understand the use of a Hall probe to measure magnetic flux density
Force on a Current‑Carrying Conductor – Using a Hall Probe to Measure Magnetic Flux Density
Learning Objective
Explain how a Hall probe can be used to determine the magnetic flux density B in a region where a straight conductor carrying current I experiences a magnetic force, and use this information to calculate the force.
1. Theoretical Background
1.1 Magnetic force on a current‑carrying conductor (Cambridge 20.2)
Vector form (Lorentz‑force law)
\[
\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}
\]
- I – current (A); direction given by the right‑hand rule for positive charge flow.
- \(\mathbf{L}\) – vector whose magnitude equals the length of wire in the field and whose direction is that of the conventional current.
- \(\mathbf{B}\) – magnetic flux density vector (T).
Scalar magnitude
\[
F = I\,L\,B\sin\theta
\]
where \(\theta\) is the angle between \(\mathbf{L}\) and \(\mathbf{B}\).
Special cases
- Perpendicular arrangement (\(\theta = 90^{\circ}\)): \(\sin\theta = 1\) → \(F = I\,L\,B\).
- Parallel arrangement (\(\theta = 0^{\circ}\) or \(180^{\circ}\)): \(\sin\theta = 0\) → \(F = 0\) (no magnetic force).
Fleming’s left‑hand rule (AO3)
- Fore‑finger points in the direction of \(\mathbf{B}\) (from north to south).
- Middle finger points in the direction of conventional current \(\mathbf{I}\).
- Thumb points in the direction of the magnetic force \(\mathbf{F}\) on the conductor.
1.2 The Hall effect (Cambridge 20.3)
- A thin semiconductor plate (the Hall sensor) carries a current IH along the x‑axis.
- When a magnetic field B is applied perpendicular to the plate (along the z‑axis), charge carriers experience a magnetic force that pushes them toward one side, creating a transverse electric field.
- The resulting potential difference across the plate is the Hall voltage VH.
The basic Hall‑voltage equation (derived for a uniformly magnetised, thin plate) is
\[
V{H}= \frac{I{H}\,B}{n\,q\,t}
\]
where
- n – charge‑carrier density (m\(^{-3}\)).
- q – elementary charge (\(\pm1.602\times10^{-19}\) C). The sign of VH tells whether the dominant carriers are electrons (negative) or holes (positive).
- t – thickness of the Hall plate (m).
Re‑arranging gives the magnetic flux density
\[
B = \frac{V{H}\,n\,q\,t}{I{H}}
\]
Assumptions required by the syllabus
- The plate is thin enough that B can be considered uniform across its thickness.
- The magnetic field is uniform over the active area of the sensor.
1.3 Using a calibrated Hall probe
A commercial Hall probe contains a Hall sensor and built‑in electronics that convert the measured Hall voltage into a direct read‑out of B. The underlying principle is the equation above.
- Zero the probe in a field‑free region (or use the “null” function) to remove any offset voltage.
- Orient the probe so that the sensor surface is perpendicular to the expected direction of B. Mis‑alignment introduces a factor \(\cos\theta\) (see error table).
- Read the value of B from the display or data logger. Most probes also indicate the sign of B, which corresponds to the direction of the field relative to the sensor’s reference axis.
2. Experimental Procedure – Determining the Magnetic Force
This practical links the Hall‑probe measurement of B to the mechanical force on a current‑carrying wire.
- Mount a straight copper wire of known length L between two fixed supports so that it lies in a region of uniform magnetic field.
- Connect the wire to a variable DC power supply. Measure the current I with an ammeter (or a digital multimeter set to the appropriate range).
- Place the calibrated Hall probe at the exact location of the wire (or as close as possible) and record the magnetic flux density B. Ensure the probe’s sensor plane is perpendicular to the field.
- Measure the vertical deflection d of the wire (e.g., with a ruler, a calibrated scale, or a laser‑pointer method). If the wire is suspended as a cantilever, use the torque‑balance method to relate d to the force.
- Calculate the magnetic force using the scalar form (θ = 90°):
\[
F_{\text{calc}} = I\,L\,B
\]
Compare this value with the force obtained from the measured deflection (e.g., \(F = k\,d\) where k is the effective spring constant of the support).
3. Worked Example – Non‑perpendicular Geometry
Suppose the wire is placed at an angle of \(\theta = 30^{\circ}\) to the magnetic field.
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| Current | I | 3.0 | A |
| Length of wire in field | L | 0.200 | m |
| Magnetic flux density (Hall probe) | B | 0.040 | T |
| Angle between L and B | \(\theta\) | 30 | ° |
\[
F = I\,L\,B\sin\theta = (3.0)(0.200)(0.040)\sin30^{\circ}=0.0048\ \text{N}
\]
4. Sample Calculation – Perpendicular Case
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| Current through the wire | I | 2.5 | A |
| Length of wire within the field | L | 0.150 | m |
| Magnetic flux density (Hall probe reading) | B | 0.035 | T |
\[
F = I\,L\,B = (2.5)(0.150)(0.035)=0.0131\ \text{N}
\]
5. Sources of Error and Mitigation (AO3)
| Error Source | Effect on Measurement | Mitigation |
|---|---|---|
| Probe mis‑alignment | Measured \(B_{\text{meas}} = B\cos\theta\) (under‑estimate if \(\theta\neq0\)) | Use a spirit level or the probe’s built‑in alignment guide; record the angle and apply a \(\cos\theta\) correction. |
| Temperature drift | Sensitivity of the Hall sensor changes → systematic error in B | Allow the probe to equilibrate, apply the manufacturer’s temperature‑compensation factor, or perform a calibration at the experimental temperature. |
| Current measurement uncertainty | Incorrect I → error in calculated force | Use four‑wire (Kelvin) connections; verify the ammeter range and zero the meter before each reading. |
| Non‑uniform magnetic field over length L | Using a single B value may misrepresent the average field | Take several Hall‑probe readings along the wire and use the arithmetic mean, or restrict the experiment to the uniform region of a Helmholtz pair. |
| Sign of Hall voltage ignored | Direction of B (and therefore direction of F) may be assigned incorrectly | Check the probe’s calibration sheet; note whether the sensor reports positive or negative B and use Fleming’s left‑hand rule accordingly. |
6. Summary
- The magnetic force on a straight conductor is given by the vector law \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\); its magnitude is \(F = I L B \sin\theta\). For a perpendicular arrangement this reduces to \(F = I L B\).
- The Hall effect provides a direct electrical method for measuring magnetic flux density: \(V{H}= I{H}B/(nqt)\). The sign of \(V_{H}\) reveals the dominant charge‑carrier type.
- A calibrated Hall probe incorporates this principle. After zeroing, proper alignment, and temperature stabilisation, it yields a reliable value of B.
- Combining the measured B with the known current and length of the wire allows calculation of the magnetic force, which can be compared with a mechanical measurement to demonstrate the link between theory (AO1) and practical skills (AO2‑AO3).
