Magnetic field due to finite coil along axis¶

Using the Biot—Savart law, it is possible to obtain the formula for the magnetic flux density at any point on the axis of a coil. It is directly proportional to the coil’s turn count and current and inversely proportional to its length, and also depends on the position of the measurement point relative to the coil ends.

Conditions:

The point of measurement must lie on the axis of the coil.
The medium is a vacuum.
The \(z\)-axis is the axis of rotation and is oriented according to the right-hand side rule.

Links:

Physics LibreTexts — Solenoids and Toroids. Equation (12.7.4).

magnetic_flux_density¶: Magnitude of magnetic_flux_density.

Symbol:: B
Latex:: \(B\)
Dimension:: magnetic_density

current¶: current flowing through the wire.

Symbol:: I
Latex:: \(I\)
Dimension:: current

turn_count¶: Number of turns in the coil. See positive_number.

Symbol:: N
Latex:: \(N\)
Dimension:: dimensionless

coil_length¶: length of the coil.

Symbol:: l
Latex:: \(\ell\)
Dimension:: length

first_angle¶: Acute angle between the coil axis (or side) and the vector from the measuring point and the first end of the coil (that has a smaller \(z\) coordinate).

Symbol:: phi_1
Latex:: \(\varphi_{1}\)
Dimension:: angle

second_angle¶: Acute angle between the coil axis (or side) and the vector from the measuring point and the second end of the coil (that has a greater \(z\) coordinate).

Symbol:: phi_2
Latex:: \(\varphi_{2}\)
Dimension:: angle

law¶

B = mu_0 * I * N / (2 * l) * (cos(phi_1) + cos(phi_2))

Latex:: \[B = \frac{\mu_0 I N}{2 \ell} \left(\cos{\left(\varphi_{1} \right)} + \cos{\left(\varphi_{2} \right)}\right)\]