# Hall Effect- a different outlook for seeing Hall coefficient

Hall effect is an extraordinary phenomena which involves generation of a subsidiary Electric field $V_H$ when a electric field and magnetic field is applied to a conductor. You can get a basic introduction from wiki. If you are not familiar with it

But familiar Hall effect introduces it through a bound rectangular boundary, which limits the current destity in y direction to zero in steady state. In brief the Hall coefficient is defined as $R_H=\frac{E_y}{J_x B}=\frac{1}{ne}$

where ”$J_x$” is the current density of the carrier electrons, and $E_y$ is the induced electric field,” n” is the carrier charge density and e is the carrier charge.

But consider the Hall effect in a 2D ring.The electric field applied in the ends if inner and outer ring and the magnetic field in the transverse direction.Then solving the Lorentz force equation along with Drude theory model (eqn 1)we get a 2 coupled equation,which can be solved directly

$\frac{dp}{dt}=-e(E+v\times B)-\frac{p}{\tau}$—–(1)

which gives from drude theory (see Ascroft Mermin)

$\sigma_0 E_x=\omega_c \tau J_y+J_x ----(2) \\ \sigma_0 E_y=-\omega_c \tau J_x+J_y----(3)$

Solving the two equations by multiplying the equations successively by $p_x,p_y$ and adding/sutracting we get

$\sigma_0 (E\times J)_z=\omega_c \tau J.J$—-(4)

and

$\sigma_0 (E. J)_{xy}= J.J$—-(5)

where $\sigma_0 (E\times J)_z=E_xJ_y-E_yJ_x$—-(6)

$\sigma_0 (E. J)_{xy}=J_xE_x+J_yE_y$—-(7)

So we can define $tan (\theta_H)=\omega_c\tau$

which in common terms is the hall angle

where $\omega_c=\frac{eH}{m}$

Therefore the relation can be written in a more generally

$\frac{tan (\theta_H)}{H}=\frac{\sigma}{ne} \\using \\ \sigma=\frac{ne^2\tau}{m}$

The hall coefficient would then be defined by

$R_H=\frac{tan (\theta_H)}{\sigma H}=\frac{1}{ne}$

The angle between the net electric field and the net current is given by the angle $tan (\theta)$

This to me is a more general statement for Hall coefficient which reduces to the normal hall coefficient if we consider appropriate boundaries and noticing tan(theta) must be dimensionless and considering $E_x< we get

$R_H=\frac{E_y}{J_x H}=\frac{1}{ne}$

which is the usual hall-coeficient.

Therefore the definition of Hall coefficient should be the tangent of Hall angle between the net Electric field and current in the conductor per unit applied magnetic field per unit conductivity.

Also this would help to resolve some issues regarding energy conservation in hall effect.

Suggestions are welcome.

Aritra Kundu

PS:here is an interesting article regarding hall effect in circular geometry http://www.jstor.org/stable/983911?seq=4