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Difficulty: Very hardMagnetic Force and Electromagnetism

Match each physical electromagnetic configuration listed under Column I with its corresponding net magnetic force or motion characteristic listed under Column II.

  • A charged particle projected parallel to the lines of a uniform magnetic fieldExperiences zero magnetic force (F=0F = 0) and maintains its initial linear trajectory
  • Two long, straight parallel conductors carrying electric currents in opposite directionsExperience a mutually repulsive force per unit length given by FL=μ0I1I22πd\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}
  • A current-carrying rectangular wire coil positioned with its plane parallel to a uniform magnetic fieldExperiences maximum net magnetic torque (τ=NIAB\tau = N I A B) while the net translational magnetic force is zero
  • A charged particle injected perpendicularly into a uniform magnetic fieldFollows a uniform circular path due to a constant magnetic centripetal force (F=qvBF = qvB)

Answer

1. Charged particle moving parallel to magnetic field lines →\rightarrow Experiences zero magnetic force (F=0F = 0) and maintains its initial linear trajectory.
2. Parallel conductors carrying opposite currents →\rightarrow Experience a mutually repulsive force per unit length given by FL=μ0I1I22πd\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}.
3. Current-carrying rectangular coil parallel to magnetic field →\rightarrow Experiences maximum net magnetic torque (τ=NIAB\tau = N I A B) while net translational force is zero.
4. Charged particle injected perpendicularly into magnetic field →\rightarrow Follows a uniform circular path due to a constant magnetic centripetal force (F=qvBF = qvB).
Each electromagnetic system is correctly paired based on the vector cross-product rules governing magnetic force (F=q(v×B)F = q(\mathbf{v} \times \mathbf{B}) and F=I(L×B)F = I(\mathbf{L} \times \mathbf{B})). Parallel motion produces zero force due to zero angle; opposite currents repel per unit length according to Ampère's law; a parallel loop experiences maximal couple/torque without linear displacement; and perpendicular particle velocity results in a constant centripetal deflection into circular motion.

Step-by-Step Solution

1
Analyze the angle θ\theta between velocity and magnetic field for a parallel moving charge
Since velocity is parallel to the field, θ=0∘\theta = 0^\circ. The magnetic Lorentz force formula F=qvBsin⁡θF = qvB\sin\theta yields F=0F = 0, meaning no deflection occurs.
Magnetic forces require a non-zero perpendicular component of motion relative to the magnetic field direction.
2
Determine the interaction force between parallel conductors with opposite currents
By applying the magnetic field rule for long straight conductors and Fleming's left-hand rule for the resulting force, currents flowing in opposite directions repel each other with force per unit length FL=μ0I1I22πd\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}.
Opposite current directions generate magnetic field lines between the wires that reinforce each other, creating a high-density magnetic field region that pushes the conductors apart.
3
Evaluate forces and torque on a rectangular coil aligned parallel to magnetic field lines
The forces on opposite arms are equal in magnitude and opposite in direction, canceling out net linear translation (Fnet=0F_{\text{net}} = 0). However, they act along different lines of action, producing a maximum torque τ=NIABcos⁡0∘=NIAB\tau = N I A B \cos 0^\circ = N I A B.
Maximum torque occurs when the plane of the coil is parallel to the field because the lever arm for the magnetic forces on the side conductors is at its maximum length.
4
Evaluate the trajectory of a charged particle entering perpendicularly into a magnetic field
At θ=90∘\theta = 90^\circ, sin⁡90∘=1\sin 90^\circ = 1, giving a constant magnetic force F=qvBF = qvB. Because vector force is perpendicular to velocity at every instant, it changes only the direction of velocity, driving the particle into a circular orbit of radius r=mvqBr = \frac{mv}{qB}.
A constant magnitude force acting perpendicular to the instantaneous velocity vectors fulfills the exact condition for centripetal acceleration.

Key Concept

Magnetic forces on moving charges and current-carrying conductors under specific geometric alignments and current configurations
Estimated Time:3m 0s
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Topics

Magnetic Force and Electromagnetism

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