Current and Magnetic Fields: Biot-Savart Law, Ampère's Law, and Applications

Current and Magnetic Fields

Introduction

This section covers the fundamental laws governing magnetic fields produced by electric currents, including the Biot-Savart Law and Ampère's Law. Applications to wires, coils, and solenoids are discussed, along with the forces between parallel currents.

Biot-Savart Law

Definition and Formula

The Biot-Savart Law describes the magnetic field generated at a point by a small segment of current-carrying wire. It is fundamental for calculating magnetic fields in various geometries.

• Scalar form: $dB = \frac{\mu_0}{4\pi} \frac{I dL \sin \theta}{r^2}$ where $dB$ is the infinitesimal magnetic field, $I$ is current, $dL$ is the length element, $\theta$ is the angle between $dL$ and the position vector $\vec{r}$, and $r$ is the distance from the wire to the point.

• Vector form: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{L} \times \hat{r})}{r^2}$

• Direction: The direction of $\vec{B}$ is given by the right-hand rule: point your thumb in the direction of current, and your fingers curl in the direction of the magnetic field lines.

Example: Magnetic Field of a Circular Coil

For a circular coil of radius $R$ carrying current $I$, the magnetic field on the axis (z-axis) is calculated by integrating the Biot-Savart Law.

• Field on axis: $B_z = \frac{\mu_0}{2} \frac{IR^2}{(R^2 + z^2)^{3/2}}$

• At center of coil ($z=0$): $B_z = \frac{\mu_0 I}{2R}$

• Far from coil ($z \gg R$): $B_z \approx \frac{\mu_0 IR^2}{2z^3}$

Example: The field at the center of a current loop is used in magnetic resonance imaging (MRI) and particle accelerators.

Example: Magnetic Field of a Long Straight Wire

The magnetic field at a distance $r$ from a long, straight wire carrying current $I$ is:

• $B = \frac{\mu_0 I}{2\pi r}$

• The field lines form concentric circles around the wire, with direction given by the right-hand rule.

Example: This formula is used to determine the field near power transmission lines.

Force Between Parallel Currents

Parallel wires carrying currents exert forces on each other due to their magnetic fields.

• Magnetic field at wire 2 due to wire 1: $B_1 = \frac{\mu_0 I_1}{2\pi d}$

• Force on wire 2: $F_{12} = I_2 L B_1 = \frac{\mu_0 I_1 I_2 L}{2\pi d}$

• Direction: Parallel currents attract; antiparallel currents repel.

Example: This principle is used to define the ampere, the SI unit of current.

Ampère's Law

Definition and Formula

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

• $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

• $I_{enc} = \iint \vec{J} \cdot d\vec{A}$, where $\vec{J}$ is the current density.

• Useful for calculating $\vec{B}$ in highly symmetric situations (e.g., infinite wire, solenoid, toroid).

Application: Infinite Straight Wire

• For a wire carrying current $I$ into the page, the field at radius $r$ is: $B = \frac{\mu_0 I}{2\pi r}$

• Direction is determined by the right-hand rule around the wire.

Application: Cylindrical Conductor

• Inside the conductor ($r < R$): $B = \frac{\mu_0 I r}{2\pi R^2}$

• Outside the conductor ($r > R$): $B = \frac{\mu_0 I}{2\pi r}$

Example: Used in analyzing current distribution in thick wires and cables.

Application: Solenoid

A solenoid is a coil of wire wrapped in a cylindrical shape. The magnetic field inside an ideal solenoid is uniform and parallel to its axis.

• $B = \mu_0 n I$

• Where $n$ is the number of turns per unit length, $I$ is the current.

• Direction is given by the right-hand rule: curl fingers in direction of current, thumb points in direction of $\vec{B}$ inside.

Example: Solenoids are used in electromagnets and inductors.

Application: Toroid

A toroid is a coil shaped like a doughnut. The magnetic field inside a toroid is confined within its core and can be calculated using Ampère's Law. (See textbook example 28.10 for details.)

Summary Table: Key Magnetic Field Formulas

Situation	Magnetic Field Formula	Direction
Infinitesimal wire segment (Biot-Savart)	$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{L} \times \hat{r})}{r^2}$	Right-hand rule
Long straight wire	$B = \frac{\mu_0 I}{2\pi r}$	Concentric circles (RHR)
Circular coil (axis)	$B_z = \frac{\mu_0}{2} \frac{IR^2}{(R^2 + z^2)^{3/2}}$	Along axis
Circular coil (center)	$B_z = \frac{\mu_0 I}{2R}$	Perpendicular to plane
Solenoid (ideal)	$B = \mu_0 n I$	Along axis
Cylindrical conductor ($r < R$)	$B = \frac{\mu_0 I r}{2\pi R^2}$	Concentric circles
Cylindrical conductor ($r > R$)	$B = \frac{\mu_0 I}{2\pi r}$	Concentric circles

Additional info:

• The notes also reference the magnetic field of a toroid, which is covered in standard physics textbooks and is an application of Ampère's Law.

• All equations use $\mu_0$, the permeability of free space: $\mu_0 = 4\pi \times 10^{-7} \ \mathrm{T \cdot m/A}$.

• Magnetic field lines always form closed loops and never begin or end in space.