Hall effect.

• Hall effect is the production of voltage across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current

The above figure shows a conductor placed in a magnetic field (B) along the z-axis. The current (I) flows through it along the x-axis

Hall voltage $(V_H)$ is developed along y-axis with electric field intensity $E_H$.

At Equilibrium,

(Force due to Hall voltage on charge carriers)=(Force due to magnetic field)

$qE_H = Bqv$

Where,

q = Magnitude of current

v = Drift velocity.

$E_H = \frac{V_H}{d}$

But, $\frac{V_H}{d} =Bv$

$IeV_H = B.v.d….(i)$

Let n.e be the charge density,

Now, I=(n.e)(w.d).v

$v = \frac{I}{n.e.w.d} ......(ii)$

From (i) and (ii)

$V_H = B.\frac({I}{n.e.w.d})d$

$V_H = \frac{BI}{new}$

This is the required expression for Hall voltage.

If $J = \frac{I}{wd}= Current \ \ density$

Hall voltage can also be written as

$V_H = \frac{BJd}{ne}$