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Drift Velocity and Ohm’s Law

Conductors and Insulators

An atom consists of a central positively charged core known as nucleus. The nucleus is surrounded by a number of electrons revolving around it in particular circular orbits. In some substances, the electrons in the outer most orbits are loosely  bound to the nucleus. These electrons may leave the atom and become ‘free electrons’. The motion of free electrons is random. Due to the motion of free electrons, charge is carried from one end of the substance to the other. In certain substances, the orbital electrons are strongly bound to the nucleus. In these substances, free electrons are available only in small number. So, charge cannot flow easily from one end of the substance to the other. Depending upon the capacity to allow the passage of the charge, the substances are generally classified into two categories.

(i) conductors      (ii) insulators

Conductors are those substances through which electric charge can pass easily.

Metals such as silver, iron, copper, gold, aluminum, brass, human body, earth, aqueous solutions of salts, acids and bases are some example of conductors. Among metals, silver is the best conductor of electricity. Conductors contain large number of free electrons. Due to the repulsion between free electrons, they get evenly scattered throughout the conductor. Therefore, no portion of the conductor has a accumulation of electrons (i.e., charges).

Insulators (also called dielectric) are those substances through which electric charge cannot pass easily.

Quartz, glass, wood, dry air, gases, dry silk, pure water, bakelite, mica, amber, ebonite, sulphur, wax, oils etc. are some examples of insulators. Insulators contain a negligible number of free electrons. In any region of insulators happens to have an accumulation of electrons, they will remain localised in that region. It may be mentioned here that the conductivity of any substance is affected by temperature. On heating, the insulators tend to become conductors.

Apart from the above two categories, some substances like germanium, silicon, etc., have been classified into a third category known as semi-conductors. They possess conductivity lying in between that of conductors and insulators. No substance is a perfect conductor or a perfect insulator. The difference between conductors and insulators is only of the degree. Under suitable conditions, both conductors and insulators can be electrified.

Refer this Simulation for Insulator and Conductor

I've combined two animations to show how charges move on an insulator (latex balloon, and on a conductor, mylar balloon.

Drift Velocity

The velocity with which the free electrons are drifted towards the positive terminal, under the action of the applied electric field, is called the drift velocity of the free electrons.

A good conductor is that which contains a large number of free electrons, i.e., the electrons which can be moved easily from one point to the other. Assuming that one free electron per atom is available; the number of free electrons per cubic meter of copper is of the order of 1026. All the free electrons move with all sorts of velocities in all possible directions. These velocities are due to their thermal energies. As a result of this, random distribution of velocities, there is no average velocity of electrons. If u1, u2,....,uN are the velocities of various electrons, average velocity u is,

u = u1+u2+...+uN/N = 0

Hence, there is no net transference of charge across any cross-section of the conductor. In other words, there is no current flowing through the conductor. When the ends of the conductor are connected to a source of e.m.f. having potential difference ‘V’, an electric field E (= – V/l), where ‘l’ is the strength of the conductor, is set up in the conductor. Free electrons move in a direction opposite to ‘E’. The electrons are constantly colliding with each other. They get accelerated in between two consecutive collisions and lose all their energy on next collision. Again, they get accelerated and this process is repeated.

Force on free electron = -eE

So, acceleration, a = -eE/m = – e/m (- V/l)

a = eV/ml                …... (1)

Let τ1, τ2, ….,τN be the average time intervals between two consecutive collisions of electrons numbered as 1,2, …. n respectively. Velocities acquired by various electrons (in the presence of electric field) are

v1 = u1 + aτ1

v2 = u2 + aτ2

..................

.................

vN = uN + aτN

Therefore, average velocity ‘v’ with which an electron moves or drifts due to electric field, is given by

v = v1+v2+......+vN/N

or, v = (u1+a τ1) + (u2 + aτ2) + …...+ (uN + aτN)/N

= [u1+u2+......+uN/N]  + [a(τ1+τ2+....+τN)/N]

= 0 + aτ

Or, v = aτ

Or, v = (eV/ml)τ                  …... (2)

Here, ‘τ’ is the average time interval between any two consecutive collisions and is called relaxation time.

It may be noted that the order of  drift velocity ‘v’ is very small (approximately 1 mm s-1).

Electric Current and Drift Velocity

Consider a conductor having its two ends raised to potentials V1 and V2 (V1 >V2). If ‘n’ is the number of free electrons per unit volume of the conductor, then the number ‘N’ of electrons crossing AB in ‘t’ second is given by,

N = n  (volume of cylinder)

= nAvt

Here, ‘A’ is the area of cross-section of the conductor. Also if ‘e’ is the charge on one electron, then charge ‘q’ crossing AB in ‘t’ second will be,

q = N.e = nAvte

Therefore, the electric current ‘i’ flowing through the conductor is given by

i = q/t = nAvte/t = nAve    …... (3)

Since n, A and e are constant.

i ∝ v

Thus, the electric current flowing through a conductor is proportional to the drift velocity of the electrons.

Ohm’s Law

From equation (2), v = aτ

substituting for ‘a’ from equation (1), we get,

v = (eV/ml)τ

Substituting for v in equation (3), we get

i = nAe [(eV/ml)τ]

Or, i = (nAe2τ/ml) V

Or, i = (1/R) V

Here, 1/R = (nAe2/ml)τ

Since, n,A,m,l and τ are all constant quantities,

i ∝ V

Thus, at constant temperature current flowing through a conductor of uniform area of cross-section, is proportional to the difference of potential across its terminals. This is Ohm’s law for conductors.

Therefore, R = (ml/nAe2) (1/τ)

’R’ is known as the resistance of the conductor.

Refer this video to know more about on Ohm’s Law.

Drift velocity of the electrons should not be confused with the velocity with which electricity is conducted.

Drift velocity of electrons is of the order of a few mm per second while conduction of electricity, which is in the form of an electric field or a wave-motion, takes place with velocity of light.

The charge carriers in solids are mostly the free electrons.

In the case of liquids, the charge carriers are the positive and negative ions.

The charge carriers, in the case of gases, are the ions.

The charge carriers in the case of vacuum tubes are the thermions or electrons emitted by the filament.

Ohm’s law refers to the proportion relation between voltage and current. It also applies to the specific equation V=IR, which is valid when considering circuits that contain simple resistors (whose resistance is independent of voltage and current).

?Circuits or components that obey the relation V=IR are known as ohmic and have current-voltage plots that are linear and pass through the origin

Material that obeys Ohm's Law is called "ohmic" or "linear"  because the potential difference across it varies linearly with the current.

Voltage drives current while resistance impedes it.

There are components and circuits that are non-ohmic; their I-V plots are not linear and/or don't pass through the origin.

An increase of potential energy in a circuit causes a charge to move from a lower to a higher potential (ie. voltage). Note the difference between potential energy and potential.

Though the Ohm’s Law is quite useful and extremely important, yet it has certain restrictions. We enlist below some of the limitations of this law:

(a) As ids cussed above, this law applies only to conductors and moreover, only at a constant temperature. It is a known fact that the resistance of a conductor increases with temperature. Hence, when the temperature changes, the graph of the conductor would not be linear.

(b) Insulators do not conduct electricity and hence the ohm’s law does not apply to them. But, despite of this fact, when an extremely high voltage is exerted on an insulator, it leads to a dielectric breakdown and current starts flowing suddenly.

(c) Ohm’s law does not apply to semiconductors. The material begins to conduct properly only beyond a specific voltage.

(d) The relationship between V and I depends on the sign of V for the same absolute value of V.

(e) The relation between V and I is non-unique, that is, for the same current I, there is more than one value of voltage V.

An electric current of 16 A exists in a metal wire of cross section 10-6 m2 and length 1 m. Assuming one free electron per atom. The drift speed of the free electron in the wire will be (0.00x) m/s. Find value of x.

(Density of metal = 5103 kg/m3, atomic weight = 60)

Solution:

We know that, drift velocity, vd = i/neA

n = (5103/60)  6.02 1026

Thus, vd = [(1660) / (51036.0210261.610-1910-6)]

= 210-3 m/s

Thus, from the above observation we conclude that, the drift speed of the free electron in the wire will be 210-3 m/s.

Question 1

The drift velocity of mobile charge carriers in electric circuits is

(a) very fast; less than but very close to the speed of light

(b) fast; faster than the fastest car but nowhere near the speed of light

(c) slow; slower than Michael Jackson runs the 220-meters

(d) very slow; slower than a snail

Question 2

If an electric circuit could be compared to a water circuit at a water park, then the current would be analogous to the ____.

(a) water pressure          (b) gallons of water flowing down slide per minute

(c) water                        (d) bottom of the slide

Question 3

A certain electrical circuit contains a battery with three cells, wires and a light bulb. Which of the following would cause the bulb to shine less brightly? Choose all that apply.

(a) increase the voltage of the battery (add another cell)

(b) decrease the voltage of the battery (remove a cell)

(c) decrease the resistance of the circuit

(d) increase the resistance of the circuit

Question 4

Which of the following will cause the current through an electrical circuit to decrease? Choose all that apply.

(a) decrease the voltage              (b) decrease the resistance

(c)  increase the voltage            (d) increase the resistance

Question 5

If the resistance of a circuit were tripled, then the current through the circuit would be ____.

(a) one-third as much

(b) three times as much

(c) unchanged

(d) None of these

Q.1
Q.2
Q.3
Q.4
Q.5

d

b

b & d

a & d

a

Related Resources:-

You might like to Resistance and Resistivity.

For getting an idea of the type of questions asked, refer the  Previous Year Question Papers.