CUET UG Physics Booster Test 2-Gauss’s Law and Field Line Topology

Electrostatic field lines originate from positive charges and terminate on negative charges. Therefore, electric charges act as sources and sinks of electric flux. Magnetic field lines, however, always form continuous closed loops and never begin or end at any point. This distinction leads to the difference between Gauss's law for electricity and Gauss's law for magnetism. Gauss's law for electricity allows non-zero net electric flux when electric charge is enclosed. In contrast, Gauss's law for magnetism states that the net magnetic flux through every closed surface is zero because isolated magnetic poles do not exist. Statement 4 is incorrect because isolated electric charges exist, whereas isolated magnetic poles have not been observed. Thus, Statements 1, 2 and 3 are correct.

A closed Gaussian surface enclosing a bar magnet contains both the north and south poles of the magnet. Magnetic field lines emerge from the north pole and enter the south pole externally, then continue through the magnet to complete closed loops. As a result, the number of magnetic field lines entering the surface equals the number leaving it. Therefore, the net magnetic flux through the closed surface is zero. The statement that the surface encloses a net magnetic monopole is incorrect because isolated magnetic poles have never been observed experimentally. A bar magnet always contains both poles together. Thus, the correct answer is C. The surface encloses a net magnetic monopole.

Magnetic flux through a small surface element is defined as ΔφB = B · ΔS Since the expression involves a dot product, the resulting quantity is a scalar. The magnitude of flux depends on both the magnetic field strength and the angle between the magnetic field vector and the area vector. Mathematically, ΔφB = BΔS cosθ Thus, the flux changes as the orientation of the surface changes. Statement 3 is incorrect because an open surface can have non-zero magnetic flux whenever the magnetic field has a component normal to the surface. Therefore, Statements 1, 2 and 4 are correct.

According to Gauss's law for magnetism, ∮ B · dS = 0 Therefore, the net magnetic flux through any closed surface enclosing a bar magnet or a current loop is zero. Hence, Item 1 corresponds to "Zero" and Item 4 corresponds to "Always zero magnetic flux." For an electric dipole enclosed within a Gaussian surface, equal positive and negative charges are present. The net enclosed charge is therefore zero. Hence, Item 2 corresponds to "Net enclosed charge is zero." For a closed surface enclosing a positive charge +q, Gauss's law for electricity gives ΦE = q/ε₀ Hence, Item 3 corresponds to "q/ε₀." Thus, the correct matching is: 1 → d 2 → c 3 → a 4 → b

Even if a Gaussian surface is drawn around the north-pole region of a bar magnet, magnetic field lines do not begin or end there. The lines continue through the magnet and form complete closed loops. According to Gauss's law for magnetism, ∮ B · dS = 0 This means the total outward magnetic flux equals the total inward magnetic flux. Consequently, the number of field lines leaving the surface must equal the number entering it. Therefore, there is no net magnetic flux through the surface. Thus, the correct answer is B. Equal, Equal.

Gauss's law for magnetism states that the total magnetic flux through any closed surface is zero: ∮ B · dS = 0 This law is a direct consequence of the non-existence of isolated magnetic monopoles. Since magnetic field lines always form closed loops, every magnetic field line entering a closed surface must also leave it. The law does not depend on the geometry of the surface. Whether the surface is spherical, cubical, cylindrical, irregular, or encloses magnetic materials, the total magnetic flux through the closed surface remains zero. Thus, the correct answer is B. is true for any closed surface regardless of shape.

According to Gauss's law for magnetism, ∮ B · dS = 0 The cube forms a completely closed surface. In a uniform magnetic field, magnetic field lines enter some faces of the cube and leave through other faces. The inward flux and outward flux have equal magnitudes but opposite signs. Therefore, the total magnetic flux through the entire closed cube is zero. The side length and magnetic field strength are irrelevant because Gauss's law guarantees zero net magnetic flux through every closed surface. Thus, the correct answer is C. 0 Wb.

Perfectly parallel and equally spaced magnetic field lines represent a uniform magnetic field. In such a region, the field strength and direction remain constant. Because magnetic field lines neither originate nor terminate within the region, there are no magnetic sources or sinks present. This observation is consistent with the non-existence of magnetic monopoles. A plane perpendicular to the field would actually experience maximum flux rather than zero flux. Furthermore, Gauss's law remains completely valid in a uniform magnetic field. Thus, the correct answer is B. there are no magnetic sources or sinks within the region.

A diagram showing magnetic field lines emerging radially from a single point resembles the electric field of a positive electric charge. Such a pattern would imply the existence of an isolated magnetic pole or magnetic monopole. However, according to Gauss's law for magnetism, ∮ B · dS = 0 the net magnetic flux through any closed surface must always be zero. Therefore, magnetic field lines cannot originate from a single isolated point and diverge outward indefinitely. Such a diagram directly contradicts the experimentally established absence of magnetic monopoles. Thus, the correct answer is C. net magnetic flux over any closed surface must be zero.

Electrostatic field lines cannot form closed loops because they begin on positive charges and terminate on negative charges. Therefore, Statement 1 is correct. If magnetic field lines intersected, two different magnetic field directions would exist at the same point, which is impossible. Hence Statement 3 is correct. According to Ampere's law, closed magnetostatic field loops are associated with electric currents. Therefore Statement 4 is also correct. Statement 2 is considered incorrect in this context because a closed magnetostatic field loop around empty space requires an enclosed current-carrying conductor; empty space by itself cannot act as the source of the magnetic field. Thus, Statements 1, 3 and 4 are correct.

A toroid may be considered a solenoid bent into the shape of a circle. The magnetic field inside a toroid is confined almost entirely within the core region enclosed by its windings. The field lines form concentric closed loops within the toroid. According to Ampère's circuital law: ∮B⋅dl=μ_0I_(enc) Every magnetic field loop inside the toroid encloses the current carried by the windings. Therefore, it is incorrect to state that the enclosed region is completely devoid of current. The confinement of magnetic field lines inside the toroid makes the external magnetic field nearly zero. This is one of the major practical advantages of toroidal coils.

Magnetic fields are produced by moving electric charges. Around a current-carrying conductor, magnetic field lines form concentric closed loops. Ampère's circuital law relates the magnetic field around a closed path to the current enclosed: ∮B⋅dl=μ_0I_(enc) Therefore, a closed magnetic field loop is associated with a region through which current passes. Such loops represent the magnetic field produced by moving charges. Statement 1 is incorrect because magnetostatic field loops are associated with currents rather than completely empty regions. Statement 3 is also incorrect because Gauss's law of magnetism does not require magnetic monopoles. In fact, the absence of monopoles is one of the reasons the law predicts zero net magnetic flux through a closed surface.

In a long solenoid, the magnetic field inside is nearly uniform. Therefore, the field lines inside are parallel and equally spaced. Near the ends of a solenoid, magnetic field lines spread outward and curve away from the axis. These lines eventually connect and form complete closed loops, reflecting the absence of magnetic monopoles. If field lines outside a solenoid were perfectly straight and never curved back, they would not form closed loops. This would contradict the fundamental nature of magnetic field lines and violate Ampère's law as well as Gauss's law of magnetism. Hence: 1 → d 2 → a 3 → c 4 → b

Magnetic field lines emerge from the north pole and enter the south pole outside the magnet. To complete the loop, the field lines continue inside the magnet from south to north. Therefore, the direction of the magnetic field inside a bar magnet is: S→N According to Gauss's law of magnetism: ∮B⋅dS=0 This law applies to every closed surface, including a surface drawn around the north pole region of a magnet. The total magnetic flux entering the surface equals the total flux leaving it, giving zero net flux. Thus, the correct combination is: S to N and Zero.

Fringing refers to the spreading of magnetic field lines near the edges of magnetic pole pieces. In practice, magnetic field lines cannot remain perfectly straight all the way to the edges because magnetic field lines must form continuous closed loops. If the field lines remained perfectly parallel everywhere and terminated abruptly at the pole boundaries, the magnetic field pattern would contradict the fundamental properties of magnetic fields. Therefore, fringing is a natural consequence of the requirement that magnetic field lines curve and connect to form closed loops. It is not an experimental defect and cannot be completely eliminated. The phenomenon ensures consistency with the basic laws of magnetism, including Ampère's law and Gauss's law of magnetism.

Magnetic flux through an area is given by: Φ=BA Given: B=2.0 TA=0.1 m^2 Therefore, Φ_(total)=2.0×0.1=0.20 Wb Since 5% of the flux fringes outward, Φ_(fringe)=0.05×0.20Φ_(fringe)=0.01 Wb Hence, the straight flux connecting the two facing pole surfaces is: Φ_(straight)=0.20-0.01Φ_(straight)=0.19 Wb Unit Verification T×m^2=Wb Thus, the directly connecting magnetic flux equals 0.19 Wb.

In the hypothetical extension of Gauss's law for magnetism including magnetic monopoles, ∮B⋅dS=μ_0q_(m,enc) where q_(m,enc)represents the net magnetic charge enclosed. The surface contains: +q_m and -q_m Hence, q_(m,enc)=+q_m-q_m=0 Substituting: ∮B⋅dS=μ_0(0)=0 Thus even in a hypothetical universe containing magnetic monopoles, a closed surface enclosing equal positive and negative magnetic charges would have zero net magnetic flux.

Gauss's law for electrostatics is: ∮E⋅dS=q_(enc)/ε_0 The enclosed electric charge acts as the source of electric flux. If magnetic monopoles existed, a modified magnetic Gauss law would be: ∮B⋅dS=μ_0q_m Here, the enclosed magnetic charge q_m would serve exactly the same role as electric charge q in electrostatics. Thus: • Electric charge → Source of electric flux • Magnetic charge → Source of magnetic flux Therefore q_m is directly analogous to the enclosed electric charge q.

Karl Friedrich Gauss made extraordinary contributions to mathematics, astronomy, physics and geodesy. He displayed exceptional mathematical ability from childhood and is widely regarded as a mathematical prodigy. Gauss collaborated with the physicist Wilhelm Weber. Together they worked on investigations of terrestrial magnetism and communication systems. In 1833 they developed an early electric telegraph connecting laboratory buildings in Göttingen. His mathematical investigations of curved surfaces led to the concept of intrinsic geometry. These ideas later became the foundation upon which Bernhard Riemann developed Riemannian geometry, which ultimately became important in Einstein's general theory of relativity. Therefore all four statements are correct.

The scientific understanding of magnetism in terms of moving electric charges emerged during the nineteenth century through the work of scientists such as Ørsted, Ampère, Faraday and Maxwell. Before 1800, magnets were widely used in navigation and other applications, but their fundamental origin was not understood. Thus the statement claiming that a satisfactory moving-charge explanation existed before 1800 is historically incorrect. NCERT emphasizes that technological applications often precede complete scientific understanding. Magnets had been used for roughly two millennia before the development of modern electromagnetic theory. The first electric telegraph developed by Gauss and Weber in 1833 represents an example of scientific understanding leading to technological advancement.