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Current Electricity
(Electric Current, Thermal and Chemical Effect
of Current)
ELECTRIC CURRENT
Consider the ends of the conductor be connected to a battery, i.e., an electric field is maintained within the conductor. Now the field acts on the electrons and gives them a resultant motion in the direction of (^) Er because a free charge in electric field experiences a force. The flow of electrons constitutes an electric current. The time rate of flow of charge through any cross section is called current. If a charge q passes through an area in time (^) t , then the average electric current through the area in this time is defined as
av
q i t
Now, the instantaneous current is given by
t 0
q dq i lim t dt
The SI unit of current is ampere. If one coulomb of charge crosses an area in one second, the current is one
ampere. For transient currnet
dq i dt
(^) while for steady current
q i t
The conventional current is in opposite direction to the direction of movment of electrons.
CURRENT DENSITY
The current density j at a point is defined as a vector having magnitude equal to current-per unit area surrounding that point and normal to the direction of charge flow, i.e., direction in which current passes through that point. If (^) S
uur be the area vector corresponding to area (^) S , then (^) i j. S
r uur
The total current through finite surface area S is
s
i (^) j. S
r uur
, If current i is uniformally distributed over an areas and perpendicular to it then i j. S
ruur
DRIFT VELOCITY
we know that a conductor contains a large number of free electrons or conduction electrons. When electrons leave their atoms and become free, the atoms of the conductor become positively charged and are called positive ions. So, the remaining material is a collection of relatively positive ions known as lattice. In the absence of any external electric field, the electric current through this area is zero, otherwise the conductor will not remain equipotential. When an electric field is established between the two ends of the conductor, the free electrons experience an electric force opposite to the field. Due to this force, the motion of electrons is accelerated. The field does not give an accelerated motion to the electrons but it simply gives them a small constant [1]
[2] Current Electricity
velocity along the conductor which is superimposed on the random motion of the electrons. So, the electrons drift slowly opposite to the applied field. The net transfer of electrons across a cross section results in current. If the electron drifts a distance l in a long time t. we define drift velocity as
vd (^) d
l ...(1)
The drift velocity is the average uniform velocity by free electrons inside a conductor by the application of an electric field. where e is charge of electron with mass m. [Q force on electron due to electric field, F = e E and acceleration , a = F/m = (e E/m)]
d
eE v. m
(^) in time between two successive collision. ...(3)
RELATIONSHIP BETWEEN CURRENT DENSITY AND DRIFT
An electric field is maintained between the two ends of a conductor towards the left. The electrons move towards the right. Let the drift velocity of the electrons be vd. Suppose there are n charge carriers per unit volume and each charge carrier has a charge e. In time dt, the electron advance a distance l which is given by l = vd dt Now calculate the number of electrons crossing the length l of the conductor in time dt. This will be equal to the number of electrons contained in a volume Al, i.e. Avd dt. number of electrons = volume × number of electrons per unit volume = A vd dt × n Hence charge crossing in time dt = number of electrons × charge on the electron = Avd dt n e
Further, current
chargecrossing in timedt i time dt
E
l
V (^) d
d d
Av dt n e i Av ne dt
So i^ ^ neAvd ...(1) The current density is given by
j i neAvd A A
or j^ ^ n e vd ...(2) Mobility of free electron is defined in a conductor is drift velocity acquired per unit electric field strength.
applied across the conductor,
Vd Vd E E
or i (neA)E
OHM’S LAW
The current flowing through a conductor is always directly proportional to the potential difference across its two ends. V i or^ V Ri
[4] Current Electricity
If l = 1 and A = 1 , then R Therefore, specific resistance of the material of a conductor is equal to the resistance offered by the wire of unit length and unit area of corss section of the material of wire. Its unit is ohm-metre. This is constant for a material. The reciprocal of resistivity of the material of a conductor is called as conductivity 1 j E
The unit of conductivity is ohm–1^ metre –1^ m (^) ^1. Good conductors of electricity have large conductivity than insulators.
FACTORS AFFECTING ELECTRICAL RESISTIVITY
The drift velocity vd in magnitude of electrons is given by
d
eE v m
^
The current flowing through the conductor due to drift of electrons is given by i n A e vd
eE nAe E^2 nAe m m
^
If V be the potential difference applied across the two ends of the conductor, then V E l
From eqs. (2) and (3) we get, nAe V^2 i m
l
or R = (^2)
V m or R i ne A A
^ ^ ^
^ ^ ^
l l ...(4)
where (^2)
m resistivity ne
The resistivity of the material of a conductor depends upon the following factors: (i) It is inversely proportional to the number of free electrons per unit volume n of the conductor, i.e., depends on the nature of material. (ii) It is inversely proportional to the average relaxation time of free electrons in the conductor. As is a function of temperature and hence the resistivity of a conductor depends on its temperature. The resistivity increases with the increase in temperature of conductor.
TEMPERATURE DEPENDENCE OF RESISTIVITY
Small temperature variations, the variation of resistivity can be expressed as
T T 0 1 T T 0
where (^) Tand (^) T 0 are the resistivities at temperature T and T 0 respectively and (^) is temperature coefficient of resistivity. The resistance of a conductor is given by
Current Electricity [5]
R
A
^ ^
l (T) = Resistivity at temperature T
(T ) 0 = Resistivity at temperature of T^0. The resistance depends on the length and area of cross section besides resistivity. When the temperature increases, the length and area of cross section also increases are quite small and the factor (l/A) may be treated as constant. Therefore, R R(T) Restance at temperature T..
Now, R T T T 0 (^1) T T 0 R(T) = Resistance at temperature T 0. where (^) is known as temperature coefficient of resistance.
Grouping of Resistance
(A) RESISTANCES IN SERIES
In the shown figure. shows the series combination of three resistors having resistanceR 1 and R 3 and. A battery of e.m.f. E is connected across this combination. In this combination, the same current i is flowing through each resistor. Let V 1 , V 2 and V 3 be the potential differences across R 1 , R 2 and R 3 respectively. Now according to Ohn’s V 1 = i R 1 , V 2 = i R 2 and V 3 = i R (^3) Further V = V 1 +V 2 +V 3 v
R 1 R 2 R 3
i
E
iR 1 iR 2 iR 3
i R 1 R 2 R 3 ...(1)
If R 2 be the equivalent resistance of series combination, then the potential difference V across the combination will be V = i R (^) S ...(2) Compairing eqs. (1) and 2), we get
RS R 1 R 2 R 3 ...(3) In series combination, the following points should be remembered (i) The current is same in every part of the circuit (ii) The total reistance of the circuit is equqal to the sum of indivd resistances connected in the circuit. (iii) The total resistance of series combination is more than the greatest resistance of the circuit. (iv) The potential difference across any resistor is proportional to its resistance, i.e. v 1 : v 2 : v 3 = R^1 : R^2 : R^3
(B) RESISTANCE IN PARALLEL
Fig. shows a parallel combination of three resistors having resistance R 1 , R 2 and R 3 battery of e.m.f. E is connected points A and B. Let i be the current from the battery and i 1 , i 2 and i 3 be the currents through resistance R 1 , R 2 and R 3 respectively. Then i = i 1 + i 2 + i 3 ...(5) As shown in the figure, the potential difference across each resistance is V. Applying Ohm’s law, we have V = i 1 R 1 = i 2 R 2 = i 3 R 3
or 1 1
V
i R
2
V
i R
(^) and 3 3
V
i R
Current Electricity [7]
(ii) If r >> R, i.e., the effective internal resistance is far far greater than external resistance, then R can be neglected in comparison to hr, then nE E i nR r
The current in the circuit is the same as due to a single cell, so n of useful (iii) If in series grouping of n cells, s cells are reversed, then Eeq (^) n s E sE (^) n 2s E Total resistance of the circuit = (R+n r)
n 2s E i R n r
(2) Parallel grouping
Total Resistance of the circuit^ ^ R^ ^ ^ r / n [Q R and (r/n) are in series]
Now, current in the cirut,
E
i r R n
^
(i) If R>> (r/n), i.e. (r/n) can be neglected in comparison to R, then E i R
E
E
r
R
Therefore, the current in the circuit is equal to the circuit current due to a single cell. (ii) If (r/n) >> R, i.e., R can be neglected in comarison to (r/n), then nE i r
Therefore, if the effective internal resistance is greater than the external resistance, the current in the circuit is equal to n time the circuit current due to a single cell. (3) Mixed Grouping
Total resistance of circuit
nr R m
^ ^
E E
upto n
R
P
The current i in the circuit is given by
i nE^ nmE^ NE R nr / m mR nr mR nr
The current i in the circuit will be maximum when the factor ( m R+ n r) in the denominator is minimum. The denominator is minimum when mR = n r
nr R m
^ ^
Hence current will be maximum when external resistance is equal to the total internal resistance of all the cells.
KIRCHHOFF’S LAW
Ohmm;s law is unable to give current in complicated cirucit. Kirchhoff’s in1842, gave two general laws which are extremely useful in electrical circuits. There are.
[8] Current Electricity
(i) The algebraic sum of the currents at any junction in a circuit is zero, i.e.
i^ ^0 This means that there is no accumulation of electric charge at any point in the circuit.
i (^1)
i (^2)
i (^4)
i (^3)
i (^5)
i (^6)
O
(ii) In any closed circuit,the algebraic sum of the products of the current and resistance of each part of the circuit is equal to the total emf in the circuit i.e.,
iR^ E The product of current and resistance is taken as positive when we traverse in the direction of current. The emf is taken positive when we traverese from negative to positive electrode through electrolyte. Let us apply Kirchhoff’s second law to figure shown For the mesh ACDBA, i 1 R 1 – i 2 R 2 = E 1 –E 2 ...(i) For the mesh EFDCE i 2 R 2 +(i 1 + i 2 ) R 3 = E 2 ...(ii)
R (^1) P (^1)
i 2
i + i 1 2
A (^) B
C D
E F
From FFBAE, i 1 R 1 +(i 1 + i 2 ) R 3 = E 1 ...(iii)
CONDITION OF BALANCE IN WHEATSTONE’S BRIDGE
When there is no deflection in the galvanometer, the bridge is known as balanced. The condition of balance is given by
[10] Current Electricity
The workdone by electric field is converted in thermal energy of resistor through the collisions with ions or atoms. The thermal energy is generally referred to as heat produced in resistor. So, the amount of heat produced (H) is
given by H W i Rt^2 joule
In calorie, the heat produced is given by 2
i Rt
H calorie This is expression for joule’s law of heating...(4)
JOULE’S LAWS OF HEATING
Joule’s laws :
(a) The heat produced in a given resistor in a given time is proportional to the square of current flowing in it, i.e.,
H i 2 ...(1)
(b) The heat produced in a given resistor in a given time by a given current is directly proportional to the resistance, i.e.,
H R ...(2)
(c) The heat produced in a given resistor by a given current is proportional to time t for which the current is passed, i.e.,
H t ...(3)
ELECTRIC POWER
The electric power is defined as the rate at which work is done by the source of e.m.f. in maintaining the current in an electric circuit. If an amount of work W is done in maintaining electric current in a circuit for a time t, then electric power is given by
W
P
t
Let a current i ampere flows through a conductor for a time t second under a potential difference V volt. The workdone for maintaining the current is given by W = V i t joule ...(2) So, the power of an electric circuit is one watt when one ampere current flows through it under a potential difference of one watt. 1 watt = 1 joule/sec. The bigger units of electric power are 1 kW = 10 3 W and 1 MW = 10 6 W Commercial unit of power is horse power (HP). 1 HP = 746 watt. Other expression for power are :
P i R^2 and
2
V
P
R
2
P Vi i R^2 V
R
Current Electricity [11]
(i) When resistances are connected in series.
In this case, the current in each resistance will be the same. Hence from eq. (3), we have
P V and^ p^ R
This shows that in series connections, the potential difference and power consumed will be more in larger resistance.
(ii) When resistances are connected in parallel.
In this case, the potential difference V across each resistance is same. Hence from eq. (3), we have
P
R and^
i
R
This shows that in parallel connections, the current and power consumed will be more in smaller resistance.
APPLICATIONS OF HEATING EFFECT OF CURRENT
(1) Series combination of bulbs :
Consider a series combination of three bulbs of powers P 1 , P 2 and P 3 which are manufactured for working on a supply of V volt. The resistances of these bulbs are respectively.
2 2 1 2 1 2
V V
R R
P P
and
2 3 3
V
R
P
total resistance,^ R^ ^ R 1 ^ R 2 R 3 ...(2)
Effective power
2 2 2 2
1 2 3
V V V V
P P P P
or 1 2 3
P P P P ...(3)
Current through each bulb
1 2 3
V
i
R R R ...(4)
The brightness of these bulbs are
2 2
H 1 i R 1 , H 2 i R 2 and^
2
H 3 i R 3 ...(5)
This shows that the bulb with highest resistance will glow with maximum brightness. Further
R ,
P
therefore,
the bulb of lowest power or wattage will have highest resistance and will glow with maximum brightness.
(2) Parallel combination of bulbs :
Consider a parallel combination of three bulbs of powers P 1 , P 2 and P 3 respectively which are manufactured for working on a supply voltage V volt. In this case, we have
2 2 (^1 2 ) 1
V V
R R
P P
and
2 3 3
V
R
P
Current Electricity [13]
Thus the temperature at which the thermo e.m.f. is zero is known as inversion temperature or temperature of inversion. Beyond this temperature the e.m.f. again increases but in the reverse direction. The temperature of inversion depends upon (i) the nature of materials forming the thermocouple (ii) the temperature of cold junction. The thermo e.m.f., e varies with temperature according to the following equation.
e aT bT 2 ...(1)
de
a bT
dT
at T = T (^) n, e is maximum, i.e.,
de
dT
= 0. Thus
0 a 2 bTn
or
n^
a
T
b
Further at T Ti , e 0.Thus from equation (1)
0 ^2
aTi bTi
i^
a
T
b
From equations (2) and (3)
Ti 2 Tn
Thus the inversion temperature T (^) i is as much above the neutral temperature as the temperature of the cold junction (0°C) is below it. Ti is therefore not a constant for the given thermocouple but depends upon the temperature of the cold junction. If T 0 be the temperature of cold junction, then
Ti Tn Tn T 0 or Ti 2 Tn T 0
0
n^ i
T T
T
Peltier’s Effect :
Peltier discovered an effect which is the converse of Seebeck effect. When a current is passed across the junction of two dissimilar metals, heat is evolved at one junction and absorbed at the other, i.e., one junction is heated and the other is cooled. This effect is known as Peltier effect.
Peltier Coefficient :
The amount of heat (in joules) absorbed or evolved at a junction of two different metals when one coulomb of charge flows at the junction is called the Peltier coefficient. It is denoted by
Peltier heat
charge flowing
H
Q
[14] Current Electricity
This coefficient is not constant but varies as the absolute temperature of the junction. It also depends on the metal used.
If a charge q coulomb passes across a junction having a peltier coefficient volt, then the energy absorbed or
evolved at the junction = q joule.
If V be the junctional P.D. in volt, then energy absorbed or evolved = Vq joule
q Vq
V
Hence the Peltier coefficient expressed in joule per coulomb is numerically equal to the junctional P.D. in volt.
Thomson effect :
Thomson observed that when two parts of a single conductor are maintained at different temperatures and a current is passed through it, heat may be absorbed or evolved in different sections of may be absorbed or evolved in different sections of the conductor. This effect is called Thomson effect. According to Thomson effect, heat is absorbed or evolved in excess of Joule heat when a current is passed through an unequally heated conductor.
Thomson coefficient :
Thomson coefficient is defined as the amount of heat evolved or absorbed when a unit positive charge is passed
through a part of the wire whose ends are maintained at a unit temperature difference. This is denoted by .
Let a charge Q is passed through a small part of the wire having a temperature difference T between the
ends. Thomson heat is
H Q (^) T
or (^)
H
Q T
CHEMICAL EFFECT OF ELECTRIC CURRENT
It has been observed that some liquids allow the passage of current through them while some do not show such behaviour. On the basis of their electrical behaviour liquids can be divided into the following three categories : (i) The liquids which do not allow the current to pass through them. For example distilled water, vegetable oil etc. (ii) The liquids which allow the current to pass through them but do not dissociate into ions. For example, mercury. (iii) The liquids which allow current to pass through them and also dissociate into ions. For example salt solutions, acid and bases. Such liquids are called electrolytes. Thus when a current is passed through an electrolyte, it dissociates into ions. This is known as chemical effect of current.
FARADAY’S LAWS OF ELECTROLYSIS
The relation between quantity of electric charge passed and the amount of ion deposited at the electrode is given by Faraday’s laws of electrolysis. There are two laws :
Faraday’s first law :
According to Faraday’s first law, the mass of the substance deposited or liberated in electrolysis is directly proportional to the charge passed through the electrolyte.
[16] Current Electricity
Faraday constant =^
= 96500 C/gram equivalent. The charge of 1 mole of electrons is called one faraday. So one faraday = N (^) A × e = (6.023 × 10 23 ) × (1.602 × 10 –19^ C) = 96500 C. Therefore, faraday is unit of charge (1 faraday = 96500 C) while the quantity charge per mole of electrons is called Faraday constant (F = 96500 C/mole or 1 faraday).
