Question

A current I flows to the right through a rectangular bar of conducting material, in the presence of a uniform magnetic field B pointing out of the page.

a) If the moving charges are positive, in which direction are they deflected by the magnetic field? This deflection results in an accumulation of charge on the upper and lower surfaces of the bar, which in turn produces an electric force to counteract the magnetic force. Equilibrium occurs when the two exactly cancel. (This phenomenon is called the Hall effect).

b) Find the resulting potential difference (the Hall voltage) between the top and bottom of the bar, in terms of B, v (the speed of the charges), and the relevant dimensions of the bar.

c) How would your analysis change if the moving charges were negative? [The Hall effect is the classic way of determining the sign of the mobile charge carriers].

Answer 1

(a)

Here the magnetic field is out of the page. Let the out of page is positive z direction. Then

Current flow is to the right. Therefore the velocity of the charged particles is toward right. Then

Then force is given by

                                

here negative sign indicates that the positive charges deflect downward. Hence the bottom plate acquires positive charge.

(b)

For the system to be in equilibrium, the force on the bar due to the magnetic field must be equal to the force due to the electric field produced by the accumulation ofpositive charges in at the bottom of the bar.

Therefore,

or

If t is the thickness of the bar, then the Potential difference across the ends of the bar is

Therefore resulting potential difference between the top and bottom of the bar is with bottom plate is at positive potential and the upper plate is at negative potential.

(c)

If the moving charges were negative, then for the current flowing toward right, charge flows toward left. Then the deflection of charged particles is still toward bottom of the rod.

The potential difference will be the same but the bottom plate is at negative potential and the upper plate is at positive potential.