Problem 27 - Master Resource Book in JEE Main Physics

Question

Consider a wire carrying a steady current, \(I\) placed in a uniform magnetic field \(\mathbf{B}\) perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do no work. This implies that, \(\quad\) [NCERT Exemplar] (a) motion of charges inside the conductor is unaffected by B since they do not absorb energy (b) some charges inside the wire move to the surface as a result of B (c) if the wire moves under the influence of \(\mathrm{B}\), no work is done by the force (d) if the wire moves under the influence of \(\mathrm{B}\), no work is done by the magnetic force on the ions, assumed fixed within the wire

Step-by-Step Solution

Answer

Options (a) and (d) are correct.

Step 1: Understand the Concept of Magnetic Force

The magnetic force is given by the formula \( \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \), where \( q \) is the charge, \( \mathbf{v} \) is the velocity of the charge, and \( \mathbf{B} \) is the magnetic field. It's important to note that this force acts perpendicular to the direction of motion, which means it does not do work on the charge since work is the dot product of force and displacement along the same line.

Step 2: Examine How Magnetic Force Affects Charges in the Wire

Charges inside the wire move due to electric fields, and the magnetic field exerts a force perpendicular to their velocity. However, this force does not change the kinetic energy of the charges because it does no work, meaning option (a) is correct since the motion of charges is unaffected energetically. However, the direction can change depending on the field and current.

Step 3: Evaluate Surface Charge Movement Possibility

The magnetic field does change the distribution of moving charges due to the Lorentz force, pushing them sideways, creating a potential difference between the edges of the wire. This can cause charges to move towards the surface, indicating that option (b) could be feasible.

Step 4: Analyze Work Done by Force on the Wire

When a wire moves within a magnetic field at a velocity, the force is perpendicular to its movement; therefore, external work must be done to move the wire against any magnetic effects. However, this work is not done through magnetic forces acting on individual charges within the wire; they do not change kinetic energy as per the principle that magnetic fields do no work, supporting option (d).

Key Concepts

Lorentz Force

The Lorentz Force is a fundamental concept when discussing the behavior of charges in a magnetic field. This force is described by the equation \[\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\] where \(q\) is the charge, \(\mathbf{v}\) is the velocity of the charge, and \(\mathbf{B}\) is the magnetic field. The interesting part of this equation is that the force - is perpendicular to both the velocity of the charge and the magnetic field, - does not change the kinetic energy of the particles because it acts at a right angle to their motion.This perpendicular nature means the Lorentz Force cannot do work because work is defined as the product of force in the direction of movement and displacement. Therefore, this force can change the direction in which a charge is moving, but it won't affect its speed. In essence, it feeds into the path without affecting energy exchange. This explains why motion of charges is energetically unaffected by magnetic fields, as mentioned in the original exercise options.

Work Done by Magnetic Fields

A critical aspect of magnetic fields is that they do no work on moving charges. To understand this, we need to recall a fundamental principle from physics: work is defined as the dot product of force and displacement. This means that for work to occur, the force applied must have a component along the direction of the object's movement. In magnetic forces, as shown by the Lorentz Force law, the force is always perpendicular to the direction of motion. Therefore, there is no component of force in the direction of displacement, so: - magnetic fields do not add or subtract energy from charges, - they only change the direction of their movement. This is why, even when a wire moves under the influence of a magnetic field, the work required to move it comes from an external source and not from the magnetic force itself. It also clarifies why options around magnetic force not doing work on the moving wire or its ions are correct.

Motion of Charges in Magnetic Field

The movement of charged particles in a magnetic field is intriguing due to its directional change effects. Because of the Lorentz Force, when a charged particle moves through a magnetic field: - its path becomes a curved trajectory, such as a circle or spiral, depending on its motion constraints, - the speed remains constant, but the path changes. Within a conductor, as in the original exercise scenario, this can lead to a phenomenon called the Hall Effect, where charges accumulate on one side of the conductor, creating a voltage difference. This effect evidences that while individual charges are pushed, no work is done on the charges energetically. Instead, their paths are redirected, possibly redistributing them to the surface. Hence, it aligns with the notion of charges possibly being pushed to edges or surfaces under the influence of a magnetic field, as outlined in the exercise's options.