Magnetic Forces, Current-Carrying Wires, and Magnetic Properties of Matter

Magnetic Forces Between Parallel Current-Carrying Wires

Current in Same Direction

When two parallel wires carry electric currents in the same direction, each wire produces a magnetic field that exerts a force on the other wire. The direction of the force can be determined using the right-hand rule.

• Attraction: Parallel currents attract each other.

• Magnetic Field: The magnetic field produced by one wire at the location of the other is given by: $B = \frac{\mu_0 I}{2\pi d}$ where $I$ is the current and $d$ is the distance between wires.

• Force per Unit Length: $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$

• Example: Two wires carrying 5 A each, separated by 2 cm, will attract each other with a calculable force per unit length.

Current in Opposite Directions

If the currents are in opposite directions, the wires repel each other.

• Repulsion: Opposite currents repel.

• Direction: The force direction is reversed compared to the same direction case.

Forces on Current Loops

Interactions Between Loops

Current loops interact similarly to parallel wires. The force can be visualized in two equivalent ways:

• Parallel Currents: Attract each other.

• Opposite Currents: Repel each other.

• Magnetic Poles: Opposite poles attract, like poles repel.

Torque on a Current Loop in a Uniform Magnetic Field

Square Current Loop

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

• Forces: Forces on opposite sides of the loop cancel, but the other two sides create a torque.

• Magnitude of Force: $F = I l B$ where $l$ is the length of the side perpendicular to the field.

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

• Magnetic Dipole Moment: $\mu = IA$

• Generalization: The result holds for loops of any shape.

Applications: Electric Motor

Simple Electric Motor

An electric motor uses the torque on a current-carrying loop in a magnetic field to produce rotation. The commutator reverses the current every half cycle to maintain continuous rotation.

• Upward and Downward Forces: The sides of the loop experience forces in opposite directions, causing rotation.

Atomic Magnets and Electron Spin

Atomic Origin of Magnetism

The magnetic properties of materials originate from the motion of electrons in atoms.

• Orbital Motion: Electrons orbiting the nucleus act as tiny current loops, creating magnetic dipole moments.

• Electron Spin: Each electron has an inherent magnetic moment due to its quantum property called spin.

Magnetic Properties of Matter

Random Arrangement

In most materials, atomic magnetic moments are randomly oriented, resulting in no net magnetization.

Ferromagnetism

Some materials, such as iron, have atomic moments that tend to align in the same direction, leading to strong magnetization.

• Domains: Ferromagnetic materials are divided into regions called domains, within which moments are aligned.

• Random Domains: In an unmagnetized sample, domains are randomly oriented, so the net magnetization is zero.

Induced Magnetic Dipoles and Magnetism

When a ferromagnetic material is placed in an external magnetic field, domains align with the field, inducing a net magnetic dipole moment.

• Induced Dipole: The induced dipole always has its opposite pole facing the source of the field (e.g., a solenoid).

• Permanent Magnet: Some domains may remain aligned after the external field is removed, making the object a permanent magnet.

General Principles of Magnetism

Magnetic Fields and Forces

• Magnetism: Fundamentally, magnetism is the interaction between moving charges.

• Biot-Savart Law: The magnetic field due to a moving point charge is $\vec{B} = \frac{\mu_0}{4\pi} \frac{q \vec{v} \times \hat{r}}{r^2}$

• Ampère’s Law: For symmetric situations, $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{\text{through}}$

• Magnetic Force on a Charge: $\vec{F} = q \vec{v} \times \vec{B}$

• Magnetic Force on a Wire: $\vec{F} = I \vec{l} \times \vec{B}$

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

Applications

Right-Hand Rule

• Point your right thumb in the direction of current $I$; your fingers curl in the direction of $\vec{B}$.

• For a dipole, $\vec{B}$ emerges from the north pole side.

Charged-Particle Motion

• No force if $\vec{v}$ is parallel to $\vec{B}$.

• Circular motion at the cyclotron frequency if $\vec{v}$ is perpendicular to $\vec{B}$: $f_{\text{cyc}} = \frac{qB}{2\pi m}$

Summary Table: Magnetic Interactions

Situation	Force Direction	Result
Parallel currents (same direction)	Attract	Wires move together
Parallel currents (opposite direction)	Repel	Wires move apart
Current loop in uniform $\vec{B}$	Torque aligns loop	Rotation
Ferromagnetic domains in $\vec{B}$	Align with field	Induced magnetism

Additional info: These notes cover core concepts from Chapter 29: The Magnetic Field, including forces between wires, current loops, atomic origins of magnetism, and applications such as electric motors and ferromagnetism.