Quantitative Analysis of Hall Effect

We assume free electron model of the metal.
Let vd be the constant drift velocity of the charge carriers. The magnetic deflecting force that causes the charge carriers to drift towards the right edge of the strip is q (v_d x B).

The charge carriers do not build up without limit on the right edge of the metal strip. This is because the displacement of the charges towards right sets up a transverse Hall Electric field EH which opposes further motion of the charges to the right. The Hall Electric Field EH is related to Hall potential difference V_xy by
                    E_H = v_xy/d

The force on a charge q due to this field in qE_H.
Eventually the equilibrium is reached when the deflecting magnetic force on the charge carriers is balanced by the oppositely directed Hall electric force. Thus the net force on the charges becomes zero.
        qE_H + q(v_d x B)
or            E_H = - V_d x B
Thus If the charge carrier is positive, it moves with a drift velocity vd parallel to the direction of current. If it is negative, its direction of motion is opposite to the direction.
If v_d and B are at right angles, Eq. (1) in magnetic only is
        E_H = v_d x B
Now, the drift speed vd of the charges is related to the current-density J by
        v_d = J/nq

Here, n = number of charge-carriers per unit volume.
From Eqs. (2) and (3),
        E_H = JB/nq or    n = JB/qE_H
Thus, the value of n can be found out.

Hall Coefficient

    E_H, J and B can be experimentally measured.
    E_H = V_xy /d. Also, J= i/A where I is the current in the plate and A the cross-sectional area. Thus from Eq. (4), 1/nq can be known. This factor is called the ‘Hall coefficient” (R_H).
        1/nq = E_H/JB = R_H

Applications of Hall Effect

1.    Hall effect gives information about the sign of charge carriers in electrical conductors.
2.    Measurement of Hall coefficient gives us the number of current carriers is positive or negative.
3.    The mobility of carriers can be measured by the conductivity of the material and Hall coefficient.
Mobility (μ) = σ R_H
4.    Hall effect may be used to measure the flux density of a magnetic field. A simple instrument used for this purpose is known as Hall probe.

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