Magnetic Fields: Properties, Forces, and Applications

Magnetic Fields

Magnets and Magnetic Dipoles

Magnets are objects that produce magnetic fields, which exert forces on other magnets and moving charges. All magnets have at least one north pole and one south pole. Magnetic field lines emerge from north poles and enter through south poles, forming closed loops.

• Magnetic Dipole: A simple magnet is a dipole, with field lines looping from north to south.

• Magnetic Monopoles: No magnetic monopoles have been observed; breaking a magnet results in two smaller dipoles.

• Earth's Magnetic Field: Near the Earth's surface, the field resembles a dipole. Away from Earth, the field is distorted by the solar wind.

• Magnetic Pole Reversals: Evidence from ocean floor minerals shows Earth's magnetic field has reversed over geological time.

Origin of Magnetic Fields

Magnetic fields arise from two sources:

• Moving Electric Charges: Electric currents generate magnetic fields.

• Intrinsic Magnetic Fields: Some particles (e.g., electrons) possess intrinsic magnetic moments due to quantum properties.

Magnetic Force on a Moving Charge

The force experienced by a charged particle in a magnetic field depends on its charge, velocity, and the magnetic field:

• Direction: The force is perpendicular to both the velocity (v) and the magnetic field (B).

• Magnitude: Proportional to the charge (q), speed (v), and the sine of the angle between v and B.

Formula: $\vec{F}_B = q(\vec{v} \times \vec{B})$ Magnitude: $F_B = qBv\sin\theta$

• Right-Hand Rule: Used to determine the direction of the force. For positive charges, the force follows the direction of v × B; for negative charges, it is opposite.

• Units: The unit of magnetic field is the tesla (T), where $1\,\text{T} = 1\,\text{N}/(\text{A} \cdot \text{m})$. $1\,\text{Gauss} = 10^{-4}\,\text{T}$.

Motion of Charged Particles in Uniform Magnetic Fields

Charged particles moving in a uniform magnetic field experience a force that can cause circular or helical motion.

• Circular Motion: If velocity is perpendicular to B, the particle moves in a circle.

• Helical Motion: If velocity has both perpendicular and parallel components to B, the path is a helix.

• Radius of Path: $r = \frac{mv}{qB}$

• Angular Speed: $\omega = \frac{v}{r} = \frac{qB}{m}$

• Cyclotron Frequency: $f = \frac{\omega}{2\pi} = \frac{qB}{2\pi m}$

• Work Done: Magnetic forces do no work, as the force is always perpendicular to velocity: $dW = \vec{F} \cdot d\vec{l} = q(\vec{v} \times \vec{B}) \cdot \vec{v} dt = 0$

Motion in Crossed Electric and Magnetic Fields

When both electric (E) and magnetic (B) fields are present, the total force on a charged particle is:

Lorentz Force Law: $\vec{F} = q\vec{E} + q(\vec{v} \times \vec{B})$

• Velocity Selector: In crossed fields, the speed can be set so that the net force is zero: $v = \frac{E}{B}$

• Application: Used in mass spectrometers to select particles of specific velocity.

Magnetic Flux and Gauss's Law for Magnetism

Magnetic flux quantifies the amount of magnetic field passing through a given area.

• Definition: $d\Phi_B = (B \cos \theta) dA = B_\perp dA = \vec{B} \cdot d\vec{A}$

• Total Flux: $\Phi_B = \iint \vec{B} \cdot d\vec{A}$

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

Example Calculation:

• For a circular area of radius 6.50 cm in the xy-plane and $B = 0.230\,\text{T}$:

  • (a) $B$ along $+z$: $\Phi_B = BA = (0.230\,\text{T}) \pi (0.065\,\text{m})^2 = 3.05 \times 10^{-3}\,\text{Wb}$

  • (b) $B$ at $53.1^\circ$ from $+z$: $\Phi_B = BA \cos 53.1^\circ = 1.83 \times 10^{-3}\,\text{Wb}$

  • (c) $B$ along $+y$: $\Phi_B = 0$

Gauss's Law for Magnetism: $\iint_{\text{closed surface}} \vec{B} \cdot d\vec{A} = 0$

• No net magnetic flux through a closed surface; reflects the absence of magnetic monopoles.

Magnetic Force on a Current-Carrying Wire

Current-carrying wires in magnetic fields experience forces due to the motion of charges.

• Force on a Segment: $d\vec{F} = I (d\vec{L} \times \vec{B})$

• Force on a Straight Wire: $\vec{F} = I (\vec{L} \times \vec{B})$

• Magnitude: $F = ILB \sin \theta$

• Direction: Given by the right-hand rule.

Example:

• 20.0 cm × 30.0 cm loop, $I = 1.0\,\text{A}$, $B = 2.5\,\text{T}$ to the right:

  • Left side: $F$ into page

  • Right side: $F$ out of page

  • Top and bottom: no force

  • Magnitude: $F = (1.0\,\text{A})(0.20\,\text{m})(2.5\,\text{T}) \sin 90^\circ = 0.50\,\text{N}$

  • Net force on loop: $F_{\text{net}} = 0$

Torque on a Current Loop and Magnetic Dipole Moment

While the net force on a current loop in a uniform magnetic field is zero, a net torque can act on the loop.

• Torque on Loop: $\tau = NIAB \sin \theta$

• Vector Form: $\vec{\tau} = I \vec{A} \times \vec{B}$

• Magnetic Dipole Moment: $\vec{\mu} = I \vec{A}$

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

• Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$

• Right-Hand Rule: Direction of $\vec{A}$ is found by curling fingers in current direction; thumb points in $\vec{A}$ direction.

Example:

• Rectangular coil: $5.4\,\text{cm} \times 8.5\,\text{cm}$, $N = 25$, $I = 15\,\text{mA}$

  • $\mu = IA = (15 \times 10^{-3}\,\text{A})(0.054\,\text{m})(0.085\,\text{m}) = 6.89 \times 10^{-5}\,\text{A} \cdot \text{m}^2$

  • Total: $\mu_{\text{total}} = N\mu = 1.72 \times 10^{-3}\,\text{A} \cdot \text{m}^2$

  • Torque for $B = 0.350\,\text{T}$ parallel to plane: $\tau = \mu_{\text{total}} B = 6.02 \times 10^{-4}\,\text{Nm}$

  • Torque for $B$ perpendicular: $\tau = 0$

Summary Table: Key Magnetic Field Equations

Concept	Equation	Description
Magnetic Force (charge)	$\vec{F}_B = q(\vec{v} \times \vec{B})$	Force on moving charge in magnetic field
Magnetic Force (wire)	$d\vec{F} = I(d\vec{L} \times \vec{B})$	Force on wire segment
Torque on Loop	$\tau = NIAB \sin \theta$	Torque on current loop
Magnetic Dipole Moment	$\mu = IA$	Current × area
Energy of Dipole	$U = -\vec{\mu} \cdot \vec{B}$	Potential energy in field
Magnetic Flux	$\Phi_B = \iint \vec{B} \cdot d\vec{A}$	Field through area
Gauss's Law (Magnetism)	$\iint_{\text{closed}} \vec{B} \cdot d\vec{A} = 0$	No net flux through closed surface

Additional info:

• These notes cover topics from Chapter 27 (Magnetic Field and Magnetic Forces) and Chapter 28 (Sources of Magnetic Field) in a typical college physics curriculum.

• Applications include mass spectrometry, electric motors, and magnetic storage devices.