Crossed Fields: The Hall Effect — Question 3

Question 3

Changing strip width has different Hall consequences at fixed current and fixed current density

A homogeneous single-carrier Hall strip has density n, signed charge q and fixed thickness t. Its width w is doubled while the perpendicular field B is held fixed.

1. Compare the Hall voltages before and after widening when total current I is maintained.

2. Compare them when current density Jx is maintained instead, and explain the apparent difference.

Question 3 - Solution

Changing strip width has different Hall consequences at fixed current and fixed current density

(a) Method and calculation

$$V_H=-\frac{IB}{nqt},\quad I\text{ fixed}.$$$$w\prime=2w\Rightarrow v_d\prime=v_d/2,\quad V_H\prime=V_H.$$

(b) Interpretation and further calculation

$$E_y=J_xB/(nq),\quad V_H=-E_yw\quad(J_x\text{ fixed}).$$$$w\prime=2w\Rightarrow I\prime=2I,\quad V_H\prime=2V_H.$$

Physical check

The width cancels at fixed total current because a wider strip has a proportionally smaller drift speed.