0.12 Magnetic force on a conductor  (Page 2/5)

Direction of magnetic force

The magnetic force in the downward direction tends to drift electron in downward direction following a parabolic path. This drifting polarizes the conductor strip electrically. We know that each infinitesimally small element of the conductor is electrically neutral. But, there is accumulation of negative charge at lower edge as electrons drift down due to magnetic force. Correspondingly, there is accumulation of positive charge at the upper edge as there is depletion of electrons exposing immobile positive atoms in that region. The process of polarization, however, continues only momentarily. At any moment, the opposite polarity of charges at the edges sets up an electric field. In this case, the electric field is directed from upper (positive edge) to lower edge (negative edge). This electric field, in turn, pulls electron upward.

Polarization of charges and electric field

The dynamic condition is brought under equilibrium when electric force equals magnetic force. Let “E” be the electric field at equilibrium,

$$eE=e{v}_{d}B$$ $$\Rightarrow v_d=\frac{E}{B}$$

where $v_d$ is the drift velocity. Once the equilibrium is reached, electrons keep moving with the drift velocity as they would have moved in the absence of magnetic field. Here, the opposite edges of the conductor strip function as infinite charged plates. The electric field, E, is given as :

$$E=\frac{V}{a}$$

where, “a” is the width of the conductor strip and “V” is the electrical potential difference between the edges of conductor strip. This potential difference between the edges is known as Hall’s potential. We can measure it by connecting a voltmeter to the edges of the conductor strip.

Numbers of free electrons per unit volume

The “Hall effect” can be used to measure numbers of electrons per unit volume in a conductor. We know that the drift velocity of an electron is :

$$v_d=\frac{I}{neA}$$

where “n” is numbers of free electrons per unit volume and “A” is the cross section area of the strip. Substituting in the equation of equilibrium, we have :

$$v_d=\frac{E}{B}$$ $$\frac{I}{neA}=\frac{E}{B}$$

Substituting for “E”, we have :

$$\Rightarrow \frac{I}{neA}=\frac{V}{aB}$$ $$\Rightarrow n=\frac{IaB}{eAV}$$

Also, the area A is product of width and thickness, A = ab. Hence,

$$\Rightarrow n=\frac{IB}{ebV}$$

The quantities in the right hand expression are either known or measurable. Thus, we are able to measure the numbers of free (conduction) electrons per unit volume using Hall’s effect.

Drift velocity

Use of Hall’s effect allows measurement of drift velocity as well. The magnitude of drift velocity is about 0.0003 m/s, which is quite a small value that can be measured in the laboratory. The determination of drift velocity uses a very simple technique based on the detection of Hall’s effect.

The idea here is to move the conductor strip carrying current in the direction opposite to the direction of drift velocity i.e. in the direction of conventional current in the presence of uniform magnetic field. The motion of conductor is adjusted such that the relative drift velocity of electron with respect to stationary magnetic field is zero. In this case, speed of conductor strip is equal to the drift speed of electron. Also, the magnetic force is zero as relative velocity of electrons with respect to stationary magnetic field is zero. In turn, there is no drifting of electron towards the edge of the conductor and the Hall potential is zero. Thus, we are able to detect when the velocity of conductor strip equals drift velocity of electron.