**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.