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:

Quantity	Symbol	Value	Unit
Current through the wire	$I$	2.5	A
Length of wire in field	$L$	0.150	m
Magnetic flux density (Hall probe reading)	$B$	0.035	T

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).