CUET UG Physics Booster Test 3-Gauss’s Law and Field Line Topology

According to Gauss's law for electricity, ΦE = qenclosed/ε₀ An electric dipole consists of equal positive and negative charges. Therefore, the net enclosed charge is zero, giving zero net electric flux through any closed surface enclosing the entire dipole. According to Gauss's law for magnetism, ∮ B·dS = 0 Magnetic field lines form continuous closed loops and there are no isolated magnetic monopoles. Hence, the net magnetic flux through any closed surface enclosing a complete bar magnet is also zero. Therefore, both the net electric flux through a surface enclosing a dipole and the net magnetic flux through a surface enclosing a bar magnet are zero.

A surface enclosing the north-pole region of a bar magnet still has zero net magnetic flux because magnetic field lines continue through the interior of the magnet and form closed loops. Therefore 1 → d. A surface enclosing the entire bar magnet has equal numbers of field lines entering and leaving, giving zero net magnetic flux. Therefore 2 → c. A surface enclosing empty space near a magnet has field lines passing through it without any magnetic source enclosed, so the net magnetic flux remains zero. Therefore 3 → a. A surface enclosing an electric dipole contains equal positive and negative charges, resulting in zero net enclosed charge and zero net electric flux. Therefore 4 → b. Thus, the correct matching is 1-d, 2-c, 3-a, 4-b.

Magnetic flux is defined as φB = B·S = BS cos θ where θ is the angle between the magnetic field and the area vector. When a surface is parallel to the magnetic field, the area vector is perpendicular to the field. Therefore, θ = 90° and cos 90° = 0 Hence, φB = 0 The flux is therefore zero because no magnetic field lines pass normally through the surface.

Gauss's law for magnetism states ∮ B·dS = 0 This law remains valid regardless of the shape of the closed surface or the complexity of the magnetic configuration enclosed within it. Whether the surface contains bar magnets, current loops, electromagnets, or a combination of all of them, the total magnetic flux through the closed surface remains zero because magnetic field lines form closed loops and magnetic monopoles do not exist. Therefore, the total magnetic flux summation is always zero.

Magnetic field lines emerge from the north pole outside a magnet and enter the south pole. Inside the magnet, the lines continue from the south pole back to the north pole, completing closed loops. Therefore, even if a Gaussian surface surrounds only the north-pole region, magnetic field lines that appear to leave the surface are balanced by lines entering through another part of the surface. This ensures that the total magnetic flux remains zero. This behavior directly follows from the absence of magnetic monopoles and Gauss's law for magnetism.

According to Gauss's law for magnetism, ∮ B·dS = 0 for every closed surface. A Gaussian sphere is a closed surface, and therefore the total magnetic flux through it must always be zero. Doubling the radius changes the surface area of the sphere, but it does not change the fundamental law governing magnetic flux. Since magnetic field lines form closed loops and magnetic monopoles do not exist, the number of field lines entering the sphere equals the number leaving it. Therefore, regardless of whether the sphere has radius R or 2R, the net magnetic flux remains zero. Thus, the correct answer is B. 0 Wb.

If magnetic monopoles existed, magnetic field lines could begin or end on magnetic charges just as electric field lines begin or end on electric charges. In such a case, Gauss's law for magnetism would be modified to include enclosed magnetic charge. The flux through a closed surface would no longer necessarily be zero. A closed surface enclosing a magnetic monopole would have a non-zero net magnetic flux. Therefore, the statement that Gauss's law would remain ∮B·dS = 0 would be incorrect. Thus, the correct answer is C.

Magnetic field lines form continuous closed loops and do not begin or end at any point. This is a consequence of the absence of magnetic monopoles. A closed loop of magnetostatic field lines is associated with electric currents, consistent with Ampere's law. Furthermore, because magnetic monopoles do not exist, no closed surface can have a net excess of magnetic field lines leaving it. This is precisely the meaning of Gauss's law for magnetism. Statement 4 is not universally correct because magnetic fields can be produced by current loops without requiring a physical north-south pole pair as a separate configuration. Therefore, Statements 1, 2 and 3 are correct. Thus, the correct answer is C. 1, 2 and 3 are correct.

A diagram showing magnetic field lines radiating outward from a single point resembles the electric field pattern of an isolated positive electric charge. Such a magnetic diagram would imply the existence of a magnetic monopole. Since magnetic monopoles have not been observed experimentally, the diagram violates Gauss's law for magnetism, which requires the net magnetic flux through any closed surface to be zero. However, the same field-line pattern is perfectly valid for an isolated electric charge because electric charges act as sources and sinks of electric field lines. Therefore, Statements 2, 3 and 4 are correct. Thus, the correct answer is D. 2, 3 and 4 are correct.

At any point in space, the magnetic field has a unique direction. If two magnetic field lines intersect, a particle placed at the intersection would experience two different magnetic field directions simultaneously, which is impossible. Therefore, intersecting field lines imply an ambiguous field direction. A closed magnetostatic loop existing entirely in empty space without any enclosed current would contradict Ampere's law. In magnetostatics, closed magnetic field loops are associated with currents that generate them. Therefore, the two implications are: Intersection of field lines → Ambiguous direction Closed magnetostatic loop in empty space → Violation of Ampere's law Thus, the correct answer is D.

A toroid is formed by bending a solenoid into a circular shape. The magnetic field lines remain confined within the toroidal core, making the external magnetic field ideally zero. Inside a long solenoid, the magnetic field is strong and nearly uniform because the field lines are parallel and closely spaced. Outside the solenoid, magnetic field lines spread out, curve around, and eventually re-enter the opposite end to form closed loops. Therefore: 1 → d 2 → b 3 → a 4 → c Thus, the correct matching is A. 1-d, 2-b, 3-a, 4-c.

According to Ampere's circuital law, ∮ B·dl = μ₀Ienc A circular magnetic field line is characteristic of the field surrounding a current-carrying conductor. The existence of a closed magnetic loop implies that there is a net electric current enclosed by the loop. Without an enclosed current, a perfect isolated circular magnetostatic field line would not satisfy the principles of magnetostatics. Thus, the correct answer is B. there must be a net electric current passing through the circular area.

For a finite solenoid, magnetic field lines emerge from one end and return through the surrounding space to the other end. This ensures that the field lines form closed loops. If the field lines remained perfectly straight and confined at the ends, they would not properly circulate around the current distribution. Such a configuration would contradict the implications of Ampere's law regarding magnetic fields generated by currents. However, the net magnetic flux through any closed surface would still remain zero, so Gauss's law for magnetism would not necessarily be violated. Thus, the correct answer is C. it would be consistent with Gauss's law but violate Ampere's law.

Outside a bar magnet, magnetic field lines travel from the north pole to the south pole. Inside the magnet, they continue from the south pole back to the north pole, completing continuous closed loops. Because of this continuity, every magnetic field line leaving a closed surface is balanced by another entering it. Therefore, the net magnetic flux through a closed surface surrounding either pole remains zero. This is consistent with Gauss's law for magnetism and the absence of magnetic monopoles. Thus, the correct answer is B. the net flux around either pole remains exactly zero.

Magnetic flux is given by Φ = BA Initial flux: Φ = 1.5 × 0.04 Φ = 0.06 Wb Effective area after fringing: A' = 0.04 × 1.20 A' = 0.048 m² Since flux remains constant, B' = Φ/A' B' = 0.06/0.048 B' = 1.25 T Thus, the average magnetic field decreases because the same flux is distributed over a larger area. Therefore, the correct answer is A. 1.25 T.

Fringing refers to the spreading of field lines near the edges of a region where a field is approximately uniform. This phenomenon occurs not only in magnetic fields but also in electric fields, such as between capacitor plates. Near pole pieces, magnetic field lines spread outward at the edges instead of remaining perfectly straight. This fringing makes the field slightly non-uniform. If field lines remained perfectly straight right up to the ends, they would not properly close into loops, conflicting with the requirements of magnetostatics. Therefore, the statement that fringing applies only to magnetic fields is incorrect. Thus, the correct answer is C. Fringing only applies to magnetic fields, not electrostatic fields.

If magnetic monopoles existed, Gauss's law for magnetism would become ∮ B·dS = μ₀qm where qm represents the enclosed magnetic charge. In such a universe, isolated north and south poles could exist independently. Magnetic field lines would be able to originate from positive monopoles and terminate on negative monopoles, similar to electric field lines. An enclosed magnetic dipole would still have equal and opposite magnetic charges, giving a net enclosed magnetic charge of zero. Therefore, the right-hand side would remain zero for a complete dipole. Hence all four statements are correct. Thus, the correct answer is D. 1, 2, 3 and 4 are correct.

In the hypothetical monopole version of Gauss's law, ∮ B·dS = μ₀qm the quantity qm plays the same role that electric charge q plays in Gauss's law for electricity. A positive magnetic charge enclosed by a surface would produce a net outward magnetic flux. However, the existence of monopoles would not eliminate magnetic dipoles. Dipoles could still exist as combinations of positive and negative magnetic charges. Also, μ₀ remains part of the equation and is not replaced by qm. Therefore, Statements 1 and 2 are correct. Thus, the correct answer is A. 1 and 2 are correct.

Karl Friedrich Gauss was one of the greatest mathematicians in history and also made major contributions to astronomy, physics and geodesy. NCERT notes that Gauss, together with Wilhelm Weber, built one of the earliest electric telegraph systems in 1833. This work demonstrated practical electrical communication long before modern telecommunications developed. Gauss was not associated with quantum mechanics, biology, chemistry or fluid dynamics as described in the other options. Thus, the correct answer is A. Mathematics and Astronomy, Electric telegraph.

Magnetic materials aligning north-south have been known since ancient civilizations through the use of lodestones and compasses. Scientific understanding of magnetism in terms of moving electric charges emerged only after the discoveries of Oersted, Ampere and others during the nineteenth century. Gauss and Weber jointly constructed one of the earliest electric telegraphs. The Meissner effect was discovered by Meissner and Ochsenfeld and is named after Meissner. Therefore: 1 → c 2 → a 3 → d 4 → b Thus, the correct matching is C. 1-c, 2-a, 3-d, 4-b.