Week 6 Lec. 2

Magnetic Force on Moving Charges

Force on a Charged Particle in a Magnetic Field

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

• Formula: The force is given by $\vec{F} = q\vec{v} \times \vec{B}$, where q is the charge, v is the velocity vector, and B is the magnetic field vector.

• Direction: Determined by the right-hand rule for positive charges and left-hand rule for negative charges.

• Magnitude: $F = |q|vB\sin\theta$, where $\theta$ is the angle between v and B.

• Example: A particle with charge $q = -1\ \text{C}$ moving in the positive z-direction at $5\ \text{m/s}$ in a magnetic field $\vec{B} = 3\hat{x} + 4\hat{y}\ \text{T}$ experiences a force calculated using the cross product.

Application: The path of a charged particle in a magnetic field is curved, as shown in the example below.

Path of a Charge in a Magnetic Field

Direction and Magnitude of Magnetic Fields

When a charged particle enters a region with a magnetic field, its trajectory is influenced by the field's direction and strength. The curvature of the path provides information about the field.

• Direction: The direction of the magnetic field can be deduced from the curvature of the particle's path using the right-hand rule.

• Magnitude: The radius of curvature is inversely proportional to the magnetic field strength; a tighter curve indicates a stronger field.

• Example: Comparing two chambers, if the path curves more in chamber 1 than in chamber 2, then $|B_1| > |B_2|$.

Electric Current and Magnetic Fields

Historical Discovery and Empirical Evidence

Electric currents produce magnetic fields, a phenomenon first observed by Hans Oersted in 1820. The orientation and direction of current flow affect the magnetic field produced.

• Oersted's Experiment: A compass needle placed near a current-carrying wire deflects, indicating the presence of a magnetic field.

• Direction of Deflection: Depends on the direction of current flow.

• Key Concept: There is a direct relationship between electric current and magnetic field generation.

Magnetic Force on a Current-Carrying Conductor

Force on a Straight Wire

A current-carrying wire placed in a magnetic field experiences a force. This force is the result of the collective action of the magnetic force on each moving charge within the wire.

• Formula: $\vec{F} = I\vec{L} \times \vec{B}$, where I is the current, L is the length vector of the wire, and B is the magnetic field.

• Direction: Perpendicular to both the wire and the magnetic field, given by the right-hand rule.

• Magnitude: $F = ILB\sin\theta$.

• Example: The force on a wire segment in a uniform magnetic field can be measured experimentally.

Force on a Non-Straight Conductor

For conductors that are not straight, the force is calculated by integrating over the length of the conductor:

• Formula: $\vec{F} = I \int d\vec{s} \times \vec{B}$, where $d\vec{s}$ is an infinitesimal segment of the wire.

• Application: Used for loops, arcs, and other shapes.

Direction of Magnetic Force

Right-Hand Rule

The direction of the magnetic force on a current-carrying conductor is always perpendicular to both the current and the magnetic field. The right-hand rule is used to determine this direction.

• Right-Hand Rule: Point your thumb in the direction of current, fingers in the direction of the magnetic field, and your palm points in the direction of the force.

• Example: For a wire at an angle $\phi$ to the field, $F = ILB\sin\phi$.

Magnetic Force and Torque on a Current Loop

Force and Torque on a Current Loop

A current loop in a uniform magnetic field experiences no net force, but it does experience a torque. This torque is responsible for the operation of electric motors.

• Net Force: The translational forces on opposite sides of the loop cancel out.

• Torque Formula: $\tau = IAB\sin\theta$, where A is the area of the loop and $\theta$ is the angle between the normal to the loop and the magnetic field.

• Application: The torque causes the loop to rotate, converting electrical energy into mechanical energy.

Summary Table: Magnetic Force on Conductors

Situation	Formula	Direction
Moving charge in B	$\vec{F} = q\vec{v} \times \vec{B}$	Perpendicular to v and B
Straight wire in B	$\vec{F} = I\vec{L} \times \vec{B}$	Perpendicular to I and B
Current loop in B	$\tau = IAB\sin\theta$	Causes rotation (torque)

Additional info: The notes also reference the Hall effect, mass spectrometer, and magnetic flux, which are related topics in the study of magnetic fields and forces on charges and currents.