Chapter 4: Moving Charges and Magnetism Case Study

Case Study Questions

Case Study 1

A rectangular loop of wire is placed in a uniform magnetic field. The plane of the loop is initially perpendicular to the magnetic field. The loop is then rotated about an axis parallel to the field at a constant angular speed. The magnetic flux through the loop changes with time.

Questions:

1. What is the expression for the magnetic flux through the loop as a function of time?
2. How does the induced emf in the loop vary with time?
3. Explain the significance of Lenz’s law in this context.

Answers:

1. The magnetic flux through the loop as a function of time is given by:Φ(t) = B × A × cos(ωt), where B is the magnetic field strength, A is the area of the loop, and ω is the angular speed.
2. The induced emf in the loop varies with time as:ε(t) = -dΦ/dt = B × A × ω × sin(ωt).
3. Lenz’s law states that the direction of the induced emf and hence the induced current in the loop will be such that it opposes the change in magnetic flux that produced it. This ensures the conservation of energy.

Case Study 2

Consider a long straight wire carrying a steady current I. A small compass is placed near the wire, and it is observed that the compass needle aligns itself perpendicular to the wire when the current is switched on.

Questions:

1. What is the direction of the magnetic field produced by the current-carrying wire?
2. How does the strength of the magnetic field vary with distance from the wire?
3. Explain how the right-hand thumb rule is applied to determine the direction of the magnetic field.

Answers:

1. The direction of the magnetic field produced by the current-carrying wire is tangential to the circular field lines surrounding the wire and follows the right-hand thumb rule.
2. The strength of the magnetic field varies inversely with the distance from the wire. It is given by:B = (μ₀I)/(2πr), where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
3. The right-hand thumb rule states that if you grasp the wire with your right hand such that the thumb points in the direction of the current, the fingers curl in the direction of the magnetic field lines.

Case Study 3

A charged particle enters a region of uniform magnetic field with its velocity perpendicular to the field. The particle undergoes circular motion due to the magnetic force acting as a centripetal force.

Questions:

1. Derive the expression for the radius of the circular path of the particle.
2. What factors affect the radius of the circular path?
3. Explain why the speed of the particle remains constant in the magnetic field.

Answers:

1. The radius of the circular path is given by:r = (mv)/(qB), where m is the mass of the particle, v is its speed, q is its charge, and B is the magnetic field strength.
2. The factors affecting the radius of the circular path are the mass and speed of the particle, its charge, and the strength of the magnetic field.
3. The speed of the particle remains constant because the magnetic force acts perpendicular to the velocity, doing no work on the particle, and hence does not change its kinetic energy.

Case Study 4

In an experiment, a solenoid with a large number of turns per unit length is connected to a battery and a switch. When the switch is closed, a current flows through the solenoid, creating a magnetic field inside it.

Questions:

1. Describe the magnetic field inside the solenoid.
2. How can the strength of the magnetic field inside the solenoid be increased?
3. Explain the role of the core material, if any, placed inside the solenoid.

Answers:

1. The magnetic field inside the solenoid is uniform and parallel to the axis of the solenoid. The field lines are straight and equally spaced.
2. The strength of the magnetic field inside the solenoid can be increased by increasing the current through the solenoid, increasing the number of turns per unit length, or by using a core material with high magnetic permeability.
3. The core material placed inside the solenoid enhances the magnetic field by providing a path of low reluctance for the magnetic flux, thereby increasing the field strength inside the solenoid.

Case Study 5

A rectangular loop of wire is moving with a constant velocity through a uniform magnetic field directed perpendicular to the plane of the loop. The loop enters and exits the magnetic field region.

Questions:

1. What is the emf induced in the loop while it is entirely inside the magnetic field region?
2. Describe the variation of induced emf as the loop enters and exits the magnetic field region.
3. Explain how Faraday’s law of electromagnetic induction applies to this situation.

Answers:

1. The emf induced in the loop while it is entirely inside the magnetic field region is zero because the magnetic flux through the loop is constant.
2. As the loop enters the magnetic field region, the induced emf increases to a maximum and then decreases to zero when the loop is fully inside the field. As the loop exits the magnetic field, the induced emf increases to a maximum in the opposite direction and then decreases to zero when the loop is completely out of the field.
3. Faraday’s law of electromagnetic induction states that the induced emf in a loop is equal to the negative rate of change of magnetic flux through the loop. This applies to the situation as the change in flux as the loop enters and exits the magnetic field induces an emf in the loop.