How do you calculate the Hall voltage?
1 Answer
The **Hall voltage (\(V_H\))** is a measurable potential difference generated across a conductor or semiconductor when it is placed in a magnetic field, and current flows through it. This phenomenon, called the **Hall Effect**, occurs because of the Lorentz force acting on charge carriers (electrons or holes).
The Hall voltage is calculated using the formula:
\[
V_H = \frac{B \cdot I}{n \cdot q \cdot t}
\]
Where:
- \(V_H\) = Hall voltage (volts, \(V\))
- \(B\) = Magnetic flux density (teslas, \(T\))
- \(I\) = Current passing through the conductor (amperes, \(A\))
- \(n\) = Charge carrier density (number of charge carriers per unit volume, \(m^{-3}\))
- \(q\) = Charge of an individual carrier (\(q = 1.6 \times 10^{-19}\, C\) for electrons)
- \(t\) = Thickness of the material perpendicular to the current and magnetic field (meters, \(m\))
### Explanation of Parameters
1. **Magnetic Flux Density (\(B\))**: The strength of the magnetic field applied perpendicular to the current flow.
2. **Current (\(I\))**: The electric current passing through the conductor or semiconductor.
3. **Charge Carrier Density (\(n\))**: Determines how many charge carriers are available per unit volume.
4. **Thickness (\(t\))**: The distance across which the Hall voltage is measured.
5. **Charge (\(q\))**: The fundamental charge of the particle involved (e.g., electron or hole).
### Simplified Formula for Practical Use
In many practical cases, the Hall coefficient (\(R_H\)) is used to simplify calculations:
\[
V_H = R_H \cdot \frac{I \cdot B}{t}
\]
Where \(R_H = \frac{1}{n \cdot q}\) is the Hall coefficient, determined experimentally for the material.
---
### Steps to Calculate Hall Voltage:
1. **Measure or determine the parameters**:
- Magnetic flux density (\(B\))
- Current (\(I\))
- Thickness (\(t\))
- Charge carrier density (\(n\)) (or use the Hall coefficient if given)
2. **Substitute values** into the Hall voltage formula.
3. **Solve for \(V_H\)**.
---
### Example Calculation:
Suppose:
- \(B = 0.5\, T\) (magnetic field strength)
- \(I = 2\, A\) (current)
- \(t = 0.01\, m\) (thickness)
- \(n = 8 \times 10^{28}\, m^{-3}\) (electron density in copper)
\[
V_H = \frac{B \cdot I}{n \cdot q \cdot t}
\]
Substitute values:
\[
V_H = \frac{0.5 \cdot 2}{(8 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot 0.01}
\]
\[
V_H = \frac{1}{1.28 \times 10^{11}} = 7.81 \times 10^{-12}\, V
\]
Thus, the Hall voltage is \(7.81 \, \text{pV}\) (picoVolts), a small value typical for metals like copper.
---
In semiconductors, the Hall voltage is much larger due to the lower charge carrier density (\(n\)), which makes it easier to measure and utilize in devices like Hall sensors.
The Hall voltage is calculated using the formula:
\[
V_H = \frac{B \cdot I}{n \cdot q \cdot t}
\]
Where:
- \(V_H\) = Hall voltage (volts, \(V\))
- \(B\) = Magnetic flux density (teslas, \(T\))
- \(I\) = Current passing through the conductor (amperes, \(A\))
- \(n\) = Charge carrier density (number of charge carriers per unit volume, \(m^{-3}\))
- \(q\) = Charge of an individual carrier (\(q = 1.6 \times 10^{-19}\, C\) for electrons)
- \(t\) = Thickness of the material perpendicular to the current and magnetic field (meters, \(m\))
### Explanation of Parameters
1. **Magnetic Flux Density (\(B\))**: The strength of the magnetic field applied perpendicular to the current flow.
2. **Current (\(I\))**: The electric current passing through the conductor or semiconductor.
3. **Charge Carrier Density (\(n\))**: Determines how many charge carriers are available per unit volume.
4. **Thickness (\(t\))**: The distance across which the Hall voltage is measured.
5. **Charge (\(q\))**: The fundamental charge of the particle involved (e.g., electron or hole).
### Simplified Formula for Practical Use
In many practical cases, the Hall coefficient (\(R_H\)) is used to simplify calculations:
\[
V_H = R_H \cdot \frac{I \cdot B}{t}
\]
Where \(R_H = \frac{1}{n \cdot q}\) is the Hall coefficient, determined experimentally for the material.
---
### Steps to Calculate Hall Voltage:
1. **Measure or determine the parameters**:
- Magnetic flux density (\(B\))
- Current (\(I\))
- Thickness (\(t\))
- Charge carrier density (\(n\)) (or use the Hall coefficient if given)
2. **Substitute values** into the Hall voltage formula.
3. **Solve for \(V_H\)**.
---
### Example Calculation:
Suppose:
- \(B = 0.5\, T\) (magnetic field strength)
- \(I = 2\, A\) (current)
- \(t = 0.01\, m\) (thickness)
- \(n = 8 \times 10^{28}\, m^{-3}\) (electron density in copper)
\[
V_H = \frac{B \cdot I}{n \cdot q \cdot t}
\]
Substitute values:
\[
V_H = \frac{0.5 \cdot 2}{(8 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot 0.01}
\]
\[
V_H = \frac{1}{1.28 \times 10^{11}} = 7.81 \times 10^{-12}\, V
\]
Thus, the Hall voltage is \(7.81 \, \text{pV}\) (picoVolts), a small value typical for metals like copper.
---
In semiconductors, the Hall voltage is much larger due to the lower charge carrier density (\(n\)), which makes it easier to measure and utilize in devices like Hall sensors.