Hall Effect Derivation

Before moving on to Hall effect derivation, students must note that the Hall effect is the production of the voltage difference. It is caused across an electric conductor and is transverse to this electric current. It essentially refers to the product of magnetic induction and current density when a magnetic field works perpendicular to the current flow associated with a thin film. 

What is the Hall Effect?

The Hall Effect, also called the Hall–Heroult effect, is the influence of the transverse component of an electric current on the magnetic field in a transverse Hall bar. When a current flows in a plane perpendicular to the direction of the magnetic field, the Lorentz force acts on the magnetic charges (magnetic flux) enclosed by the transverse current. 

The Lorentz force is the force experienced by a magnetic charge when acted upon by an external force field, which in this case is the transverse electric current. Since the electric current has a certain polarity, the Lorentz force pushes the enclosed magnetic charge into a direction parallel to the electric current.

This motion of the magnetic charge produces a magnetic field that is perpendicular to the current, and by Faraday's law, the resultant Lorentz force produces a tangential magnetic field. Thus, the Hall effect is also referred to as the transverse Hall–Heroult effect.

The Hall voltage generated in a magnetic field is called a Hall effect. In physics, a Hall voltage is a voltage generated when current flows perpendicular to a magnetic field. It is proportional to the current density, hence inversely proportional to the sample thickness (the Hall effect does not work with planar devices).

History

The Hall Effect was discovered independently in 1890 by Orest Chwolson and Edwin Powell in the U.S. and in Russia by Vladimir I. Vernadsky. Edwin Powell and Orest Chwolson reported their findings of the Hall effect in Nature in October 1890, where they described the effect as 'Hall's Law.’ 

But the term Hall effect was first suggested by Orest Chwolson, who used the term Hall law in a paper published in 1892. The paper was not well received by the Physical Society in New York, due to the unorthodox terminology that Chwolson had used. After further investigation by E.M. Purcell and Joseph Henry, a second paper by Chwolson was accepted by the Physical Society, and the terminology Hall Effect was used.

Theory 

The Hall Effect can be defined by two variables:

• The electric current through a current-carrying conductor (the input).

• The magnetic field surrounding the conductor (the input) and one observable variable the Hall voltage that develops across a Hall plate (the output).

Magnetic Field 

In the Hall Effect, an electric current passes through a current-carrying conductor. The magnetic field surrounding the conductor is inversely proportional to the radius of the conductor. The current through the conductor can be thought of as two fluxes (current flowing in opposite directions) in quadrature; the product of the fluxes gives the total flux through the conductor.

If the conductor is thin and the total flux is very small, the magnetic field around the conductor is approximately uniform. This gives an approximate formula for the magnetic field, that when multiplied by a constant, produces the Hall voltage measured across the conductor

Voltage 

When an electric current passes through a conductor, it produces an electrostatic field around the conductor due to the electrical charges of the current. The charges of the current produce a potential difference across the conductor. If the conductor is made up of one or more conductors with currents in the same direction, it is a common case that a Hall voltage develops across the conductor, and is measured as the output. 

Electromagnetic Effect 

The Hall Effect is an electromagnetic phenomenon, and it can be applied to many electromagnetic devices. This is because the current-carrying conductor, the Hall plate, and the magnetic field all have an electrical charge and produce an electric field in the space surrounding them. For example, a magnet produces a magnetic field, and when a current of electrons passes through the magnetic field, the electrons change direction and produce an electric field in the surrounding space. 

Electronic Effect 

When electrons pass through the conductor, the movement of electrons produces a voltage between the electrons and the conductor. This is because they are all negative, and if they were charged in a similar way to a battery, they would repel each other and form a positive space between them and the conductor.

The voltage that is produced by this is proportional to the current through the conductor. This current can be obtained by measuring the Hall voltage and applying the formula.

What are the Applications of Hall Effect? 

Understanding this concept in its initial level involves an explanation of the scope of practical application that Hall effect derivation has. These are as follows:

• Hall effect helps in measuring the magnetic field around an electrical charge, and thus qualifies as a magnetometer. 

• Hall effect formula enables one to determine whether a material serves as a semiconductor or an insulator. 

Hall effect definition finds immense application in integrated circuits (ICs) in the form of Hall effect sensors.

• Semiconductor Hall devices 

The Hall devices are used to measure the current and potential in a circuit. This is useful in many applications such as in integrated circuit design and testing. Common implementations include high-side and low-side Hall devices.

• Hall Effect sensors 

These are small solid-state semiconductor devices used to measure magnetic fields, such as those produced by Earth's magnetic field, and used in a number of other applications, such as sensing for vehicles, aerospace and electronic devices. Hall sensors are also a popular alternative to reed switches for use in small appliances such as microwave ovens, microwave bread toasters and cordless vacuum cleaners.

• Magnetoresistivity and Magnetoelectric effect 

In the metallic conductor, electrons are free to move from one point to another. Due to this freedom of movement, the conductor can be polarized by the magnetic field. The movement of electrons causes a change in the resistance of the conductor which can be measured by a Hall voltmeter. This change in resistance is dependent on the direction and strength of the electric field. In this effect, the resistivity of the conductor changes and the magnitude of resistance changes with the direction and the strength of the magnetic field.

• Electro-conductive materials for electromagnetic measurement 

The ability of the Hall Effect to measure the electric field enables its application in measuring the resistivity and the surface conductivity of non-metallic materials. Metallic materials do not have the magnetic field to induce an electromotive force but metallic wires can have current flowing through them.

The magnitude of the voltage is proportional to the strength of the electric field, and hence this enables the development of high-resolution Hall voltmeters.

With a brief light shed on its applications, let us move on to how you can make the Hall effect derivation from scratch.

How to Make Hall Effect Derivation? 

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Hall effect physics involves a metal body that contains a single form of charge carriers, like electrons. Also, the metal warrants a lack of movement of charges along the y-axis.

Therefore, one has to consider the following components of Hall effect expression components to have a better understanding of the derivation – 

However, the ‘I’ component within the Hall effect calculation stands for –nevA. In this case, ‘I’ stands for an electric current, ‘n’ signifies the number of electrons per unit volume, and ‘A’ is the conductor’s cross-sectional area. 

Therefore, the Hall effect derivation refers to the following – 

\[eE_{H}Bev\frac{evH}{d}=BevV_{H}=Bvd\]

However, this derivation stipulates that the force is downward facing because of the magnetic field (equal to the upward electric force), in the case of equilibrium. 

Another way to find the exact value of VH is through the following equation – 

\[V_{H}=\frac{-Bi}{net} \frac{EH}{JB}=-\frac{1}{ne}\]

This particular equation takes the help of Hall effect coefficient derivation, which is – 

\[\frac{EH}{JB}\]

Besides, the Hall coefficient (RH) implies the ratio between the product of current density and magnetic field and the induced electric field. 

However, the derivation of RH takes into account the factors as stated below – 

• E = electric field. 

• V = drift velocity. 

• μ = mobility of the hole. 

Therefore,

\[R_{H}=-\frac{1}{ne}\mu=\frac{v}{E}=\frac{J}{neE}=\sigma R_{H}=\frac{RH}{\rho}(v)\]

Also, you should be aware of the fact that the Hall angle in the Hall effect stands for the angle between electric field and drift velocity. It is essentially the ratio between density (signified by the x-axis) and current density (denoted by the y-axis). 

Hall resistance or \[R=\frac{VH}{i}-\frac{B}{net}\]