Chapter 27 – Magnetic Fields and Magnetic Forces

Magnetic Force on a Charged Particle

The force experienced by a charged particle moving in a magnetic field is a fundamental concept in electromagnetism. This force is always perpendicular to both the velocity of the particle and the direction of the magnetic field.

• Magnetic Force Formula: The force F on a particle of charge q moving with velocity \vec{v} in a magnetic field \vec{B} is given by:

• Direction: Determined by the right-hand rule.

• Magnitude: , where θ is the angle between \vec{v} and \vec{B}.

• Example: An electron moving perpendicular to a uniform magnetic field experiences a force that causes it to move in a circular path.

Motion of Charged Particles: Velocity Selector and Mass Spectrometer

Charged particles in combined electric and magnetic fields can be filtered or analyzed based on their velocity or mass-to-charge ratio.

• Velocity Selector: Uses perpendicular electric and magnetic fields so that only particles with velocity pass through undeflected.

• Mass Spectrometer: Measures the radius of curvature of a particle's path in a magnetic field to determine its mass-to-charge ratio.

• Example: In a velocity selector, if V/m and T, only particles with m/s pass through straight.

Magnetic Force on a Current-Carrying Conductor

A current-carrying wire in a magnetic field experiences a force due to the collective effect of the magnetic forces on moving charges within the wire.

• Formula:

• Where: I is current, \vec{L} is the length vector in the direction of current, \vec{B} is the magnetic field.

• Example: A straight wire of length 0.5 m carrying 2 A perpendicular to a 0.3 T field experiences a force N.

Force and Torque on a Current Loop

A current loop in a magnetic field experiences a torque, which is the basis for electric motors and galvanometers.

• Torque Formula:

• Magnetic Dipole Moment: , where A is the area vector of the loop.

• Example: A rectangular loop in a uniform field experiences a torque that tends to align the loop's plane perpendicular to the field.

Chapter 28 – Sources of Magnetic Field

Magnetic Field of a Moving Charge

A moving point charge produces a magnetic field described by the Biot-Savart Law for a point charge.

• Formula:

• Where: is the permeability of free space, is the unit vector from the charge to the field point, is the distance.

• Example: An electron moving in a straight line creates a circular magnetic field around its path.

Magnetic Field of a Current Element, Wire, or Pair of Wires

Current-carrying wires generate magnetic fields, which can be calculated using the Biot-Savart Law or Ampère’s Law for symmetric cases.

• Biot-Savart Law (for a current element):

• Long Straight Wire: at distance r from the wire.

• Parallel Wires: Two parallel currents attract if in the same direction, repel if opposite.

• Example: The field at 5 cm from a wire carrying 10 A is T.

Magnetic Field of a Current Loop

A current loop produces a magnetic field along its axis, which is strongest at the center.

• On-axis Field at Center:

• Where: R is the radius of the loop.

• Example: A 1 A current in a 10 cm loop produces T at the center.

Ampère’s Law

Ampère’s Law relates the integrated magnetic field around a closed loop to the total current passing through the loop.

• Formula:

• Application: Useful for calculating fields in symmetric situations (e.g., inside a solenoid, around a wire).

• Example: The field inside a long solenoid is , where n is turns per unit length.

Chapter 29 – Electromagnetic Induction

Faraday’s Law and Lenz’s Law

Changing magnetic flux through a loop induces an electromotive force (emf). The direction of the induced emf opposes the change in flux (Lenz’s Law).

• Faraday’s Law:

• Magnetic Flux:

• Lenz’s Law: The induced emf creates a current whose magnetic field opposes the change in the original magnetic flux.

• Example: If the magnetic field through a loop increases, the induced current flows to produce a field opposing the increase.

Motional emf

A conductor moving through a magnetic field experiences an emf due to the motion of charges in the field.

• Formula for a straight conductor of length l moving at velocity v perpendicular to B:

• Example: A rod 0.5 m long moving at 3 m/s in a 0.2 T field has V induced across its ends.

Induced Magnetic Fields

Changing magnetic fields induce electric fields, which in turn can create currents and new magnetic fields. This is the basis for electromagnetic waves and many electrical devices.

• Key Point: The induced electric field is non-conservative and forms closed loops.

• Example: In a transformer, a changing current in the primary coil induces a voltage in the secondary coil via changing magnetic flux.

Additional info: This guide expands on the study questions by providing definitions, equations, and examples for each topic, ensuring a comprehensive review for exam preparation.