Question

Explain the origin of resistaance in a conductor

Answer 1

In electrostatic condition, the electric field is zero everywhere within the conductor, and there is no current. It does not mean that all charges within the conductor are at rest. In metals such as copper or aluminium, some of the electrons are free to move within the conducting material. These free electrons move randomly in all directions. But, the electrons do not escape from the material as they are attracted to the positive ions of the material. Since the motion of the electrons is random, there is no net flow in any direction and hence the current is zero.

When the same metal is in an electric field, the electrons are subjected to a steady force given by F = qE. If the electrons were moving in vacuum, the steady force would cause a steady acceleration in the direction of F  an the electrons speed would have gone up. But in a conductor, the electrons undergo frequent collisions with the ions of the material. Hence, the direction of the electrons undergoes a random change. The net effect is that in addition to the random motion of the electrons, there is also a very slow net motion or drift of the electrons. This drift or flow with a constant velocity (in a direction opposite to E) is called the drift velocity V_D.

To find the relationship between current and drift velocity, consider a conductor of length l and area of cross-section A. If V is the potential difference across the ends of the conductor, then the strength of the electric field is

E = V/I

The acceleration acquired by each electron due to the electric field is

where is the coulomb's force experienced by each electron and m is the mass of the electron.

- eE

The drift velocity of the electrons is given by

where t, the relaxation time, is the average time that an electron spends between two collisions. It is of the order of 10^-14 s.

CE

Now the volume of the conductor is equal to Al and if n is the number of free electrons per unit volume, then, the total number of free electrons in the conductor will be equal to n Al.

Hence, the total charge q = - n Ale

The time taken by free electrons to cross the conductor is

1.9,"Ale x Vd

where we have substituted for t and q.

1 = _nAeVd

For a given conductor, I a V_d

A small value of drift velocity 10^-5m/sec produces a large amount of current, as there are a large number of free electrons in a conductor.

The drift velocity of the electrons V_d is (using E = V / l)

Also I = - neAV_d

On substituting for V_d in the above expression we get,

or

ne2τ

From this, the resistance R can be identified as

ne2τ A

where,

R : resistance of the conductor

m : mass of the electrons

l : length of the conductor

n : density of free electrons in the conductor

e : electronic charge

A : area of cross-section

t : relaxation time

Using the equation R-ρ--, we get for the resistivity ρ of the material ρ