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 are electrons and is a steady-state condition with no movement of charges in the 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 is Hall voltage
  • EH is Hall field
  • v is drift velocity
  • d is the width of the metal slab
  • B is the magnetic field
  • Bev is a force acting on an electron
\(I=-nevA\)

Where,

  • I is an electric current
  • n is no.of electrons per unit volume
  • A is the 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 the induced electric field and to the product of applied magnetic field and current density. In semiconductors, RH is positive for the 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 is an electric field
  • v is drift velocity
  • Ris the Hall coefficient
  • 𝛍is the mobility of the 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 is Hall resistance

Hall Effect Derivation in Semiconductors

In semiconductors, electrons and holes contribute to different concentration and mobilities which makes it difficult for the explanation of the Hall coefficient given above. Therefore, for the simple explanation of a moderate magnetic field, the 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 is electron concentration
  • p is hole concentration
  • 𝛍is the mobility of electron
  • 𝛍is the mobility of 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.

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