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:

• \( \mu_0 \) 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 \( \vec{B}_1, \vec{B}_2, \dots \) 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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