Sources of Magnetic Fields - Biot Savart

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Textbook

• University Physics Volume 2: Chapter 12 - Sources of Magnetic Fields

Magnetic field due to a section of a wire

The magnetic field produced by a small segment of a current-carrying wire can be calculated using the Biot-Savart Law. This law relates the magnetic field ${\vec {dB}}$ produced by an infinitesimal section of current-carrying wire $d{\vec {l}}$ to the current $I$ and the distance $r$ from the segment:

$d{\vec {B}}={\frac {\mu _{0}}{4\pi }}{\frac {Id{\vec {l}}\times {\hat {r}}}{r^{2}}}$

Where:

• $\mu _{0}=4\pi \times 10^{-7}\,{\text{T}}\cdot {\text{m/A}}$ is the permeability of free space,
• $d{\vec {l}}$ is the infinitesimal vector length of the wire,
• ${\hat {r}}$ is the unit vector pointing from the wire segment to the point of interest,
• $r$ is the distance from the wire segment to the point of interest.

Magnetic field due to long straight wire

For an infinitely long, straight wire carrying a current $I$, the magnetic field at a distance $r$ from the wire is given by Ampère’s Law:

$B={\frac {\mu _{0}I}{2\pi r}}$

Where:

• $B$ is the magnitude of the magnetic field,
• $r$ is the distance from the wire,
• $I$ is the current in the wire.

The magnetic field forms concentric circles around the wire, and its direction can be determined using the right-hand rule.

Magnetic field due to circular arc of wire

For a current-carrying circular arc of radius $R$ subtending an angle $\theta$ at the center, the magnetic field at the center of the arc is given by:

$B={\frac {\mu _{0}I\theta }{4\pi R}}$

Where:

• $I$ is the current through the arc,
• $\theta$ is the angle subtended by the arc at the center (in radians),
• $R$ is the radius of the arc.

This formula is derived from the Biot-Savart Law for a symmetric circular geometry.

Adding up fields

When calculating the total magnetic field from multiple current elements, use the principle of superposition. The total magnetic field ${\vec {B}}_{\text{total}}$ is the vector sum of the individual fields ${\vec {B}}_{1}$, ${\vec {B}}_{2}$, ... from each current element:

${\vec {B}}_{\text{total}}={\vec {B}}_{1}+{\vec {B}}_{2}+\dots$

This requires adding the magnetic field vectors, taking into account both their magnitudes and directions.

Force between parallel wires

Two parallel wires carrying currents $I_{1}$ and $I_{2}$, separated by a distance $r$, exert a force on each other due to the magnetic fields they produce. The force per unit length between the wires is given by:

$F_{\text{per unit length}}={\frac {\mu _{0}I_{1}I_{2}}{2\pi r}}$

Where:

• $I_{1}$ and $I_{2}$ are the currents in the wires,
• $r$ is the separation between the wires.

The force is attractive if the currents are in the same direction and repulsive if the currents are in opposite directions.

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