CUET UG Physics Booster Test 2-Current Elements

In the Biot-Savart law, the source of the magnetic field is the vector quantity I dl. Hence, Vector source → I dl. The total magnetic field at a point is obtained by adding the contributions of all current elements vectorially. Therefore, Total magnetic field → Vector sum of individual source fields. The Biot-Savart law is applicable for steady currents. Thus, Steady current → Assumed condition for Biot-Savart calculation. Finally, I dl itself represents an infinitesimal current element. Therefore: 1-c,  2-d,  3-b,  4-a Hence, Option A is correct.

In the Biot-Savart law, the displacement vector r extends from the current element to the observation point P. Its magnitude appears in the denominator: dB∝1/r^2 Hence, it determines the inverse-square dependence of the magnetic field. The vector form of the law is: dB=μ_0/4πI(dl×r)/r^3 Therefore, r is involved in the cross product. However, r is not always perpendicular to dl. The angle between them may have any value from 0^∘to 180^∘. Thus, statements 1, 2 and 4 are correct.

The Biot-Savart law gives: dB∝I dl/r^2 New values: I^'=3Idl^'=2dlr^'=r/2 Therefore: dB^'∝(3I)(2dl)/(r/2)^2dB^'∝6I dl/r^2/4dB^'∝24I dl/r^2 Hence: dB^'/dB=24 Therefore, Option C is correct.

The Biot-Savart law contains the distance dependence: dB∝1/r^2 If the distance becomes: r^'=xr then: dB^'∝1/(xr)^2dB^'=1/x^2(1/r^2) Therefore: dB^'=x^(-2)dB Thus, increasing the distance by a factor x decreases the magnetic field by a factor x^2. Hence, Option B is correct.

The magnitude of a cross product is: ∣dl×r∣=dl r sin⁡θ The resulting vector is perpendicular to the plane containing dl and r. According to the right-hand screw rule, if the rotation from the first vector to the second vector is anticlockwise, the resulting vector points outward from the plane (towards the observer), not away from the observer. Therefore, the statement that an anticlockwise movement produces a vector directed away from you is incorrect. Hence, Option C is the incorrect statement.

In the Biot-Savart law, the magnetic field direction is determined by the cross product: dB∝dl×r The right-hand screw rule is used to determine the sense of the resulting vector. If the rotation from dl to r is clockwise, the screw advances away from the observer. Hence, the magnetic field vector points into the plane of the page. This direction is conventionally represented by a cross (×). If the rotation were anticlockwise, the field would point outward toward the observer and would be represented by a dot (∙). Therefore, clockwise rotation produces a magnetic field directed away from you. Hence, Option B is correct.

Maxwell's electromagnetic theory predicts that electromagnetic waves travel with speed: c=1/√(μ_0ε_0) This can be rewritten as: c=(μ_0ε_0)^(-1/2) Comparing with: c=(xμ_0ε_0)^(-y) we obtain: x=1 and y=1/2 This equation provided one of the strongest theoretical arguments that light itself is an electromagnetic wave. Hence, Option A is correct.

The permeability of free space is denoted by: μ_0 and appears in the Biot-Savart law: dB=μ_0/4πIdlsin⁡θ/r^2 This form is valid when the medium is vacuum (free space). When magnetic fields exist in another medium, the magnetic permeability of that medium replaces the permeability of free space. Therefore, the field formulation adapts according to the properties of the medium. Statement 2 is incorrect because μ_0 is associated with magnetic fields, whereas electrostatics primarily involves the permittivity of free space ε_0. Hence: • Statement 1 is correct. • Statement 3 is correct. • Statement 4 is correct. • Statement 2 is incorrect. Therefore, the correct answer is C. 1, 3 and 4 are correct.

Electrostatic fields arise from electric charge q, which is a scalar quantity. Hence: • Statement 1 is correct. • Statement 4 is correct. Magnetic fields arise from current elements represented by: I dl which possess both magnitude and direction. Therefore, the magnetic source is vector in nature. Thus, statement 2 is correct. Statement 3 is incorrect because electric charge remains a scalar quantity even at the macroscopic level. Therefore, statements 1, 2 and 4 are correct.

Coulomb's law for the electric field due to a point charge is: E=1/4πε_0q/r^2 Thus: E∝1/r^2 Similarly, the Biot-Savart law for a current element is: dB=μ_0/4πIdlsin⁡θ/r^2 Thus: dB∝1/r^2 Therefore, both electric and magnetic fields exhibit inverse-square distance dependence in their fundamental source laws. Hence, Option A is correct.

The magnetic field on the axis of a circular loop is: B=μ_0IR^2/2(R^2+x^2)^(3/2) At the center (x, 0): B_0=μ_0I/2R Therefore, B/B_0=R^3/(R^2+x^2)^(3/2) Given: R=3 cmx=4 cmR^2+x^2=3^2+4^2=25(R^2+x^2)^(3/2)=25^(3/2)=125R^3=27 Thus, B/B_0=27/125=0.216 Hence, B=0.216B_0 Therefore, Option A is correct.

Consider two diametrically opposite current elements on a circular loop. Each current element produces a magnetic field at a point on the axis. The components perpendicular to the axis are equal in magnitude but opposite in direction. Therefore, these perpendicular components cancel each other. The components along the axis point in the same direction and hence add together. This symmetry is the reason why the resultant magnetic field on the axis of a circular loop is directed entirely along the axis. Hence, the cancellation occurs due to the contribution of diametrically opposite current elements. Therefore, Option B is correct.

The magnetic field at the center of a circular loop is: B=μ_0I/2R Hence, B∝I/R If current becomes: I^'=xI and radius becomes: R^'=yR then B^'=μ_0(xI)/2(yR)B^'=x/y(μ_0I/2R)B^'=x/yB Therefore, Option A is correct.

For a tightly wound coil having N turns, the magnetic field at the center is: B=μ_0NI/2R This equation shows that the magnetic field is directly proportional to the number of turns. Therefore, increasing N increases the magnetic field. Since the coil is tightly wound, all turns may be considered to have approximately the same radius. The magnetic field contributions from all turns add according to the principle of superposition. Hence, the statement that the magnetic field decreases as N increases is incorrect. Therefore, Option C is correct.

A circular current loop produces a magnetic field pattern similar to that of a bar magnet. Using the right-hand thumb rule, one face of the loop acts as a north pole while the opposite face acts as a south pole. Magnetic field lines emerge from the north face and enter the south face, just as they do for a bar magnet. Therefore, when describing the two sides of a current loop, they can be regarded as magnetic north and south poles. Hence, the upper side and lower side can respectively be considered a north pole and a south pole. Therefore, Option A is correct.

The right-hand thumb rule is used to determine the direction of the magnetic field produced by a current-carrying circular loop. The fingers of the right hand are curled around the loop in the direction of the current. When this is done, the extended thumb points along the direction of the magnetic field through the axis of the loop. The thumb does not indicate the direction of the electric field. Instead, it indicates the magnetic field direction and helps identify the north pole of the loop. Therefore, statements 2, 3 and 4 are correct while statement 1 is incorrect.

According to the Biot-Savart law: dB=μ_0/4πIdlsin⁡θ/r^2 For the straight segments connected to the arc, the displacement vector r from the current element to the center lies along the same line as dl. Therefore: θ=0^∘ and dl×r=0 Since: sin⁡0^∘=0 the magnetic field contribution from these straight portions is zero. Hence, statements 1, 2 and 3 are correct while statement 4 is incorrect.

The Biot-Savart law is: dB=μ_0/4πIdlsin⁡θ/r^2 When the observation point lies on the extension of the current element, θ=0^∘ Therefore, sin⁡0^∘=0 and dB=0 Hence, magnetic field contribution becomes zero. When r is perpendicular to dl, θ=90^∘ and the field becomes maximum rather than zero. Therefore, statements 1, 3 and 4 are correct.

For every current element on a semi-circular arc, the displacement vector from the element to the center is radial. The current element dl is tangential to the arc and therefore perpendicular to r. As a result, dl×r has the same direction for all current elements. Therefore, every elemental magnetic field contribution points either into or out of the plane and all contributions add constructively. This is why the magnetic field due to a semi-circular arc can be obtained by summing all elemental contributions exactly as in the circular-loop derivation. Hence, Option A is correct.

The Biot-Savart law gives: dB=μ_0/4πIΔxsin⁡θ/r^2 Given: I=5 AΔx=2 cm=2×10^(-2) mr=1 mθ=90^∘ Substituting: dB=10^(-7)×(5)(2×10^(-2))(1)/1^2dB=10^(-7)×10^(-1)dB=10^(-8) T Therefore, dB=1×10^(-8) T Hence, Option D is correct.