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

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

Where,

VH: Hall voltage

EH: Hall field

v: drift velocity

d: width of metal slab

B: magnetic field

Bev: force acting on electron

\(I=-nevA\)

Where,

I: electric current

n: no.of electrons per unit volume

A: cross-sectional area of the conductor

\(V_{H}=\frac{-Bi}{net}\)

\(\frac{E_{H}}{JB}=-\frac{1}{ne}\)

Where,

\(\frac{E_{H}}{JB}\) : Hall coefficient (RH) it is defined as the ratio between induced electric field and to the product of applied magnetic field and current density. In semiconductors, RH is positive for hole and negative for free electrons.

\(R_{H}=-\frac{1}{ne}\)

\(\mu _{H}=\frac{v}{E}=\frac{J}{neE}=\sigma R_{H}=\frac{R_{H}}{\rho }(v)\)

Where,

E: electric field

v: drift velocity

RH: Hall coefficient

𝛍H: mobility of hole

\(\frac{J_{y}}{J_{x}}=\sigma \frac{E_{y}}{J_{x}}=\mu _{H}B_{z}=\sigma R_{H}B_{z}\)

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

\(R=\frac{V_{H}}{i}=\frac{B}{net}\)

Where,

R: Hall resistance

Hall effect derivation in semiconductors

In semiconductors, electrons and holes contribute in different concentration and mobilities which makes it difficult for the explanation of Hall coefficient given above. Therefore, for the simple explanation of moderate magnetic field, following is the Hall coefficient:

\(R_{H}=\frac{p_\mu {H}^{2}-n\mu _{e}^{2}}{e(p\mu _{H}+n\mu _{e})}\)

\(∴R_{H}=\frac{(p-nb^{2})}{e(p+nb)^{2}}\)

Where,

\(b=\frac{\mu _{e}}{\mu _{H}}\)

n: electron concentration

p: hole concentration

𝛍e: mobility of electron

𝛍H: mobility of hole

e: 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 Hall effect. Stay tuned with BYJU’S and learn various other Physics related topics.

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