understand the use of a Hall probe to measure magnetic flux density

Force on a Current‑Carrying Conductor

Learning Objective

Understand how a Hall probe can be used to measure the magnetic flux density \$B\$ in a region where a conductor carrying current \$I\$ experiences a magnetic force.

1. Theoretical Background

The magnetic force on a straight conductor of length \$L\$ carrying a current \$I\$ in a uniform magnetic field \$\mathbf{B}\$ is given by the Lorentz force law:

\$\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}\$

For a conductor oriented perpendicular to the field, the magnitude simplifies to:

\$F = I L B \sin\theta\$

where \$\theta\$ is the angle between \$\mathbf{L}\$ and \$\mathbf{B}\$. When \$\theta = 90^{\circ}\$, \$\sin\theta = 1\$ and \$F = I L B\$.

2. The Hall Effect

When a current \$I{\text{H}}\$ flows through a thin semiconductor plate placed in a magnetic field \$B\$, charge carriers experience a magnetic force that drives them to one side of the plate, creating a transverse voltage \$V{\text{H}}\$ – the Hall voltage.

The Hall voltage is related to the magnetic flux density by:

\$V{\text{H}} = \frac{I{\text{H}} B}{n q t}\$

where:

• \$n\$ = charge carrier density (m\$^{-3}\$)
• \$q\$ = elementary charge (\$1.602\times10^{-19}\,\text{C}\$)
• \$t\$ = thickness of the Hall plate (m)

Rearranging gives the magnetic flux density:

\$B = \frac{V{\text{H}} n q t}{I{\text{H}}}\$

3. Using a Hall Probe to Measure \$B\$

A Hall probe incorporates a calibrated Hall sensor and electronics that output \$B\$ directly, but the underlying principle follows the equation above. The steps are:

1. Zero the probe in a field‑free region.
2. Place the probe at the point of interest, ensuring the sensor surface is perpendicular to the expected field direction.
3. Read the magnetic flux density \$B\$ from the display or data logger.

4. Experimental Procedure for Determining the Force

The following method links the measured \$B\$ to the magnetic force on a conductor.

1. Set up a straight copper wire of known length \$L\$ between two fixed supports.
2. Connect the wire to a variable DC power supply and set a current \$I\$ (measure with an ammeter).
3. Place a calibrated Hall probe at the location of the wire to record \$B\$ (ensure the probe is oriented correctly).
4. Measure the vertical deflection \$d\$ of the wire using a ruler or a laser pointer.
5. Calculate the magnetic force using \$F = I L B\$ and compare with the force inferred from the deflection (e.g., using torque balance).

5. Sample Calculation

Given:

Force:

\$F = I L B = (2.5\ \text{A})(0.150\ \text{m})(0.035\ \text{T}) = 0.0131\ \text{N}\$

6. Sources of Error

• Misalignment of the Hall probe with the magnetic field direction (introduces \$\cos\theta\$ factor).
• Temperature drift affecting the Hall sensor’s sensitivity.
• Uncertainty in the current measurement due to contact resistance.
• Non‑uniform magnetic field over the length \$L\$ of the conductor.

7. Summary

By measuring the magnetic flux density \$B\$ with a Hall probe, the magnetic force on a current‑carrying conductor can be quantified using \$F = I L B\$. The Hall effect provides a direct, reliable method for determining \$B\$, linking the electrical measurement (Hall voltage) to the mechanical effect (force).