CUET UG Physics Booster Test 2- Current Loop Dipoles and Meters

A current-carrying conductor produces a magnetic field around itself. Therefore, conductor 'a' generates a magnetic field B_a at the position of conductor 'b'. Since conductor 'b' carries current I_b, it experiences a magnetic force due to this field. The force is given by F_(ba)=I_bL×B_a The direction of this force is determined using the right-hand rule for vector cross products. For steady currents, Newton's third law remains valid, meaning the force exerted by conductor 'a' on conductor 'b' is equal in magnitude and opposite in direction to the force exerted by conductor 'b' on conductor 'a'. Hence Statements 1, 2 and 3 are correct, while Statement 4 is incorrect.

When two long parallel conductors carry currents, each conductor produces a magnetic field that influences the other. If the currents are in the same direction, the conductors attract; if they are in opposite directions, they repel. Newton's third law is satisfied because the forces are equal and opposite. However, the magnetic fields generated by the conductors do not completely cancel everywhere in space. Depending on the direction of current and location of observation, the fields may add or partially cancel. Therefore, the statement claiming perfect structural cancellation of the magnetic field is incorrect.

The historical definition of the ampere is based on the force between two long parallel conductors. The accepted force value was 2×10^(-7) N/m for currents of 1 A in conductors separated by 1 m. The current balance was the practical instrument used to measure such forces. The year 1946 is associated with the official adoption of the theoretical ampere definition. One coulomb is the amount of charge transported by a steady current of 1 ampere in one second: Q=It=(1)(1)=1C Therefore, the correct matching is 1-c, 2-d, 3-b and 4-a.

The force per unit length between two long parallel conductors is F/L=μ_0I_1I_2/2πd Substituting I_1=8A,I_2=5A,d=0.04mF/L=4π×10^(-7)×8×5/2π×0.04=2×10^(-4)N/m For a length L=0.10mF=(2×10^(-4))(0.10)F=2×10^(-5)N Hence the force on the 10 cm section is 2.0×10^(-5)N

The torque on a current-carrying coil is τ=NIABsin⁡θ where θis the angle between the magnetic moment (normal to the coil) and the magnetic field. Given: N=20,I=12Aa=10cm=0.10mA=(0.10)^2=0.01m^2B=0.80T,θ=30^∘ Substituting, τ=20×12×0.01×0.80×sin⁡30^∘=20×12×0.01×0.80×0.5=0.96 N m Therefore, the torque experienced by the coil is 0.96 N m.

When a current-carrying rectangular loop is placed in a magnetic field, the forces acting on the opposite sides of the loop are equal in magnitude and opposite in direction. These forces do not act along the same line of action. Instead, they are separated by a perpendicular distance, thereby forming a couple. A couple produces a turning effect or torque but no net translational force. This torque tends to rotate the loop so that its magnetic moment aligns with the external magnetic field. The rotation continues until the coil reaches an equilibrium position. Therefore, the forces are non-collinear and produce rotational motion.

The magnetic dipole moment of a circular coil is m=NIA where N is the number of turns, I is the current and A is the area of the coil. For a circular coil, A=πr^2 Given: N=100,I=3.2A,r=0.1mA=π(0.1)^2A=0.0314 m^2 Therefore, m=100×3.2×0.0314m=10.048 A m^2m≈10.05 A m^2 Hence, the magnetic moment is approximately 10.05 A m^2.

The magnetic dipole moment of a current loop is defined as m=IA for a single-turn loop and m=NIA for an N-turn coil. The magnetic moment plays a role in magnetism similar to that played by electric dipole moment in electrostatics. Both quantities determine how a dipole interacts with an external field and both experience torque tending to align them with the field. The direction of magnetic moment is determined using the right-hand rule, not the left-hand rule. Furthermore, magnetic moment is an intrinsic property of the current loop and does not depend on the external magnetic field. Its dimensions are [A][L^2].

The torque on a magnetic dipole is τ=mBsin⁡θ When θ=0^∘, τ=0 and the dipole is in stable equilibrium because its potential energy is minimum. At θ=90^∘ the value of sinθbecomes unity and the torque reaches its maximum value. At θ=180^∘ the torque again becomes zero, but this corresponds to unstable equilibrium because the potential energy is maximum. A slight disturbance causes the dipole to move away from this position. Therefore, Statements 1, 2 and 3 are correct while Statement 4 is incorrect because the magnetic torque attempts to align the dipole with the field (θ=0^∘), not with θ=180^∘.

For a magnetic dipole placed in a uniform magnetic field, τ=mBsin⁡θ When θ=180^∘sin⁡180^∘=0 Therefore, τ=0 Although the torque is zero, this position corresponds to maximum potential energy and hence unstable equilibrium. Any slight displacement causes a torque that tends to move the dipole farther from the antiparallel configuration. Thus, the statement claiming that the torque is maximum at 180^∘is incorrect. The torque is actually maximum at 90^∘.

At distances much larger than the radius of a current loop, the magnetic field produced by the loop resembles that of a magnetic dipole. The axial magnetic field expression reduces to B∝1/x^3 when x≫R. This inverse cube dependence is exactly the same distance dependence observed for the electric field of an electric dipole at large distances. Because of this similarity, a current loop is regarded as the magnetic analogue of an electric dipole. The 1/x^3 factor indicates rapid weakening of the field with distance and is one of the defining characteristics of dipole fields. Therefore, the correct interpretation is that the magnetic field of a current loop decays in the same manner as the field of an electric dipole.

Many mathematical similarities exist between electrostatics and magnetism. The electric dipole moment p corresponds to the magnetic dipole moment m. Similarly, the electric field E corresponds to the magnetic field B. The electrostatic constant 1/ε_0 plays a role analogous to the magnetic constant μ_0. Furthermore, both electric dipole fields and magnetic dipole fields decrease with distance according to the inverse cube law: 1/x^3 These analogies help in understanding magnetic dipoles using concepts already familiar from electrostatics. Therefore, the correct matching is 1-c, 2-a, 3-b and 4-d.

The torque on a current-carrying coil is τ=NIABsin⁡θ where θis the angle between the magnetic moment vector and the magnetic field. In a moving coil galvanometer, a radial magnetic field is produced using specially shaped pole pieces and a soft iron core. This arrangement ensures that the magnetic field is always perpendicular to the area vector of the coil. Therefore, θ=90^∘ throughout the motion of the coil. Hence, sin⁡90^∘=1 and the torque expression becomes τ=NIAB This ensures maximum torque and a linear relationship between current and deflection.

The cylindrical soft iron core placed inside a moving coil galvanometer serves two important functions. First, because soft iron has high magnetic permeability, it concentrates magnetic flux and significantly increases the magnetic field strength inside the instrument. Second, together with the curved pole pieces, it helps create a radial magnetic field. A stronger magnetic field increases the magnetic torque acting on the coil for a given current, thereby improving the sensitivity of the galvanometer. The soft iron core does not oppose the radial field, reduce sensitivity, or eliminate the restoring torque provided by the suspension spring.

For a moving coil galvanometer in equilibrium, τ=kϕ where τis the magnetic torque, k is the torsional constant and ϕis the angular deflection. Given: τ=1.5×10^(-3) N mϕ=0.5 rad Therefore, k=τ/ϕ=1.5×10^(-3)/0.5=3.0×10^(-3) N m rad^(-1) Thus, the torsional constant of the suspension spring is 3.0×10^(-3) N m rad^(-1)

For a moving coil galvanometer, NIAB=kϕ Therefore, ϕ=(NAB/k)I This equation shows that the angular deflection is directly proportional to the current passing through the coil. Hence Statement 1 is correct. The spring constant k appears in the denominator, so increasing k reduces the deflection for the same current, making Statement 2 correct. The factor (NAB/k)remains fixed for a particular galvanometer and is therefore a constant characteristic of that instrument, making Statement 4 correct. Statement 3 is incorrect because increasing the area A increases the torque and consequently increases the deflection.

Current sensitivity is given by S_i=ϕ/I=NAB/k Thus, doubling the number of turns doubles the current sensitivity. However, increasing the number of turns requires additional wire length, which increases the resistance of the coil. Therefore, the internal resistance does not remain constant. Voltage sensitivity is S_v=ϕ/V=NAB/kR When the number of turns is doubled, both N and R increase approximately proportionally. Consequently, voltage sensitivity remains nearly unchanged. Hence Statement C is the incorrect statement.

Current sensitivity is S_i=NAB/k Hence, if N→2N then S_i→2S_i Therefore, current sensitivity doubles. Voltage sensitivity is S_v=NAB/kR When the number of turns doubles, the wire length and resistance approximately double as well. Therefore, S_v=2NAB/k(2R)=NAB/kR Thus, voltage sensitivity remains approximately unchanged. Hence the correct answer is "Doubles, Unchanged."

The effective resistance of the ammeter is R_A=R_Gr_s/R_G+r_s where R_G is the galvanometer resistance and r_s is the shunt resistance. Since R_G≫r_s the denominator becomes approximately R_G. Thus, R_A≈R_Gr_s/R_GR_A≈r_s Therefore, the total resistance is approximately equal to the shunt resistance. This is desirable because an ammeter should have very low resistance so that it does not significantly alter the circuit current.

To convert a galvanometer into a voltmeter, a large resistance called a multiplier resistance is connected in series with the galvanometer coil. This increases the total resistance of the instrument significantly. As a result, only a very small current flows through the voltmeter, minimizing disturbance to the circuit being measured. A voltmeter is always connected in parallel across the component whose potential difference is to be measured. Therefore, Statement 1 is incorrect. Statement 4 is also incorrect because ammeter conversion uses a low-resistance shunt connected in parallel, which is fundamentally different from voltmeter conversion.