Introduction of 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 \[R=\frac{VH}{i}-\frac{B}{net}\]
FAQs on Hall Effect Derivation
1. What is the Hall effect in Physics?
The Hall effect is the phenomenon where a voltage difference, known as the Hall voltage, is produced across an electrical conductor. This voltage is transverse (perpendicular) to both the electric current flowing through the conductor and an external magnetic field applied perpendicular to the current. It is caused by the Lorentz force acting on the charge carriers, which pushes them towards one side of the conductor.
2. How is the Hall Effect derived for a simple conductor as per the Class 12 syllabus?
The derivation begins by considering the forces on a charge carrier (like an electron) inside a conductor. When current flows and a magnetic field is applied perpendicularly, the magnetic Lorentz force (Fm = qvB) pushes the charges sideways. This accumulation of charge creates a transverse electric field (Hall field, EH), which exerts an opposing electric force (Fe = qEH). In equilibrium:
- Magnetic Force = Electric Force
- qvB = qEH
- This leads to EH = vB, where 'v' is the drift velocity of the charges.
3. What is the Hall Coefficient (RH), and what does its sign signify?
The Hall Coefficient (RH) is a parameter that quantifies the Hall effect in a specific material. It is defined by the formula RH = 1/(nq), where 'n' is the concentration of charge carriers and 'q' is the charge of each carrier. The sign of the Hall Coefficient is crucial as it reveals the nature of the dominant charge carriers:
- A negative sign for RH indicates that the primary charge carriers are electrons (q is negative), which is typical for metals and n-type semiconductors.
- A positive sign for RH indicates that the primary charge carriers are holes (q is positive), which is characteristic of p-type semiconductors.
4. What are some key real-world applications of the Hall effect?
The Hall effect has several important practical applications, including:
- Hall Effect Sensors: These devices are used to detect the presence and magnitude of magnetic fields. They are found in many technologies, such as car ignition systems, computer keyboards, and smartphone compasses.
- Determining Material Properties: It is used to determine the type (n-type or p-type), density, and mobility of charge carriers in semiconductor materials.
- Current Measurement: Hall effect sensors can measure the magnitude of a current in a wire without making direct contact, as the current produces a surrounding magnetic field.
5. What is the difference between Hall voltage and the potential difference from Ohm's Law?
The key difference lies in their direction and origin. The potential difference described by Ohm's Law (V=IR) is measured along the length of the conductor in the same direction as the current flow and is caused by the material's electrical resistance. In contrast, the Hall voltage is measured across the width of the conductor, perpendicular to the current flow, and is caused by the magnetic Lorentz force deflecting charge carriers.
6. How does the Hall effect help determine if a semiconductor is n-type or p-type?
The Hall effect provides a direct method to identify the type of a semiconductor by measuring the polarity of the Hall voltage. When a semiconductor is placed in the experimental setup:
- If the measured Hall voltage is negative, it implies the accumulated charges are electrons. This means electrons are the majority charge carriers, and the semiconductor is n-type.
- If the measured Hall voltage is positive, it implies the accumulated charges are positive holes. This means holes are the majority charge carriers, and the semiconductor is p-type.
7. What would happen if the magnetic field was applied parallel to the direction of current in a Hall effect experiment?
If the magnetic field were parallel to the current, the Hall effect would not occur, and the Hall voltage would be zero. This is because the Lorentz force, given by the formula F = q(v × B), depends on the cross product of the charge carriers' velocity (v) and the magnetic field (B). When these two vectors are parallel, their cross product is zero. Consequently, there is no transverse force to push the charges to the side of the conductor, and no Hall voltage can be generated.
