Sources of Magnetic Field: Moving Charges, Currents, and Magnetic Materials

Magnetism and Waves

Introduction

This study guide covers the fundamental sources of magnetic fields, including those generated by moving charges and electric currents. It also introduces the laws governing magnetic fields, such as Ampère’s Law, and discusses the properties of different magnetic materials. These concepts are essential for understanding electromagnetism in college-level physics.

Sources of Magnetic Field

Magnetic Field Generated by a Moving Charge

A moving electric charge produces a magnetic field in the surrounding space. The direction and magnitude of this field depend on the velocity of the charge and its position relative to the observation point.

• Key Formula: The magnetic field at position r due to a charge q moving with constant velocity v is given by: $B(r) = \frac{\mu_0 q \vec{v} \times \hat{r}}{4\pi r^2}$

• Direction: The field is perpendicular to both the velocity and the position vector from the charge to the observation point (right-hand rule).

• Comparison with Electric Field: The electric field from a point charge is: $E(r) = \frac{q}{4\pi\varepsilon_0 r^2} \hat{r}$ Both fields decrease with $1/r^2$, but the magnetic field depends on velocity.

• Superposition Principle: For multiple moving charges, the total magnetic field is the vector sum: $B(r) = \frac{\mu_0}{4\pi} \sum_i \frac{q_i \vec{v}_i \times (\vec{r} - \vec{r}_i)}{|\vec{r} - \vec{r}_i|^2}$

• Physical Constants:

  • Magnetic constant (permeability of free space): $\mu_0 \approx 4\pi \times 10^{-7} \ \mathrm{T \cdot m \cdot A^{-1}}$

  • Speed of light: $c = 2.998 \times 10^8 \ \mathrm{m \cdot s^{-1}}$

• Limitations: The above formula is strictly valid for non-relativistic speeds; relativistic cases require special relativity.

• Example: Two parallel moving charges exert attractive or repulsive magnetic forces depending on their relative directions.

Magnetic Field Generated by a Current

Current Elements and the Biot-Savart Law

Electric currents, which are composed of moving charges, generate magnetic fields. The Biot-Savart Law provides a quantitative description of the magnetic field produced by a current element.

• Biot-Savart Law: $dB = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ where $I$ is the current, $d\vec{l}$ is the length vector along the current, and $\hat{r}$ is the unit vector from the current element to the observation point.

• Right-Hand Rule: Thumb points in the direction of current, fingers curl in the direction of the magnetic field lines.

• Magnetic Field of a Straight Wire: For an infinitely long straight wire: $B = \frac{\mu_0 I}{2\pi r}$ where $r$ is the perpendicular distance from the wire.

• Magnetic Field of a Circular Loop: On the axis of a loop of radius $a$: $B = \frac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}}$ where $x$ is the distance along the axis from the center.

• Force Between Parallel Wires: $F_{\text{per unit length}} = \frac{\mu_0 I_1 I_2}{2\pi d}$ where $d$ is the separation between wires. Parallel currents attract; anti-parallel currents repel.

• Example: The net magnetic force on a rectangular loop near a straight wire can be zero depending on geometry and current direction.

Ampère’s Law

Integral Form and Applications

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

• Ampère’s Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

• Applications: Useful for calculating magnetic fields in symmetric situations, such as inside and outside long straight wires, solenoids, and toroids.

• Comparison with Gauss’s Law: Gauss’s Law for electricity: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$ Both laws relate fields to enclosed sources (charge for electric, current for magnetic).

• Example: For certain paths around multiple conductors, the net enclosed current may be zero, resulting in zero net magnetic field.

Magnetic Materials

Types and Properties

Materials respond differently to external magnetic fields based on their microscopic structure. The main types are paramagnetic, diamagnetic, and ferromagnetic materials.

• Paramagnetic Materials:

  • Have free microscopic magnetic moments that align with an external field.

  • Total magnetic field: $B = K_m B_0$, with $K_m > 1$ (relative permeability greater than 1).

• Diamagnetic Materials:

  • No free magnetic moments; external field induces dipoles that oppose the field.

  • Total magnetic field: $B = K_m B_0$, with $K_m < 1$.

• Ferromagnetic Materials:

  • Microscopic moments interact and can spontaneously align, forming magnetic domains even without an external field.

  • Exhibit strong, permanent magnetization (spontaneous symmetry breaking).

• Magnetic Dipole Moment of Electron: $\mu = 9.274 \times 10^{-24} \ \mathrm{J/T}$

Table: Magnetic Susceptibilities of Paramagnetic and Diamagnetic Materials at 20°C

Material	Type	Susceptibility ($\chi_m$) $\times 10^{-5}$
Iron ammonium alum	Paramagnetic	66
Uranium	Paramagnetic	40
Platinum	Paramagnetic	26
Aluminum	Paramagnetic	22
Sodium	Paramagnetic	0.72
Oxygen gas	Paramagnetic	0.19
Bismuth	Diamagnetic	-16.6
Mercury	Diamagnetic	-29
Silver	Diamagnetic	-26
Carbon (diamond)	Diamagnetic	-21
Lead	Diamagnetic	-1.8
Sodium chloride	Diamagnetic	-14
Copper	Diamagnetic	-1.0

Summary of Key Equations

• $B(r) = \frac{\mu_0 q \vec{v} \times \hat{r}}{4\pi r^2}$

• $dB = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$

• $B = \frac{\mu_0 I}{2\pi r}$ (straight wire)

• $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$ (Ampère’s Law)

• $F_{\text{per unit length}} = \frac{\mu_0 I_1 I_2}{2\pi d}$ (force between wires)

Additional info: Some context and explanations have been expanded for clarity and completeness, including the physical meaning of equations and the classification of magnetic materials.