What is the unit of Hall conductivity?
1 Answer
The **Hall conductivity** is a fundamental physical quantity that measures the ability of a material to conduct an electric current in the presence of a magnetic field, specifically in the Hall effect. The Hall effect arises when a current-carrying conductor is placed in a perpendicular magnetic field, causing a transverse voltage (the Hall voltage) to develop across the material.
### Hall Conductivity: Definition
Hall conductivity, usually denoted as \(\sigma_H\), is a measure of the transverse current density that flows in response to an applied longitudinal electric field in the presence of a magnetic field. It is defined as the ratio of the induced transverse current density \(J_y\) to the longitudinal electric field \(E_x\) in the presence of a magnetic field \(B_z\), as:
\[
\sigma_H = \frac{J_y}{E_x B_z}
\]
Where:
- \(J_y\) is the current density in the \(y\)-direction (transverse to the applied electric field),
- \(E_x\) is the electric field in the \(x\)-direction (along the direction of the current),
- \(B_z\) is the magnetic field in the \(z\)-direction (perpendicular to both the current and the electric field).
### Units of Hall Conductivity
To find the units of Hall conductivity, we first consider the units of the quantities involved:
- **Current density** \(J_y\) has units of amperes per square meter (A/m²),
- **Electric field** \(E_x\) has units of volts per meter (V/m),
- **Magnetic field** \(B_z\) has units of tesla (T).
The Hall conductivity is given by the ratio:
\[
\sigma_H = \frac{J_y}{E_x B_z}
\]
### Step-by-step unit analysis:
1. **Current density \(J_y\)** has units of amperes per square meter (A/m²).
2. **Electric field \(E_x\)** has units of volts per meter (V/m), which can be expressed as \( \text{V/m} = \text{kg} \cdot \text{m} / \text{s}^3 \cdot \text{A} \) (since 1 V = 1 kg·m²/(s³·A)).
3. **Magnetic field \(B_z\)** has units of tesla (T), and 1 T = 1 kg/(s²·A).
Now substitute the units:
\[
\sigma_H = \frac{\text{A/m}^2}{(\text{V/m})(\text{T})}
\]
Substitute the expressions for the units of V and T:
\[
\sigma_H = \frac{\text{A/m}^2}{\left( \frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{A}} \right) \cdot \left( \frac{\text{kg}}{\text{s}^2 \cdot \text{A}} \right)}
\]
Simplifying this expression gives the final unit for Hall conductivity:
\[
\sigma_H = \frac{\text{s}^3}{\text{kg} \cdot \text{m}}
\]
### Final Unit
The unit of Hall conductivity is:
\[
\text{S/m} = \text{siemens per meter}
\]
The **siemens (S)** is the unit of electrical conductance, and "per meter" comes from the fact that conductivity is the inverse of resistivity, which has units of ohm·meter (Ω·m). Thus, Hall conductivity has the same unit as electrical conductivity, but with a specific focus on the transverse behavior induced by a magnetic field.
### Hall Conductivity: Definition
Hall conductivity, usually denoted as \(\sigma_H\), is a measure of the transverse current density that flows in response to an applied longitudinal electric field in the presence of a magnetic field. It is defined as the ratio of the induced transverse current density \(J_y\) to the longitudinal electric field \(E_x\) in the presence of a magnetic field \(B_z\), as:
\[
\sigma_H = \frac{J_y}{E_x B_z}
\]
Where:
- \(J_y\) is the current density in the \(y\)-direction (transverse to the applied electric field),
- \(E_x\) is the electric field in the \(x\)-direction (along the direction of the current),
- \(B_z\) is the magnetic field in the \(z\)-direction (perpendicular to both the current and the electric field).
### Units of Hall Conductivity
To find the units of Hall conductivity, we first consider the units of the quantities involved:
- **Current density** \(J_y\) has units of amperes per square meter (A/m²),
- **Electric field** \(E_x\) has units of volts per meter (V/m),
- **Magnetic field** \(B_z\) has units of tesla (T).
The Hall conductivity is given by the ratio:
\[
\sigma_H = \frac{J_y}{E_x B_z}
\]
### Step-by-step unit analysis:
1. **Current density \(J_y\)** has units of amperes per square meter (A/m²).
2. **Electric field \(E_x\)** has units of volts per meter (V/m), which can be expressed as \( \text{V/m} = \text{kg} \cdot \text{m} / \text{s}^3 \cdot \text{A} \) (since 1 V = 1 kg·m²/(s³·A)).
3. **Magnetic field \(B_z\)** has units of tesla (T), and 1 T = 1 kg/(s²·A).
Now substitute the units:
\[
\sigma_H = \frac{\text{A/m}^2}{(\text{V/m})(\text{T})}
\]
Substitute the expressions for the units of V and T:
\[
\sigma_H = \frac{\text{A/m}^2}{\left( \frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{A}} \right) \cdot \left( \frac{\text{kg}}{\text{s}^2 \cdot \text{A}} \right)}
\]
Simplifying this expression gives the final unit for Hall conductivity:
\[
\sigma_H = \frac{\text{s}^3}{\text{kg} \cdot \text{m}}
\]
### Final Unit
The unit of Hall conductivity is:
\[
\text{S/m} = \text{siemens per meter}
\]
The **siemens (S)** is the unit of electrical conductance, and "per meter" comes from the fact that conductivity is the inverse of resistivity, which has units of ohm·meter (Ω·m). Thus, Hall conductivity has the same unit as electrical conductivity, but with a specific focus on the transverse behavior induced by a magnetic field.