 # Hall Effect Derivation

Hall effect is defined as the production of a voltage difference across an electrical conductor which is transverse to an electric current, and with respect to an applied magnetic field, it is perpendicular to the current. Edwin Hall discovered this effect in the year 1879.

Hall field is defined as the field developed across the conductor, and Hall voltage is the corresponding potential difference. This principle is observed in the charges involved in the electromagnetic fields.

## Hall Effect Derivation

Consider a metal with one type of charge carrier that is electrons and is a steady-state condition with no movement of charges in the y-axis direction. Following is the derivation of the Hall-effect:

$$\begin{array}{l}eE_{H}=Bev\frac{eV_{H}}{d}=BevV_{H}=Bvd\end{array}$$
(at equilibrium, force is downwards due to magnetic field which is equal to upward electric force)

Where,

• VH is Hall voltage
• EH is Hall field
• v is the drift velocity
• d is the width of the metal slab
• B is the magnetic field
• Bev is a force acting on an electron
$$\begin{array}{l}I=-nevA\end{array}$$

Where,

• I is an electric current
• n is no.of electrons per unit volume
• A is the cross-sectional area of the conductor
$$\begin{array}{l}V_{H}=\frac{-Bi}{net}\end{array}$$
$$\begin{array}{l}\frac{E_{H}}{JB}=-\frac{1}{ne}\end{array}$$

Where,

$$\begin{array}{l}\frac{E_{H}}{JB}: \, \textup{ Hall coefficient} \,\, (R_H) \,\, \textup{is defined as the ratio between the induced electric field and to the product of applied magnetic field and current density}.\end{array}$$

In semiconductors, RH is positive for the hole and negative for free electrons.

$$\begin{array}{l}R_{H}=-\frac{1}{ne}\end{array}$$
$$\begin{array}{l}\mu _{H}=\frac{v}{E}=\frac{J}{neE}=\sigma R_{H}=\frac{R_{H}}{\rho }(v)\end{array}$$

Where,

• E is an electric field
• v is the drift velocity
• Ris the Hall coefficient
• 𝛍is the mobility of the hole
$$\begin{array}{l}\frac{J_{y}}{J_{x}}=\sigma \frac{E_{y}}{J_{x}}=\mu _{H}B_{z}=\sigma R_{H}B_{z}\end{array}$$

The ratio between density (x-axis direction) and current density (y-axis direction) is known as the Hall angle, which measures the average number of radians due to collisions of the particles.

$$\begin{array}{l}R=\frac{V_{H}}{i}=\frac{B}{net}\end{array}$$

Where,

• R is Hall resistance

### Hall Effect Derivation in Semiconductors

In semiconductors, electrons and holes contribute to different concentrations and mobilities, making it difficult to explain the Hall coefficient given above. Therefore, for the simple explanation of a moderate magnetic field, the following is the Hall coefficient:

$$\begin{array}{l}R_{H}=\frac{p_\mu {H}^{2}-n\mu _{e}^{2}}{e(p\mu _{H}+n\mu _{e})}\end{array}$$
$$\begin{array}{l}∴R_{H}=\frac{(p-nb^{2})}{e(p+nb)^{2}}\end{array}$$

Where,

$$\begin{array}{l}b=\frac{\mu _{e}}{\mu _{H}}\end{array}$$
• n is electron concentration
• p is hole concentration
• 𝛍is the mobility of electron
• 𝛍is the mobility of the hole
• e is an elementary charge

### Applications of Hall effect

Hall effect finds many applications.

• It is used to determine if the given material is a semiconductor or insulator.
• It is used to measure the magnetic field and is known as a magnetometer
• They find applications in position sensing as they are immune to water, mud, dust, and dirt.
• They are used in integrated circuits as Hall effect sensors.

This was the derivation of the Hall effect. Stay tuned with BYJU’S and learn various other Physics-related topics.

Related Physics articles: ## Frequently Asked Questions – FAQs

Q1

### What is a semiconductor?

A semiconductor has an electrical conductivity value between that of a conductor, such as metallic gold, and an insulator, like glass. Its resistivity reduces as the temperature increases.
Q2

### What is the hall effect?

Hall effect is defined as the generation of a voltage difference across an electrical conductor, which is transverse to an electric current and with respect to an exerted magnetic field, it is perpendicular to the current.
Q3

### Who discovered the hall effect?

Edwin Hall discovered the hall effect in 1879.
Q4

### What is the hall field?

The Hall field is defined as the field developed across the conductor, and the Hall voltage is the corresponding potential difference.
Q5

### Give one application of the hall effect.

It determines whether the given material is a semiconductor or insulator.
Q6

### Which effect is observed in the charges involved in the electromagnetic fields?

The Hall effect is observed in the charges involved in the electromagnetic fields.

Q7

### What is a magnetic field?

A magnetic field is a field that shows the magnetic influence on electric currents, moving charges, and magnetic materials.

Q8

### What is a Hall effect sensor?

A Hall effect sensor is a sensor which detects the magnitude and the presence of magnetic fields using the principles of the Hall effect.

Q9

### What is a magnetometer?

A magnetometer is a device used to the magnetic field of an object.

Q10

### What is the main drawback of Hall effect sensors?

The accuracy of the measured quantities is not considered ideal as there is a chance of interferences by external magnetic fields.

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