Sources of Magnetic Fields - Biot Savart

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Textbook

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

Magnetic fields due to current in a wire

Biot-Savart: The general equation for the 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.

Biot Savart: Particular Solutions

Often, it is not necessary to integrate the general Biot-Savart Law (ask your teacher if you have to be able to), instead the particular solutions that follow can be used.

Magnetic field due to infinite 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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