Lecture Notes: Chapter 27 - Magnetism

Key Concepts

• Magnetic Fields: Sources of magnetic fields can be traced to electrical currents.

• Interaction: Currents interact with magnetic fields, and the force on a wire with current length l by an external magnetic field B can be defined mathematically.

• Moving Charges: The interaction between the current and magnetic fields can be related to moving charges, defined by the formula:

  [ F = q (v \times B) ]

  This force is always perpendicular to the magnetic field.

Motion of Charges in Magnetic Fields

• Circular Motion: If the velocity of a charged particle is perpendicular to the magnetic field, the particle exhibits circular motion.
• Spiral Trajectory: If there are two velocity components (one parallel and one perpendicular to the magnetic field), the parallel component does not affect the force. The perpendicular component leads to a spiral trajectory.

Lorentz Force

• The combined effect of electric and magnetic fields is called the Lorentz force, defined as:

  [ F = qE + q(v \times B) ]

Magnetic Dipole Moment

• Torque on a Loop of Currents: The torque on a closed loop of current is defined as:

  [ \tau = \mu \times B ]

  • Here, \mu is the magnetic dipole moment.

• Potential Energy: The potential energy of a dipole in a magnetic field is given by:

  [ U = -\mu \cdot B ]

Hall Effect

• Concept: When a current passes through a conductor in a magnetic field, charge carriers experience a force, leading to charge accumulation and creating a Hall voltage.
• Application: The Hall voltage helps distinguish between positive and negative charge carriers.

Example Problem 1: Drift Velocity

1. Given: A copper strip (1.8 cm wide, 1 mm thick) is placed in a magnetic field (1.2 T) with a steady current (15 A). Hall emf = 1.02 µV.
2. Objective: Determine the drift velocity and density of free conducting electrons.
3. Equations Used:
   • Relate forces from the magnetic field and electric field to find drift velocity.
   • Calculate the density of free electrons using current density equations.

Example Problem 2: Current-Carrying Loop in Magnetic Field

1. Setup: Analyze a current-carrying circular loop partially immersed in a magnetic field.
2. Objective: Determine net force on the loop based on angle and current.
3. Method: Use integration and geometric arguments to find the net force.

Example Problem 3: Helical Path of Electrons

1. Given: An electron enters a magnetic field (0.28 T) at 45 degrees.
2. Objective: Determine the radius and pitch of its helical path.
3. Approach: Decompose velocity and calculate using circular motion equations.

Example Problem 4: Balancing Forces with Magnetic Field

1. Setup: A conducting rod placed on a fulcrum in a magnetic field with a weight attached.
2. Objective: Find the current required to balance the rod.
3. Method: Use torque equations around the fulcrum and apply the right-hand rule to determine current direction.

Summary

• The interaction of electric currents with magnetic fields leads to various effects and applications in physics, including the behavior of charged particles in fields and the analysis of magnetic dipole moments.