# Derivation Hall Voltage due to the Hall Effect

Level 3 (with higher mathematics)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.
Updated by Alexander Fufaev on

## Video - Hall Effect and a SIMPLE Derivation of The Hall Voltage

In the lesson on the Hall effect, you learned how Hall voltage is generated. In the following we want to derive a formula for the Hall voltage $$U_{\text H}$$, which depends only on the quantities we can determine in the experiment.

We consider a Hall plate of width $$h$$, thickness $$d$$ and length $$L$$. It can be made of a metal or a semiconductor.

Then we send an electric current $$I$$ through the Hall plate. Electric current $$I$$ means that in the material - positive or negative charge carriers move in a certain direction with a drift velocity $$v$$. These charge carriers are either negatively charged (electrons with the charge $$q = -e$$) or positively charged (so-called holes with the charge $$q = +e$$).

Perpendicular to the direction of electron motion ($$v \perp \class{violet}{B}$$), a magnetic field with constant magnetic flux density $$\class{violet}{B}$$ penetrates the Hall sample.

## Electric and magnetic force inside the Hall sample

On the moving charge carriers always acts a magnetic force $$F_{\text m}$$ (Lorentz force), which deflects the charge carriers perpendicular to the magnetic field and perpendicular to the direction of motion. In this case, the magnitude of the Lorentz force is:

Formula for magnetic force
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Note! Lorentz force has a different direction, depending on whether you put for $$q$$ negative elementary charge $$-e$$ (for electrons) or positive elementary charge $$+e$$ (for holes).

Due to the deflection of the charge carriers by the Lorentz force, there is a negative charge excess at one edge of the sample and a positive charge excess at the other edge. This charge difference generates an electric field $$E$$, which exerts a electric force $$F_{\text e}$$ on all subsequent charges. It acts against the magnetic force.

The top and bottom edge of the Hall sample can be considered as plates of a parallel plate capacitor. The electric force between the plates is given by:

Formula for electric force in a plate capacitor
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The electric force is not only directed in the opposite direction to the magnetic force; it also becomes larger the more the charge carriers have been deflected by the magnetic force. After a short time a equilibrium of forces between the magnetic and the electric force is established, which is why you may equate the formulas 1 and 2:

Electric force equated with the magnetic force
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With the equilibrium of forces, the electric field stabilizes at a certain value. At the edges of the sample this E-field can be measured as Hall voltage $$U_{\text H}$$. Electric field is related to the voltage across the distance $$h$$ between the edges of the sample: $$E = \frac{U_{\text H}}{h}$$. Substitute the equation into the formula 3 for the E field. Charge $$q$$ cancels out:

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You cannot measure the average velocity $$v$$ of the charge carriers directly, so substitute it using the formula for uniform motion. If a charge carrier travels the distance $$L$$ (length of the sample) within the time $$t$$, then we can write the velocity as follows:

Velocity is distance per time
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The time $$t$$ can then be determined using the formula for electric current:

Current is charge per time
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Rearrange Eq. 6 with respect to time:

Time equals charge per current
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Substitute equation 7 into equation 5:

Velocity equals distance times current divided by charge
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Here, the amount of total charge $$Q$$ moving through the sample is the product of the number $$N$$ of moving charge carriers and their individual charge $$q$$ (depending on the type of charge, $$q$$ can be positive or negative):

Total charge is number of charge carriers miltiplied by single charge
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Substitute Eq. 9 into Eq. 8 to get the following formula for drift velocity:

Speed is distance multiplied by current divided by charge carrier number multiplied by charge
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Substituting the velocity 10 into the Hall voltage formula 4 yields:

Formula for Hall voltage via distance, current, thickness, magnetic field, number of charges and charge
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We can express the number of charge carriers $$N$$ contributing to the current with the charge carrier density $$n$$. Charge carrier density is defined as the number of charge carriers $$N$$ per volume $$V$$ of the conductor (here: Hall sample):

Charge carrier density is number of charges per volume
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Therefore, replace charge carrier number $$N$$ in equation 11 using Eq. 12 by rearranging 12 for $$N$$:

Formula for Hall voltage via distance, current, thickness, magnetic field, charge carrier density, volume and charge
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Now you can simplify the formula 13 a little bit. The volume $$V= h \, L \, d$$ is the product of height $$h$$, length $$L$$ and thickness $$d$$ of the Hall sample. Substitute it into Eq. 13 and then cancel $$L$$ and $$h$$:

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The coefficient $$\frac{1}{n \, q}$$ is called the Hall constant and is abbreviated as $$A_{\text H}$$:

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