Hall Effect

The Hall effect occurs when a current-carrying conductor is placed in a perpendicular magnetic field. The Lorentz force deflects charge carriers, building up a transverse electric field — the Hall voltage.

Setup

• Current I flows in the x-direction
• Magnetic field B points in the z-direction
• Hall voltage V_H develops in the y-direction

Hall Coefficient

The Hall coefficient characterizes the sign and density of charge carriers:

$R_H = \frac{1}{ne}$

where n is the carrier density (m⁻³) and e = 1.602×10⁻¹⁹ C. For electron carriers, R_H is negative (electrons deflect opposite to holes), but the magnitude is |R_H| = 1/(n|e|).

Hall Voltage

For a sample of thickness t (in the B-direction):

$V_H = \frac{IB}{net}$

This follows from equilibrium between the Lorentz force (evB) and the Hall electric force (eE_H).

Measuring Carrier Density

Rearranging: n = 1/(R_H · e), so a Hall measurement directly gives the carrier density.

Applications

The Hall effect is used to:

• Determine carrier type (electrons vs. holes)
• Measure carrier density in metals and semiconductors
• Build Hall-effect sensors for magnetic field measurement

Your Task

def hall_coefficient_m3_C(n_m3):
    # R_H = 1 / (n * e), e = 1.602e-19
    pass

def hall_voltage_V(I_A, B_T, n_m3, t_m):
    # V_H = I * B / (n * e * t)
    pass

def carrier_density_from_hall(R_H_m3_C):
    # n = 1 / (R_H * e)
    pass