Hall Effect is the phenomenon of production of transverse voltage in a current carrying metallic slab (strip) on applying a magnetic field along a direction perpendicular to the direction of current. The transverse voltage produced in this effect is known as Hall voltage.

A current $I$ is passed through a flat metal strip (like copper) in a direction as shown in figure.

Hall effect: Hall voltage derivation

A magnetic field $B$ is applied at right angle to the strip which is directed outward of the plane. Due to this magnetic field, the charge carriers of the metallic strip experienced a force. If $q$ charge is moving with velocity $v$ then the force experienced by the charge in the magnetic field $B$ is $Bqv$.

According to Fleming’s left hand rule, if the charge carriers are electrons then this force will act on them in the direction from $CD$ to $AG$. So, these electrons are accumulated at lower side $AG$ of the strip, leaving positive charge at its upper side $CD$ which makes the side $CD$ at positive potential and the side $AG$ at negative potential. In this way, a potential difference is set up. This particular potential difference between these two sides of the strip which opposed further flow of charges is called Hall potential difference or Hall voltage.

Expression for Hall Voltage

Let $V_H$ be the Hall voltage across the width $GD=d$ of the strip. Let $E$ be the electric field which is equal to the potential gradient. \[E=\frac{V_H}{d}\]

The electric force on each electron is, \[F=eE=e\frac{V_H}{d}\]

When the electrons are in equilibrium, the electric force is equal to the magnetic force i.e. \[e\frac{V_H}{d}=Bev\] \[V_H=Bvd\text{ __(1)}\]

The drift velocity of electron is given by, \[v=\frac{I}{nAe} \text{ __(2)}\]

Where, $A$ is the cross sectional area of the strip and $n$ is the charge carriers per unit volume.

From $\text{(1)}$ and $\text{(2)}$ \[V_H=\frac{BId}{nAe}\text{ __(3)}\]

Let $t$ be the thickness of the conductor, then \[A=d×t\]

Then, from equation $\text{(3)}$, \[V_H=\frac{BId}{ndte}\] \[V_H=\frac{BI}{net}\]

This is the expression for Hall Voltage. This relation shows that Hall voltage is greater in those material for which $n$ is smaller. The quantity $\frac{1}{ne}$ is known as Hall coefficient $(H_C)$.

Hall resistance of the metal strip is given by, \[R_H=\frac{V_H}{I}=\frac{Bd}{nAe}\]

Hall Effect is widely applied in many devices. It is also used to measure magnetic field strength.