# Hall Effect

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.

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.