Magnetic Field and Magnetic Forces: Study Notes

Magnetism and Waves

Introduction to Magnetism

Magnetism is a fundamental physical phenomenon arising from the motion of electric charges, resulting in attractive and repulsive forces between objects. It is closely related to electricity and is essential in many technological applications.

• Magnets have two poles: North (N) and South (S).

• Opposite poles attract, while like poles repel.

• Magnets can attract certain uncharged objects, such as iron.

• Magnetic phenomena are distinct from electric phenomena, but both involve forces and fields.

Examples of applications: MRI machines, compasses, electric motors, hard drives, and magnetic storage.

Magnetic Poles and Magnetic Charge

Unlike electric charges, magnetic poles always come in pairs (dipoles). Cutting a magnet in half results in two smaller magnets, each with a north and south pole. There are no isolated magnetic charges (monopoles) in nature.

• Magnetic dipole: A system with both north and south poles.

• Magnetic field lines emerge from the north pole and enter the south pole.

Relationship Between Electricity and Magnetism

Moving Charges and Magnetic Fields

Electricity and magnetism are fundamentally related. Moving electric charges generate magnetic fields, and magnetic fields exert forces on moving charges.

• Ampère's Law: Two currents exert forces on each other due to their magnetic fields.

• Faraday's Law of Induction: A changing magnetic field induces an electric field.

• Electromagnetic induction: Time-varying magnetic fields induce electric currents.

Comparison of Electric and Magnetic Forces

Phenomenon	Equation	Description
Electric field exerts force on charge	$F = qE$	Force on a stationary charge in an electric field
Magnetic field exerts force on moving charge	$F = q\vec{v} \times \vec{B}$	Force on a moving charge in a magnetic field

Magnetic Field and Magnetic Forces

Learning Outcomes

• State and apply the Lorentz force law.

• Understand Gauss’s Law for magnetism.

• Explain how magnetic fields exert forces on currents.

• Calculate force and torque on current loops, including applications such as dc motors.

Lorentz Force Law

The Lorentz force law describes the force experienced by a charged particle moving in electric and magnetic fields.

• General form: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

• If only a magnetic field is present: $\vec{F} = q\vec{v} \times \vec{B}$

• The direction of the force is given by the right-hand rule for positive charges; for negative charges, the direction is reversed.

Force on a Moving Charge in a Magnetic Field

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field.

• If the velocity is parallel to the magnetic field: $\vec{F} = 0$ (no force).

• If the velocity is perpendicular to the magnetic field: $|\vec{F}| = |q|vB$ (maximum force).

• If the velocity makes an angle $\theta$ with the field: $|\vec{F}| = |q|vB \sin \theta$

Example: A proton moving at an angle to a uniform magnetic field experiences a force calculated using $|\vec{F}| = |q|vB \sin \theta$.

Motion of Charged Particles in Magnetic Fields

Charged particles moving perpendicular to a uniform magnetic field undergo circular motion due to the constant magnitude and perpendicular direction of the magnetic force.

• Radius of circular path: $r = \frac{mv}{|q|B}$

• Frequency (cyclotron frequency): $f = \frac{|q|B}{2\pi m}$

• Period: $T = \frac{2\pi m}{|q|B}$

Applications: Velocity selectors, mass spectrometers, particle accelerators.

Magnetic Field Lines and Flux

Magnetic Field Lines

Magnetic field lines are a visual representation of the direction and strength of magnetic fields.

• Field lines are tangent to the direction of the magnetic field vector $\vec{B}$ at every point.

• Field lines never intersect or start/end; they always form closed loops.

• Field lines point in the direction a compass needle would align.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given surface.

• Definition: $\Phi_B = \int \vec{B} \cdot d\vec{A}$

• SI unit: Weber (Wb), where $1\,\text{Wb} = 1\,\text{T} \cdot \text{m}^2$

Gauss’s Law for Magnetism

Gauss’s Law for magnetism states that the net magnetic flux through any closed surface is zero, reflecting the absence of magnetic monopoles.

• Equation: $\oint_{\text{closed surface}} \vec{B} \cdot d\vec{A} = 0$

• Magnetic field lines always form closed loops; they do not begin or end at any point.

Magnetic Forces on Currents and Current Loops

Force on a Current-Carrying Wire

A wire carrying current in a magnetic field experiences a force given by:

• Equation: $\vec{F} = I\vec{L} \times \vec{B}$

• For a wire of length $\vec{L}$ in a uniform field $\vec{B}$, the force is perpendicular to both the current direction and the field.

• For non-uniform fields or curved wires, integrate: $d\vec{F} = I d\vec{l} \times \vec{B}$

Example: The net force on a wire with straight and curved segments can be found by calculating the force on each segment and summing the results.

Torque on Current Loops

A current loop in a uniform magnetic field experiences a torque that tends to align the loop’s magnetic dipole moment with the field.

• Magnetic dipole moment: $\vec{\mu} = NIA\hat{n}$, where $N$ is the number of turns, $I$ is current, $A$ is area, and $\hat{n}$ is the normal vector.

• Torque: $\vec{\tau} = \vec{\mu} \times \vec{B}$

• Potential energy: $U = -\vec{\mu} \cdot \vec{B}$

• The loop tends to rotate to minimize its potential energy, aligning $\vec{\mu}$ with $\vec{B}$.

Example: DC motors use the torque on current loops to produce rotational motion.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Lorentz force (general)	$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$	Force on a charge in electric and magnetic fields
Force on moving charge (magnetic only)	$\vec{F} = q\vec{v} \times \vec{B}$	Force on a moving charge in a magnetic field
Force on current-carrying wire	$\vec{F} = I\vec{L} \times \vec{B}$	Force on a straight wire in a magnetic field
Torque on current loop	$\vec{\tau} = \vec{\mu} \times \vec{B}$	Torque on a loop with magnetic dipole moment
Magnetic flux	$\Phi_B = \int \vec{B} \cdot d\vec{A}$	Total magnetic field through a surface
Gauss’s Law for magnetism	$\oint \vec{B} \cdot d\vec{A} = 0$	No net magnetic flux through a closed surface

Applications and Phenomena

• Velocity selector: Uses crossed electric and magnetic fields to select particles of specific velocity.

• Mass spectrometer: Measures mass of ions by analyzing their circular motion in a magnetic field.

• Particle accelerators: Use magnetic fields to steer and accelerate charged particles.

• DC motors: Convert electrical energy to mechanical rotation using torque on current loops.

Additional info: These notes expand on fragmented lecture slides and images, providing full academic context, definitions, and equations for a self-contained study guide on magnetic fields and forces.