CUET UG Physics Booster Test 3- Current Loop Dipoles and Meters

The magnetic force on a current-carrying conductor is F=ILBsin⁡θ Hence, force per unit length is F/L=IBsin⁡θ Given: I=1AB=3.0×10^(-5)T The current flows from east to west, while the magnetic field points from south to north. These directions are perpendicular, so θ=90^∘ and sin⁡90^∘=1 Thus, F/L=1×3.0×10^(-5)=3.0×10^(-5)N m^(-1) Using the right-hand rule for F=IL×B, the force is directed vertically downward. Therefore, the correct answer is 3.0×10^(-5)N m^(-1)downward.

The force per unit length between two long parallel conductors is F/L=μ_0I_1I_2/2πd The direction of the force depends on the relative directions of the currents. If the currents flow in the same direction, the conductors attract each other. If the currents flow in opposite directions, they repel each other. In this problem, conductor A carries current from south to north while conductor B carries current from north to south. Thus, the currents are antiparallel. Therefore, the conductors exert equal and opposite lateral forces on each other, resulting in repulsion.

The historical definition of the ampere is based on the magnetic force between current-carrying conductors. In practical measurements, the effect of Earth's magnetic field and other stray magnetic fields must be minimized to obtain accurate results. Since infinitely long straight conductors cannot be realized experimentally, multiturn coils are often used to generate measurable magnetic forces. A current balance is employed to measure these forces accurately. The definition is based on magnetic interactions between currents rather than electrostatic inverse-square laws. Therefore, Statements 1, 2 and 3 are correct, while Statement 4 is incorrect.

The classical definition of the ampere states that two infinitely long parallel conductors of negligible cross-section placed 1 metre apart in vacuum carrying equal currents of 1 ampere experience a force of 2×10^(-7)N/m between them. This definition is based on the magnetic interaction produced by steady currents. It does not involve electromagnetic induction or the rate of change of magnetic flux. Concepts involving changing flux belong to Faraday's law of electromagnetic induction, which is entirely different from the definition of the ampere. Therefore, Statement D is incorrect.

A current-carrying rectangular loop placed in a uniform magnetic field experiences forces on its sides. These forces are equal and opposite, so the resultant translational force on the loop is zero. However, the forces act at different points and form a couple that produces a torque. The magnetic dipole moment of the loop is m=NIA and the torque is given by τ=mBsin⁡θ Therefore, τ=NIABsin⁡θ where θis the angle between the magnetic moment vector (normal to the loop) and the magnetic field. Thus, the net force is zero while the torque is NIABsin⁡θ.

The torque on a current-carrying loop placed in a magnetic field is given by τ=NIABsin⁡θ where θis the angle between the area vector (or magnetic moment vector) and the magnetic field. When the area vector is parallel to the magnetic field, θ=0^∘ Therefore, τ=NIABsin⁡0^∘τ=0 This corresponds to a stable equilibrium position because the magnetic moment is aligned with the magnetic field. In this orientation, no turning effect acts on the loop. Hence, the torque is zero.

Classically, magnetic dipoles are associated with circulating currents. However, elementary particles such as electrons possess intrinsic magnetic moments that cannot be explained purely by classical current loops. These magnetic moments arise from quantum mechanical properties such as spin. Modern physics recognizes that electrons, protons, and several other particles possess intrinsic magnetic moments. Magnetic monopoles, on the other hand, have not been experimentally observed. Therefore, the statement regarding intrinsic magnetic moments of elementary particles is correct.

For a given length of wire, a circle encloses the maximum possible area. When a flexible current-carrying loop is placed in an external magnetic field, magnetic forces act on different parts of the wire. The system tends to arrange itself in a configuration that maximizes the magnetic flux linked with the loop. Since magnetic flux is Φ=BAcos⁡θ increasing the enclosed area increases the flux. Because a circle provides the maximum area for a fixed perimeter, the loop tends to assume a circular shape. This configuration is energetically favorable and is analogous to the behavior of soap films seeking configurations of minimum energy.

The potential energy of a magnetic dipole in a magnetic field is U=-mBcos⁡θ At θ=0^∘ the potential energy is minimum, corresponding to stable equilibrium. At θ=180^∘ the potential energy is maximum, corresponding to unstable equilibrium. To rotate the dipole from the stable position to the unstable position, external work must be done against the magnetic torque. The magnetic moment does not naturally remain at 90^∘, because that position is neither stable nor unstable equilibrium.

For a magnetic dipole in a magnetic field, τ=mBsin⁡θ At the unstable equilibrium position, θ=180^∘ Therefore, τ=mBsin⁡180^∘=0 Although the torque is zero at this exact orientation, the position is unstable because the potential energy is maximum. Any small angular displacement produces a torque that drives the dipole farther away from the equilibrium position rather than restoring it. Hence, the defining characteristic of unstable equilibrium is that the initial torque is zero, but the system moves away under slight disturbance.

For a circular current loop, the magnetic field at a very large distance behaves like a magnetic dipole field and decreases according to B∝1/x^3 The magnetic field at the center of a circular loop is B=μ_0I/2R The magnetic moment of a single-turn circular loop is m=IA Since A=πR^2 we obtain m=IπR^2 Thus, the loop area corresponds to πR^2, the magnetic moment corresponds to IπR^2, the center field corresponds to μ_0I/2R, and the far-field varies as 1/x^3. Hence the correct matching is 1-c, 2-d, 3-b and 4-a.

Electric fields originate from electric charges, which are scalar sources. In contrast, magnetic fields are generated by moving charges and current distributions. No isolated magnetic monopoles have been experimentally confirmed. Therefore, magnetic fields cannot be described as arising from scalar sources identical to electric charge points. At large distances, a current loop behaves like a magnetic dipole, and many mathematical analogies exist between electrostatics and magnetism, including the correspondence between 1/ε_0 and μ_0. Hence, Statement C is incorrect.

The cylindrical soft iron core inside a moving coil galvanometer concentrates magnetic flux because of its high magnetic permeability. This increases the magnetic field strength inside the instrument. Along with the specially shaped pole pieces, the core helps establish a radial magnetic field. A radial field ensures that the angle between the magnetic field and the plane normal remains fixed at 90^∘, thereby maintaining maximum torque. Hence, the internal core both strengthens the field and makes it radial.

The torque on the galvanometer coil is τ=NIABsin⁡θ The soft iron core and curved pole pieces create a nearly uniform radial magnetic field. Because of this arrangement, the magnetic field remains perpendicular to the area vector of the coil throughout its rotation. Therefore, θ=90^∘ at every position and sin⁡θ=1 Consequently, τ=NIAB which ensures that torque is directly proportional to current. This produces a linear scale and high sensitivity, making the galvanometer suitable for accurate current measurement.

In a moving coil galvanometer, the suspension spring provides a restoring torque proportional to the angular deflection: τ_r=kϕ where k is the torsional constant and ϕis the angular deflection. The magnetic torque acting on the coil is τ_m=NIAB Under equilibrium conditions, NIAB=kϕ The torsional constant k physically represents the restoring torque produced per unit angular twist of the spring. Torsional torque does not depend exponentially on the number of turns N; instead, N affects the magnetic torque linearly. Therefore, Statements 1, 2 and 3 are correct while Statement 4 is incorrect.

For a moving coil galvanometer operating in equilibrium, NIAB=kϕ where N is the number of turns, A is the coil area, B is the magnetic field strength, k is the torsional constant and ϕis the angular deflection. Rearranging for current, I=kϕ/NAB or I=(k/NAB)ϕ This equation shows that current is directly proportional to deflection. The proportionality constant depends on the physical construction of the galvanometer. Hence, the correct expression for current is Option D.

Current sensitivity is S_i=NAB/k Since both galvanometers have identical springs, k is the same. Therefore, S_(i2)/S_(i1)=N_2A_2B_2/N_1A_1B_1 Substituting values, =42×1.8×10^(-3)×0.50/30×3.6×10^(-3)×0.25=37.8×10^(-3)/27×10^(-3)=1.4 Hence, M_2/M_1=1.4

Voltage sensitivity is S_v=ϕ/V=NAB/kR Thus, S_(v2)/S_(v1)=N_2A_2B_2/N_1A_1B_1×R_1/R_2 From the previous question, N_2A_2B_2/N_1A_1B_1=1.4 Therefore, S_(v2)/S_(v1)=1.4×10/14=1.4×0.714≈1.0 Hence, the voltage sensitivities are equal.

To convert a galvanometer into an ammeter, a very small resistance called a shunt resistance is connected in parallel with the galvanometer coil. Because the shunt resistance is much smaller than the galvanometer resistance, most of the current bypasses the galvanometer and flows through the shunt. The equivalent resistance of two parallel resistors is less than the smaller resistor, making the total resistance slightly less than r_s. This ensures that the ammeter introduces minimal resistance into the circuit. An ammeter should minimize voltage drop rather than maximize it. Therefore, Statements 1, 2 and 3 are correct, while Statement 4 is incorrect.

An ideal ammeter should measure current without affecting the circuit. Therefore, its internal resistance should be zero. Since current must pass through the instrument, it is connected in series with the circuit element. An ideal voltmeter should measure potential difference without drawing current. Therefore, its internal resistance should be infinite. Since it measures voltage across a component, it is connected in parallel with that component. Thus the correct matching is: 1-b,2-a,3-c,4-d