# Hall effect and its experimental determination

#### Introduction

In this particular article Hall effect and its experimental determination, we are going to discuss Hall effect. we will also discuss experimental determination of Hall coefficient.

# Hall effect

In 1879 Hall discover that when a magnetic field is applied on a conducting plate in a definite (say z-directing) and current is made to flow in another direction (say x-direction) and current is made to flow in another direction (say x-direction) then in a direction perpendicular to both (y-direction) an electric field is generated as shown in fig. This effect is called the Hall effect. Hence, with the help of this effect Hall found that the change carriers in the conductor are electrons.

To understand Hall effect consider a copper plate which a current Ix is flowing in the x-direction. When a magnetic field Bz is applied on this plate (in a direction perpendicular to the plane of the paper and inwards) the moving charge carriers experience a magnetic force q(Vd × B) due to which they drift in the direction of the force. If the ends (and D of the plate are not connected to the each other externally the charge carriers accumulate one face and a deficit of charge carriers is produced on the other face.

These charge carriers cannot go on accumulating on one end because they create a transverse electric field E inside the plate which will oppose the drift of charge carriers. And When the magnetic and electrostatic forces on the charge carriers become equal the drift stops and a steady state is obtained. Consequently, the electric field E so generated is the steady state is called the Hall electric field.

### Theory and derivation

Lorentz force

F= q[E + Vd × B] = 0

where q is the charge on the charge carriers and Vd is their drift velocity

∴  E = -Vd × B

And the magnitude of Hall field E = Vd Bz     …..(1)

Due to a flowing current Ix, the current density is

Jx = nq Vd            …..(2)

where n is the number of charges carriers per unit volume. Using the value of Vd in equation (1).

E/Jx Bz = 1/nq    …..(3)

The ratio E/Jx Bz is called Hall coefficient R.

R = E/Jx Bz = 1/ nq   ….(4)

Measuring experimentally the current density Jx, magnetic field Bz and induced Hall electric field E, Hall coefficient can be determined. The determination of hall coefficient leads to the following important facts.

#### Important facts about Hall coefficient

(1) The nature of charge carriers is revealed by Hall coefficient R is negative the charge carriers in the place are electrons and if it is positive then the charge carriers are positive holes. The value of the Hall coefficient for some materials are given in the following table :

(2) If the magnitude of charge q on charge carriers is taken to be charged on an electron, the Hall coefficient R is inversely proportional to the number density n of charge carriers i.e. for a material of lesser Hall coefficient the number of charge carriers is more compared to a material having a higher value of Hall coefficient. Thus the measurement of Hall coefficient enables one to determine the number density of charge carriers.

(3) With the help of the Hall coefficient, the mobility of charge carriers can also be determined. By definition

Mobility, μ = Vd/Ex  …(5)

Where Vd is drift velocity of charge carriers due to applied electric field Ex. And now using equation (1)

μ = E/Ex. Bz = Φ/Bz

where Φ = E / Ex is called Hall angle.

Substituting the value of E/Bz from equation (4)

μ = Jx.R/Ex     …(6)

(4) Using Hall coefficient the conductivity of the material can also be determined

∴ Conductivity σ = Jx / Ex

Hence, σ = μ/R

##### Experimental determination of Hall coefficient

the experimental set-up for the determination of Hall coefficient is shown in fig. A sample of the material whose Hall coefficient is to be determined is placed between the poles of a strong magnet in such a way the section (length l× breadth b) is perpendicular to the field. And the strength of the magnetic field is determined using a gauss-meter. The t is the thickness of the sample then a current is passed between the end faces of the area (b × t). The current is measured by an ammeter connected in series with the sample. Hence due to the flow of current hall voltage V is induced between the faces of the area (l×t). This voltage is measured by a sensitive potentiometer.

because of Hall coefficient R= E/Jx Bz

where Hall electric field E= V/b

And current density Jx= Ix/bt

∴ R= V/b.(bt/Ix.Bz)

And, R= (V/Ix).(t/Bz)

Thus knowing the values of V, Ix, Bz, and t, Hall coefficient can be determined.

###### Conclusion

In this article we have discussed Hall effect in most easiest way along with its derivation and formulas.