lines, which begin and end on charges, magnetic field lines never start or stop. This is Gauss's law for magnetism:
∮B⋅dA=0
The net flux through any closed surface is zero because isolated magnetic poles (monopoles) do not exist — every north pole is paired with a south pole. So for a bar magnet the lines emerge from the north pole outside, curve around to the south pole, and continue through the interior of the magnet back to the north, closing the loop.
3. Field lines never cross. If two lines crossed, the tangent — and hence B — would have two directions at that point. Since the field has a single, unique direction everywhere, crossings are impossible.
4. Density represents strength. Where lines are crowded, the field is strong; where they spread out, it is weak. The flux through a small perpendicular area dA is dΦB=BdA, so packing more lines through the same area means a larger B (as near the poles of a magnet).
A quick picture
For a straight wire carrying current I, the field lines are concentric circles around the wire, tightly spaced close to the wire and spreading out farther away. Grip the wire with your right hand, thumb along the current — your curled fingers trace the direction of the loops. This is exactly the closed-loop, non-crossing, density-encodes-strength behaviour the rules above describe.
The abrupt-cut-off idealisations cannot be exact. The capacitor field is consistent with Gauss's law, but a loop crossing its boundary violates ∮E⋅dl=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 ∇⋅B=0) and an external Amperian loop enclosing the winding current gives ∮H⋅dl=0=Ien.
✓Final answer
Correct options: (b) and (d) — case (ii) contradicts both Gauss's law for magnetism and Ampère's law.
Sharp cut-off fields are unphysical. The solenoid idealisation (B uniform inside, exactly zero outside) violates Gauss's law for magnetism and Ampère's law. Correct options: (b) and (d).
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 qenc/ε0), so (a) is wrong. But a loop running from inside (where E=0) to outside (where E=0) would give ∮E⋅dl=0, contradicting the electrostatic condition ∮E⋅dl=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=0) with none leaving, giving net ∮B⋅dA=0 and violating Gauss's law for magnetism ∇⋅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 ∮H⋅dl=0, contradicting ∮H⋅dl=Ien. So (d) is correct.
✓Final answer
(b) case (ii) contradicts Gauss's law for magnetic fields and (d) case (ii) contradicts ∮H⋅dl=Ien. (a) is false, and (c) is false because case (i) actually violates∮E⋅dl=0 rather than agreeing with it.
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 E: a Gaussian surface drawn entirely inside, or entirely outside, the region gives flux equal to enclosed charge over ε0 — a sharp uniform-then-zero field is often still consistent with this, since it correctly reflects the enclosed charge. For 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 ∇⋅B=0, since real 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 E into a region of zero E. The electrostatic condition ∮E⋅dl=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 ∮H⋅dl=0 from that region's contribution, contradicting ∮H⋅dl=Ien 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.
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
AP EAPCET 2024Set eng-2024-05-22-AN1 markMCQ
Q.Current flows in a conductor from east to west. The direction of the magnetic field at a point below the conductor is towards
(A) North
(B) South
(C) East
(D) West
›Reveal solutionSolution
Using the right-hand (Biot–Savart circulation) rule for a straight current-carrying wire pinpoints the field direction at a point directly below it.
Concept and Intuition
The magnetic field lines around a straight current-carrying wire form concentric circles in planes perpendicular to the current, with direction given by the right-hand rule: point the thumb along the current, and the curled fingers give the field's circulation direction. Setting up coordinates makes this unambiguous instead of relying purely on visualising the curled hand.
Step-by-Step Solution
Define East =+x^, North =+y^, Up (out of the ground) =+z^.
Current flows from east to west, i.e. in the direction I^=−x^.
"Below the conductor" means at a point displaced in the −z^ direction from the wire, so the unit vector from the wire to the field point is r^=−z^.
The field circulation direction is B^∝I^×r^=(−x^)×(−z^)=x^×z^.
Using x^×y^=z^,y^×z^=x^,z^×x^=y^, we get x^×z^=−y^.
−y^ corresponds to the South direction.
Common Mistakes
Applying the right-hand rule with the current direction reversed (using "west to east" instead of "east to west").
Forgetting that "below" the wire means displaced downward (into the ground), not to one horizontal side.
✓Final answer
The correct option is (B) — South.
ANSWER: B
AP EAPCET 2023Set eng-2023-05-19-FN1 markMCQ
Q.The magnetic field lines of a bar magnet
(A) leave from the south pole of the magnet
(B) are absent inside the magnet
(C) intersect each other
(D) form continuous closed loops
›Reveal solutionSolution
This tests a fundamental property of magnetic field lines. Answer: they form
continuous closed loops.
Concept and Intuition
Unlike electric field lines (which begin and end on charges), magnetic field lines
have no isolated source or sink because there are no magnetic monopoles. Every field
line that leaves the north pole outside the magnet must re-enter through the south
pole and continue inside the magnet from south to north, closing the loop.
Step-by-Step Solution
Check each option against known facts:
(A) "leave from the south pole" — false; outside the magnet, lines emerge from the
north pole.
(B) "absent inside the magnet" — false; field lines exist and run from south to
north inside the magnet.
(C) "intersect each other" — false; two field lines can never cross (the field
direction at a point would be ambiguous otherwise).
(D) "form continuous closed loops" — true; this is a direct consequence of the
absence of magnetic monopoles (∇⋅B=0).
So (D) is the correct statement.
Common Mistakes
Thinking field lines "start" at north and "end" at south permanently, without
realizing they continue inside the magnet to close the loop.
✓Final answer
The correct option is (D) — form continuous closed loops.
ANSWER: D
AP EAPCET 2022Set eng-2022-07-06-FN1 markMCQ
Q.Among the following, the incorrect Maxwells Electromagnetic equation is
(A) ∮B⋅dl=μ0ic+μ0ε0dtdϕE
(B) ∮B⋅dA=ε0Q
(C) ∮E⋅dl=dt−dϕB
(D) ∮E⋅dA=ε0Q
›Reveal solutionSolution
Three of the four listed equations are genuine Maxwell equations stated correctly. Option (B) mislabels Gauss's law for magnetism — it should equal zero (no magnetic monopoles exist), not Q/ε0.
Concept and Intuition
Maxwell's four equations are: Gauss's law for E (∮E⋅dA=Q/ε0, since electric field lines start/end on charges), Gauss's law for B (∮B⋅dA=0, since magnetic field lines always form closed loops — there are no isolated magnetic "charges"/monopoles), Faraday's law (∮E⋅dl=−dϕB/dt, a changing magnetic flux induces an electric field), and the Ampère–Maxwell law (∮B⋅dl=μ0ic+μ0ε0dϕE/dt, currents and changing electric flux both produce magnetic fields).
Step-by-Step Solution
Compare (A) to the Ampère–Maxwell law: matches exactly (conduction current term + displacement current term). Correct.
Compare (C) to Faraday's law: matches exactly, including the minus sign (Lenz's law built in). Correct.
Compare (D) to Gauss's law for electric fields: matches exactly. Correct.
Compare (B): the left side, ∮B⋅dA, is the magnetic flux through a closed surface — by Gauss's law for magnetism this must always be zero, since magnetic monopoles don't exist. Setting it equal to Q/ε0 (which is the electric Gauss's law) is a mismatch — the equation as written is incorrect.
Common Mistakes
Assuming the equation "looks like a real Gauss's law form" and accepting it without checking which field it applies to (electric vs. magnetic).
Forgetting that the defining, unique feature of magnetism (vs. electricity) is the absence of a source term — ∮B⋅dA=0 always, regardless of any charge or current inside.
✓Final answer
The correct option is (B) — this is the incorrect equation.