Question

Two expressions were given for the Hall resistance: \(R_{H}=\frac{B_{z}}{q p} \cdot \frac{w}{l \cdot d_{t h i c k}}\) and \(R_{H}=\frac{h}{q^{2} n}\) Show that both expressions have the units of ohms.

Short answer

Both expressions for Hall resistance, when units are analyzed and simplified, result in ohms (Ω).

Step-by-step solution

Understand the Variables in the First Expression

In the first expression \( R_{H}=\frac{B_{z}}{q p} \cdot \frac{w}{l \cdot d_{thick}} \), let's identify the units of each variable: \( B_z \) (magnetic field) has units of teslas (T), \( q \) (charge) has units of coulombs (C), \( p \) (material property) is in \( m^{-3} \), \( w \) (width) and \( l \) (length) have units in meters (m), and \( d_{thick} \) (thickness) also has units of meters (m).

Analyze the Units of Each Term in the First Expression

First, find the units for \( \frac{B_{z}}{q p} \). The units of \( B_z \) are \( kg \cdot A^{-1} \cdot s^{-2} \), so \( \frac{B_z}{q} \) has \( kg \cdot A^{-1} \cdot s^{-2} \cdot C^{-1} \) simplifying to \( kg \cdot m \cdot C^{-1} \cdot s^{-1} \). The units of \( p \) are \( m^{-3} \), so \( \frac{1}{p} \) has units of \( m^3 \). Resulting units for \( \frac{B_z}{q p} \) are \( kg \cdot m^4 \cdot C^{-1} \cdot s^{-1} \).

Simplify First Expression Units Further

For the term \( \frac{w}{l \cdot d_{thick}} \), the units are \( m \cdot m^{-1} \cdot m^{-1} = m^{-1} \). Therefore, the entire first expression \( R_H \) has units \( kg \cdot m^{-1} \cdot C^{-1} \cdot s^{-1} \cdot m^4 \cdot m^{-1} \). Simplifying by cancelling, \( R_H \) yields units of \( kg \cdot m^2 \cdot C^{-1} \cdot s^{-1} \), which are \( kg \cdot m^2 \cdot s^{-3} \cdot A^{-1} \), the same as ohms.

Understand the Variables in the Second Expression

In the second expression \( R_{H}=\frac{h}{q^{2} n} \), \( h \) is Planck's constant with units of \( J \cdot s \); \( q \) is charge in coulombs (C); \( n \) is number density in \( m^{-2} \).

Analyze the Units of the Second Expression

The units of \( \frac{h}{q^2 n} \) are calculated as: \( h \) has \( kg \cdot m^2 \cdot s^{-1} \), \( q^2 \) has \( C^2 \), and \( n \) is \( m^{-2} \). Thus, \( \frac{h}{q^2 n} \) simplifies to \( kg \cdot m^2 \cdot s^{-1} \cdot C^{-2} \cdot m^2 \), simplifying to \( kg \cdot m^2 \cdot s^{-3} \cdot A^{-2} \), which matches the units for ohms.

Key concepts

Electric Fields

An electric field is a fundamental concept in physics that deals with the force a charged particle would experience in space due to other charges. It is a vector field, meaning it has both a magnitude and a direction, which points from positive to negative charges. The electric field

• is measured in volts per meter (V/m),
• expresses how a charge interacts with its environment,
• and exists between two points across which a voltage is applied.

The field is defined mathematically as:\[E = \frac{F}{q}, \]where \(E\) is the electric field, \(F\) is the force exerted by the field, and \(q\) is the charge of the particle. Electric fields play a crucial role in determining how charge carriers move through a material, influencing devices such as capacitors and semiconductors by creating a potential landscape that charges "feel."

Magnetic Fields

Magnetic fields, like electric fields, are a type of vector field associated with magnetic forces. They are generated by moving electric charges, like electric currents, or temporarily in magnetic materials. The strength and direction of a magnetic field are measured in teslas (T). Magnetic fields have a profound impact in physics tasks:

• They exert forces on other moving charges, a concept captured by the Lorentz force law.
• Magnetism and magnetic fields allow for the operation of motors and generators by converting electrical energy to mechanical motion or vice versa.

The Lorentz force is given by:\[F = q(\mathbf{v} \times \mathbf{B})\]where \(F\) is the force, \(q\) is the charge, \(\mathbf{v}\) is the velocity of a charge, and \(\mathbf{B}\) is the magnetic field. The magnetic fields intertwine with electric fields in the Hall Effect, leading to a transverse voltage across a conductor when it is exposed to a perpendicular magnetic field.

Material Properties in Physics

Material properties, vital in physics, describe how different materials react to forces and fields such as electric and magnetic fields. These properties determine the behavior of materials in practical applications and include:

• Conductivity - the ability of a material to conduct electric current, influencing how charge carriers move within it.
• Permeability - how a material responds to a magnetic field, determining its magnetization.
• Resistivity - the material's opposition to the flow of electric current, related inversely to conductivity.

Additional properties like dielectric strength and thermal conductivity may influence the suitability of a material for specific applications. These parameters are essential in selecting materials for electronic and magnetic equipment, impacting efficiency, safety, and performance. In the context of the Hall Effect, understanding material properties helps predict and utilize the effect efficiently.

Charge Carriers

Charge carriers are particles that carry electric charge through materials. In conductors, these are typically electrons due to their mobility and ability to respond to electric fields. However, in other materials, such as semiconductors, charge carriers can also include positive "holes," which are places where an electron is missing.
These carriers are critical for:

• Creating electric current, a flow of charged particles under an applied field.
• Determining the material's conductive properties through density and mobility.

The concentration of charge carriers influences the Hall voltage in the Hall Effect, with higher densities resulting in lower Hall resistance. Understanding how charge carriers behave within materials is crucial for designing and optimizing electronic devices, as it dictates how efficiently energy can be transmitted and manipulated through circuits.