Applications of Lasers in Scientific Field

Applications of Lasers in Communication, Space, Surgery, Military...

The ingenious principle of laser was first worked out in 1958 by Charles Townes of Columbia University. He made use of this principle a few years earlier in 1953 to invent a very similar device- maser where microwaves are used instead of light waves. Hence laser is an optical maser. The first successful laser using a large synthetic ruby crystal was built by Teodore H Maimann in 1960. With the discovery of laser man's control of light has been and will continue to extend to an unpredictably large and diverse number of applications in scientific field.

(i) Communication :

Since light from the laser is coherent it can theoretically carry messages in the same manner as comparatively low frequency carriers. The frequency of the laser is so high that each message or band of frequencies is a very small percentage of the carrier frequency. Hence a large number of messages can be sent. It can accommodate millions of television channels.

(ii) Space exploration :

Rockets and satellites can be efficiently controlled by laser beam.

(iii) Measurement of large distances

Distance between earth and other planets can be accurately measured by laser beam because it is highly collimated.

(iv) Welding

The laser can be used effectively for spot welding. It possible to weld a joint even after the joint has been sealed inside glass envelope.

(v) Hole drilling

A sharply focused laser beam can be used to cut holes in diamond and other hard materials,

(vi) Surgery

(a) Tooth drilling : The beam destroys by vaporization the dark decayed spot.

(b) Eye - Surgery : Eye - tumors can be removed and detached retina of human eye can be welded by low power laser beam. The Lasik surgery is nowadays used in treating the defects of eye by correcting the shape of the cornea of the eye using laser beam to get sharp image of an object exactly on the retina of the eye.

(c) Cancer treatment : Laser beam can remove some cancerous cell instantly, accurately, and without pain or bleeding.

(vii) Thermonuclear fusion

Laser can he used to trigger a thermonuclear fusion.

(viii) Military applications

The laser beam can be used to destroy an attacking missile in space. Moreover it can be used as 'death-ray gun' to destroy enemies.

(ix) Photography

Laser beam can be used to produce three dimensional images without the use of lenses. This process is called Holography.

(x) Science

(i) Chemical reactions can be accelerated by exposing it to the laser beam.

(ii) Laser can he used to study Raman effect.

Doppler Effect in Light | Redshift and Blueshift

Doppler Effect in Light

We are already familiar with the Doppler effect in sound i.e., the apparent change in frequency of sound due to the relative motion between the source and the listener in a medium. It is named so, because of Doppler who was the first to have stated a theory in the case of sound waves in the year, 1842. He pointed out the relevance of the phenomenon in the case of light.

If the source of light (source of em waves) of frequency f is stationary and the observer moves towards the source with a velocity v (v << c, the velocity of light), the apparent frequency of light as observed by the observer,

f‘ = f(1 + v/c)

The above equation is also valid in the case when the source moves towards the stationary observer. If the source moves away from the stationary observer or the observer moves away from a stationary source,

f‘ = f(1 - v/c)

Doppler effect in sound is asymmetric whereas the Doppler effect in light is symmetric. It means that the apparent frequency of sound when the source is approaching a listener at rest and the apparent frequency when the listener is approaching the source at rest are different. But, for light, the source approaching the observer and observer approaching the source exhibit exactly the same Doppler change in the frequency.

Doppler effect is a convenient tool to estimate the speed and direction of motion of stars, planets etc. in our universe relative to us. In general, if the wavelengths of light received from these objects shift towards the red end of the spectrum, these objects are moving away from the earth. This is known as Redshift. If the wavelengths of spectral lines emitted by these objects shift towards the violet end of the spectrum, these objects are moving towards the earth. In the case of stars, the red shift is observed. This shows that stars are moving away from the earth. By finding the shift, the velocity of these heavenly bodies can be estimated. When waves are received from a source which is moving in the direction of the observer, then there would be an apparent decrease in the wavelength. This is referred to as Blueshift.

The Doppler effect finds application to estimate the speed of aeroplane and automobiles, track artificial satellites, estimate the velocity and rotation of the sun etc.  

Practical applications of Joule's heating effect

 

Joule’s law of electrical heating

 

Heat produced in a resistor is directly proportional to

(i) the square of the current I,

(ii) the resistance R

(iii) the time t for which the current flows.

H = i²Rt joules

Therefore, H = VIt = (V²/R)t

 

Practical applications of Joule's heating effect

One of the inevitable consequences of the conduction of electric current through a conductor is the Joule's heating effect. It is an undesirable effect in many cases because, in these cases a part of the useful electrical energy is wasted as heat energy. For examples in electric motor, transformer, transmission lines etc., a part of the electrical energy is dissipated as heat energy. But Joule heating has also many applications. Some familiar domestic applications of joule heating effect in daily life are mentioned below:

 

1. Electric heating appliances

 

Electrical iron, electrical toaster, electrical oven, electrical kettle, electrical heater are some important electric appliances based on heating effect of electric current. The heating elements in these appliances should have (i) high melting point and (ii) high resistivity. Usually nichrome wire is used as the heating element.

 

2. Filament lamp [Incandescent electric bulb)

 

The filament of electric bulb when heated to a high temperature by electric current becomes white hot and emits light. The bulbs are usually filled with chemically inactive gases such as nitrogen, argon etc. to prolong the life of the filament. These bulbs give nearly 1 candela light energy for every watt of electric power consumed.

The material of the filament has (i) a high resistivity and (ii) high melting point. Usually tungsten (M.P = 3380°C) is used for the filament.

 

3. Safety fuse

 

The electrical wires that carry electricity to lights, fans and other electrical appliances have some resistance, although usually it is quite small. If the current is large enough, the wires will heat up and produce heat energy at a rate of i²R. One possible hazard is that the current-carrying wires become so hot as to start a fire. When a wire carries current beyond the safe limit, it is said to be overloaded. To prevent overloading fuses or circuit breakers are installed in circuits. Overloading may be due to the use of too many devices drawing current in that circuit or due to a fault somewhere in the circuit such as short circuit.

 

The safety fuse is a wire of high resistance with low melting point and made of an alloy of tin and lead (63% tin and 37% lead). It is connected in series with the electric installation. If excess current flows through the circuit the fuse wire melts and breaks the circuit. The wire melts at a constant temperature θ, when the heat produced per second by the current equals heat lost per second by radiation: i.e., i²R = H x A, where A is the surface area of the wire and H the heat radiated per unit area per second.

 

It can be shown that θ is directly proportional to specific resistance ρ, inversely proportional to the cube of the radius (i.e., r³), directly proportional to the square of the current (i²) and independent of the length (l) of the wire. The radius r of the wire can thus be calculated so that it melts at a specified value of current. The fuse wire blows at a current slightly below the rated current value. For a fuse wire, the length used does not matter, but should not be too small.

 

The fuse wire is commonly enclosed in a cartridge of porcelain or glass or related material having metal ends. The fuses for domestic purposes are rated as 1 A, 2 A, 3 A, 5 A, 10 A, 15 A, etc. In house wiring, we generally employ either 5 A or 15 A fuse, the former for lights and fans; and later for power circuits.

 

In addition to tin-lead alloy, aluminum, copper, iron, etc. are used for the fuse wire. A fuse wire has (i) high resistance (not resistivity) and (ii) low melting point.

Meter Bridge and Potentiometer

METRE BRIDGE

 

It is a simple form of Wheatstone’s bridge used to measure the resistance of a resistor.

 

Circuit Diagram of the apparatus

Procedure: To find the resistance of a conductor

 

The conductor having an unknown resistance X is connected in the left gap of the circuit and a resistance box is connected in the right gap of the circuit. A battery is connected between A and B. A galvanometer is connected between C and a jockey, a movable contact, J which can slide along the bridge wire.

 

A suitable resistance R is taken in the box and the circuit is closed. The position of the jockey is adjusted so that the galvanometer deflection is zero.

 

The balancing length AJ is measured as I₁ and BJ as I₂ = 100 – l₁. According to Wheatstone's principle, X/R = l₁r/l₂r; where r is the resistance per unit length of the wire AB.

Therefore X = R(l₁/l₂)

Thus the resistance X of the conductor is calculated. The experiment is repeated for different values of R.

The experiment can also be repeated by interchanging X and R between the gaps. If AJ = l’₁ and BJ = l’₂ = (100 — l’₁), then,

R/X = l’₁/l’₂

Therefore, X = R(l’₂/l’₁)

The average value of X is calculated.

 

Note: In the calculation of X given above, end resistances being very small are neglected. There is a contact resistance at each end of the bridge wire fixed to the copper strip. These contact resistances at both ends of the bridge wire are the end resistances. But to yield accurate result the correction has to be applied for the end resistances. This correction is called end correction. The end correction can be minimized by (i) obtaining the balancing point J at the middle of the wire and (ii) repeating the experiment by interchanging X and R between the gaps.

 

POTENTIOMETER

 

It is a device to measure potential difference. It consists of a uniform resistance wire AB, usually 10 metre long, stretched on a wooden board in a zigzag manner so that each segment is 1 m long. A metre rule kept parallel to the segments helps to measure the balancing length. A jockey J sliding over the wire can make contact with any point on the wire.

Principle

 

An accumulator is connected in series with ends A and B of the potentiometer wire with the positive of the cell connected to the terminal A. A rheostat is also included in the circuit. This is the primary circuit.

 

Let r be the resistance per unit length of the wire. When the key is closed, let i be the steady current flowing through the circuit. Potential difference (fall of potential) per unit length of the wire = ir.

 

For a length l, the fall of potential = irl

 

For a steady current, pd across the wire is proportional to the length of the wire.

 

The positive of a cell of emf E is connected to A and the negative to a jockey through a galvanometer. This is the secondary circuit. The jockey is moved along the wire and the balancing length AJ = l, where the galvanometer shows null deflection is found out. Then,

Emf of the cell, E = P.d between A and J = irl

Therefore, E ∝ l

This is the principle of the potentiometer.

To obtain balance, the Pd between A and B should be greater than the emf E of the cell in the secondary circuit.

 

Experiment 1: To compare the emfs of two cells

 

Connections:

(a) Primary circuit

An accumulator, a rheostat and a key are connected in series between the ends A and B of the potentiometer wire. The positive of the accumulator should be connected to A.

(b) Secondary circuit

The first cell of emf E₁, a galvanometer and a high resistance are connected in series between the terminal A and the jockey J. The positive terminal of the cell is connected to A.

 

Procedure

 

The primary circuit is closed. The balancing length AJ = l₁ is noted. Then

E₁ ∝ l₁

The first cell of emf E₁ is replaced by the second cell of emf E₂ in the circuit. Again the balancing length AJ = l₂ is measured.

E₂ ∝ l₂

Equations (i)/(ii),

E₁/E₂ = l₁/l₂

The experiment is repeated by adjusting the rheostat. The average value of E₁/E₂ is calculated.

 

Note: If the emf E₂ of the second cell is known, emf E₁ of the first cell can be calculated;

E₁ = E₂(l₁/l₂)

 

Experiment 2: To find the internal resistance of a cell

 

The connections are made as shown in the diagram. A resistance box R is connected across the cell through a key K. With the key K open, determine the balancing length l₁. Then the emf of the cell,

E ∝ l₁

A suitable resistance R is taken in the box and the key K is closed. The pd at the terminals of the cell falls to terminal pd, ER/(R + r), where r is the internal resistance of the cell. If l₂ is the balancing length then, ER/(R +r) ∝ l₂

Dividing eqn (1) by (2)

(R +r)/R = l₁/l₂,

r =   R(l₁ - l₂)/l₂

The experiment is repeated for different values of R. It is seen that internal resistance (r) varies with the external resistance R.

Note: Emf of a cell cannot be measured using a voltmeter. When a voltmeter is connected between the terminals of the cell, a current flows through the voltmeter. So, the voltmeter measures only the terminal pd. But a potentiometer is preferred to a voltmeter to measure the emf of a cell.

Wheatstone Bridge Principle and its Applications

 

Wheatstone Bridge

WHEATSTONE'S PRINCIPLE

The most widely accepted method of measuring resistance was developed in 1843 by Charles Wheatstone, who was considered as the first Professor of Physics at Kings College, London. The Wheatstone's bridge consists of four resistors of resistances P, Q, R and S connected to form a closed network ACBD. A cell is connected between A and B. A sensitive galvanometer of resistance G is connected between C and D. The current flowing through each branch of the circuit is shown in the diagram.

Applying Kirchhoff's rule to the mesh ACDA

i₁P + i_gG - i₂R = 0 -------------- (1)

For the mesh CBDC  

i₃Q — i₄S - i_gG = 0 ---------------- (2)

The resistance P, Q, R and S are so adjusted that the galvanometer current i_g is zero. Then the network is said to be balanced. Then,

i₁ = i₃, i₂ = i₄ and i_g = O.

From equation (1),

i₁P — i₂R = 0 ; Therefore, i₁P = i₂R

From equation (2),

i₁Q — i₂S = 0 ; Therefore, i₁Q = i₂S

Dividing eqn (3) with eqn (4)

P/Q = R/S

This is Wheatstone's principle.

 

APPLICATIONS OF WHEATSTONE'S BRIDGE

 

The basic use of Wheatstone's bridge is to find the resistance of a conductor. The conductor of unknown resistance X is connected to the fourth arm of the bridge. When the bridge is balanced,

P/Q= R/X, Therefore, X =(Q/P) x R

If the values of P, Q and R are known, the resistance X of the conductor can be calculated.

Practical devices based on Wheatstone's principle are Metre Bridge, Post Office Box, Carey Fosters Bridge (modified form of Metre Bridge), Calender and Griffith Bridge etc.

The Wheatstone principle can also be used to measure

(i) Temperature (Platinum resistance thermometer)

(ii) Temperature coefficient of resistance etc.

 

(i) Temperature measurement

The electrical resistance of a metallic wire is found to increase gradually and fairly uniformly with temperature over a wide range, and consequently based on this property a system of thermometry has been devised.

The relation between the resistance of a metallic wire and its temperature may be represented roughly by the equation

R_t = R₀(1 + αt) -------------- (1)

where R₀ is the resistance of the wire at 0°C, R_t its resistance at any temperature t°C and α its temperature coefficient of resistance. Pure platinum wire, free from alloy with carbon, silicon, tin or other impurities, when not subjected to strain possesses always the same resistance at same temperature and its variation with temperature can be represented fairly by the equation

R_t = R₀(1 + αt). Hence platinum wire is used for the resistance thermometer.

From equation (1)

α = (R_t – R₀)/(R₀ x t) ---------- (2)

The platinum resistance thermometer T is connected in the fourth arm CD of the Wheatstone's bridge. Its resistance R₀ and R₁₀₀ are measured by keeping it in melting ice and boiling water at standard pressure.

From equation (2)

α = (R₁₀₀ – R₀)/(R₀ x 100) -------- (3)

Now, the resistance R_t of the thermometer is measured by keeping it in the bath whose temperature t is to be measured. Then

α = (R_t – R₀)/(R₀ x t) ------------ (4)

From equation (3) and (4),

(R₁₀₀ – R₀)/(R₀ x 100) = (R_t – R₀)/(R₀ x t)

Therefore, i = [(R_t – R₀)/(R₁₀₀ – R₀)] x 100

 

From this equation, temperature t of the bath can be calculated.

A graph can be drawn with resistance R_t along the Y-axis and known temperature t along the X-axis. The graph is a straight line with Y-intercept R₀. From this calibration curve the unknown temperature x can be noted by measuring the resistance of the platinum resistance at the temperature x.

In the actual experiment a modified form of Wheatstone's bridge, the Calendar and Griffith bridge is used to determine the resistances of the platinum wire at 0°C, 100°C and t°C.

(ii) Temperature coefficient of resistance

By measuring the resistances R₀ and R₁₀₀ of a metallic wire at 0°C and 100°C using Wheatstone's bridge, the temperature coefficient of resistance of the material of the wire can be calculated by the equation,

α = (R₁₀₀ – R₀)/(R₀ x 100)

Ohm’s Law Statement and Explanation

Ohm’s Law Statement and Explanation

George Simon Ohm (1787-1854), a German physicist conducted a series of experiments to investigate the relation between the current passing through a conductor and potential difference between its ends. As a result of these investigations he arrived at an important conclusion which is known as Ohm's law after his name and may be stated as follows:

 

The current flowing through a conductor at constant temperature is directly proportional to the pd between its ends.

 

If I is the electric current flowing through a conductor maintained at a pd V, according to Ohm's law,

I ∝ V or V ∝ I;

i.e., V = IR;

where R is a constant known as the resistance of the conductor.

The graph connecting I and V is a straight line passing through the origin. If the temperature changes, then Ohm's law does not hold good because the resistance R of a conductor varies with change of temperature.

Unit of resistance: ohm (Ω)

Conductance (k) : It is the reciprocal of the resistance (R) of a conductor. k = 1/R

Unit: mho (Ω^-1) or siemen (S).

 

Factors on which the resistance of a conductor depends

 

The resistance R of a conductor depends on:-

(i) the length l of the conductor R ∝ I

(ii) the area of cross-section a of the conductor R ∝ 1/a

(iii) the temperature t of the conductor (The resistance of a conductor increases with rise of temperature. But for semi-conductors like Ge, Si, etc. the resistance decreases with rise of temperature).

(iv) the material of the conductor.

 

Resistivity (ρ)

 

For a given material and at a given temperature, the resistance R of a conductor is

(i) directly proportional to the length l of the conductor i.e., R ∝ l ---------- (i)

and (ii) inversely proportional to the area of cross-section a of the conductor.

i.e., R ∝ l/a ------------ (ii)

From equations (i) and (ii),

R ∝ l/a

Therefore R = ρ(l/a)

Where, ρ is a constant called resistivity or specific resistance of the material of the conductor.

When l = 1 and a = 1, then ρ = R

Hence, the resistivity of the material of a conductor is defined as the resistance of the conductor of unit length and unit area of cross-section.

Unit of resistivity : ρ = Ra/l = ohm metre (Ωm)

 

Conductivity (σ)

 

Electrical conductivity is the reciprocal of the electrical resistivity.

σ =1/ρ

Unit of conductivity, σ = 1/ρ = mho/metre (Ω^-1m^-1) or Sm^-1

 

Deduction of Ohm's law

 

Consider a conductor of length l, and cross-sectional area A containing n free electrons per unit volume. Let a pd V be applied between its ends.

The electric field along the wire E = V/l ---------- (i)

Acceleration of free electrons a= Ee/m --------- (ii)

where e = charge and m = mass of the electron.

If r is the average time interval between successive collisions of electrons with the ions of the conductor (relaxation time) the velocity acquired by the electron in time τ,

v = aτ = (Ee/m)τ, the drift velocity

 

The current through the conductor I = nAve = nA(Ee/m)τe

I = ne²Aτ/m x V/l (since, E = V/l) ---------- (iii)

Therefore, I ∝ V (since, ne²Aτ/ml = a constant)

Resistivity (ρ), Conductivity (σ) and Relaxation time (τ)

(i) From eqn. (iii), I = (ne²Aτ/m)(V/l)

Therefore, the resistance of the conductor, R = V/I = m/ne²τ x l/A

(ii) The resistivity of the material, ρ = m/ne²τ

(iii) Conductivity of the material, σ = 1/ρ = (ne²/m)τ

 

FAILURE OF OHM'S LAW

 

Ohm's law is not a basic law of nature. It simply represents the electrical behaviour exhibited by many materials. A true mathematical statement of Ohm's law is

V ∝ I

which means that a graph connecting V and I is a straight line passing through the origin. This relation holds good for metallic V conductors. Hence they are called Ohmic conductors.

 

There are many conductors in which the relation between V and I is different from the one given by Ohm's law. The conductors which does not obey the Ohm's law are known as non-Ohmic conductors. For these substance I — V graph is not a straight line passing through the origin. Circuits containing non-ohmic conductors are called non-ohmic or non-linear circuits.

 

In all non-ohmic circuits, there are some circuit elements having one or more of the following properties:

(a) Non-linear relationship between V and I

(b) The relation between V and I depends on sign of V for same magnitude of V.

(c) For the same current I, there may be more than one value of voltage V.

A few examples showing the violation of Ohm's law are given below:

(i) I — V graph for an Ohmic conductor is linear only for small currents. Even if the temperature of the conductor is kept constant, some conductors show an increase in resistivity as current increases.

(ii) A pn junction diode does not obey Ohm's law.

(iii) A thyristor or silicon controlled rectifier (SCR) does not obey Ohm's law. The SCR is a three terminal semiconductor switching device. When a pn junction is added to a junction transistor, we get a thyristor.

(iv) The flow of current through a gas in a discharge tube does not obey Ohm's law. For lower voltages, i.e., along OA Ohm's law is obeyed.

(v) The Ohm's law is not obeyed when current flows through an electrolyte if electrodes are of metals different from the one corresponding to the cations of the electrolyte.

Current Electricity Introduction

INTRODUCTION - CURRENT ELECTRICITY

The electric current is the flow of electric charges, the charge carriers. In solid conductors, the charge carriers are conduction electrons (free electrons), in electrolytes and gases, charge carriers are positive and negative ions and in semiconductors charge carriers are conduction electrons and valence electrons (holes). In a solid conductor there are a large number of free electrons, about 10²⁸/m³. These electrons are in random motion just like the molecules of a gas. The average thermal speed of these electrons is about 10⁵ ms^-1 at room temperature. But, since they are in random motion due to collision (interaction) with the atoms of the conductor, the average velocity of an electron is zero. As the net charge crossing any section of the conductor is zero, there is no net flow of charge through any section of the conductor. So the electric current through the conductor is zero.

 

Consider a battery of potential difference V connected between the ends of the conductor AB of length I. An electric field of magnitude E = V/l is established in the conductor in the direction from the positive terminal to the negative terminal. This electric field E exerts a force, F = Ee, on the electrons and gives them a net motion in a direction opposite to the direction of the electric field. There is a net flow of charge through any section of the conductor. Hence an electric current flows through the conductor.

 

Strength (Intensity) of electric current (i)

 

Intensity of an electric current through a conductor is defined as the time rate of flow of charge through any section of the conductor.

 

If a net charge dq passes through any section of a conductor in a time interval dt, the strength of the current is given by,

i = dq/dt

If the current is constant in time, then, if q is the charge that flows in the interval of time t,

i = q/t

If n is the number of electrons crossing any section of the conductor in one direction in one second,

i = ne

If an electron moves along the circumference for a circle with frequency f ,

i = fe

It is to be noted that electric current i is the same for all cross-sections of a conductor, even when the cross-sectional area may be different at different sections.

 

Unit of electric current

 

The SI unit of current is ampere (A).

1 ampere (A) = 1 coulomb/second (Cs^-1)

 

Direction of electric current

 

Although in a metal the charge carriers are electrons, in electrolytes and gases the charge carriers are positive and negative ions. A convention for labelling the direction of electric current is needed because the charges of opposite sign move in opposite directions in a given electric field. A positive charge moving in one direction is equivalent to a negative charge moving in the opposite direction. Hence for simplicity the following convention is followed.

The direction of electric current is the direction that positive charges would move, even if the actual charge carriers are negative. It is same as the direction opposite to the direction of flow of electrons (or negative charges). This is the direction of the conventional current.

In a solid, is the electrons are the charge carriers, it is more often convenient to take the direction of flow of electrons as the direction of the electric current. In this case, the direction of flow of electrons gives the direction of electronic current.

Even though we assign it a direction, the electric current is a scalar; not a vector because it does not obey the laws of vector addition. The direction merely indicates the direction of flow of charges.

Velocity of electric current

The velocity of electric current is nearly equal to the velocity of light in vacuum, 3 x 10⁸ ms^-1.

Current density at a point (j)

The electric current density at a point in a conductor is the electric current (i) flowing normally per unit area around the point.

If the current i is distributed uniformly across a conductor of cross-sectional area A; the magnitude of current density at all points on that cross-section is,

 j = i /A.

It is a vector whose direction at any point is in the direction that a positive charge carrier would move at that point. Its unit is ampere/ metre² ( Am^-2).

 

Different electric current sources

 

An electric circuit must have some source of energy to maintain a pd. The energy may be supplied by chemical reactions as in cells or by rotating a coil in a magnetic field as in generators or by converting light energy to electrical energy as in solar cells.

 

Drift velocity (v)

 

In the absence of an electric field, electrons in a conductor are in random motion due to the interaction with the electrons in the orbits of the atoms, just like the molecules of a gas. As in the case of collisions of gas molecules, there is a mean free path A and mean free time r. The speed of the electron is very high, about 10⁶ ms^-1, and is called thermal speed. Here the average velocity of an electron is zero.

When an electric field E is applied, the electrons modify their random motion in such a way that they drift slowly in the direction opposite to that of the field, with an average speed. This is because of the force Ee experienced by the electron in the electric field. Since all the electrons are drifted in the same direction, the electrons acquire a small average velocity opposite to the direction of the electric field, i.e., towards the positive terminal of the conductor. This drift is in addition to the random motion of the electrons.

The solid lines in the figure suggest a possible random path followed by an electron in the absence of an applied field and the dashed lines show how this same event might have occurred if an electric field had been applied. Note that the electron drifts steadily to the left ending at x’ rather than at x.

The drift velocity is defined as the average velocity with which free electrons are drifted under the influence of an electric field.

The drift velocity is very small and is of the order of 10^-3ms^-1. The drift velocity should not be confused with the speed of propagation of electric current along the conductor which is very nearly the speed of light. In fact the electrical impulse propagates, may be like electromagnetic waves, through the conductor with very high speed. The problem is similar to the effect of applying pressure to the end of a long tube filled with water. As soon as pressure is applied at one end, the pressure wave is transmitted rapidly along the tube and the flow of water from the other end starts instantly.

 

Relaxation time (τ)

 

When there is no external electric field, there is no drift velocity. In the presence of an external electric field E, each electron experiences an acceleration, a = F/m opposite the direction of the electric field. But this acceleration is momentary. The electrons frequently collide with vibrating atoms or ions or other electrons of the metal. After each collision each electron makes a fresh start and accelerates only to be deflected again.

The average time interval between successive collisions of electron with the atom or positive ion in the conductor is called the relaxation time.

 

Relation between drift velocity (v) and relaxation time (τ)

 

After each collision with atoms or ions, each electron makes a fresh start and accelerates from rest. The acceleration is given by,

â = — (Êe/m);

where m = mass of the electron

Negative sign shows that the electron is accelerated in a direction opposite to the direction of the external electric field. If τ is the relaxation time, the velocity attained by the electron, i.e., the drift velocity of electron,

v̂ = u + at = at = — (Êe/m)τ

Negative sign shows that the drift velocity of the electron is opposite the direction of the electric field,

Therefore, v = (Ee/m)τ

 

Note: Thermal speed (rms value of thermal speed)

 

Free electrons in a metal behave like the molecules of an ideal gas. So, we can apply the kinetic theory of gases to find the rms velocity of the electron. Thus the average kinetic energy of an electron,

(KE)-vector = (3/2)kT; But KE = (1/2)m_ev²_rms; v_rms = √3kT/m_e

 

Relation between current and drift velocity

 

Consider a conductor AB of length l and cross sectional area a. Let n be the number of free electrons per unit volume (electron density or electron concentration) of the conductor.

When it is connected to a battery, an electric field E is set up along the conductor and charge flows along the conductor with the drift velocity v.

Consider a cross section P of the conductor perpendicular to the flow of electrons through the conductor in terminal B to the positive terminal A. Q is another cross section of the conductor at a distance v from the section P. Since v, the drift velocity, is the distance travelled by an electron in one second, all free electrons of the conductor between the sections P and Q will cross the section P in 1 second.

Total number of electrons crossing the section P is 1s = volume of the conductor between P and Q x electron density = (a x l) x n = (a x v) x n = nav

Total charge crossing the section P in 1 s, i.e., strength of the electric current through the conductor,

i = nav x e = nave

 

Mobility (µ)

 

The drift velocity v of a charge carrier is directly proportional to the electric field E.

i.e., v ∝ E,

Therefore, v = µE

Where µ is a constant known as the mobility of the charge carrier.

If v is the drift velocity of electron and µ the electron mobility, the electric current through a conductor is given by, i = nave = naµEe

The current density,

j = nµEe

Unit of mobility, µ = v/E = (ms^-1)/(Vm^-1) = m²s^-1V^-1