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Current Density ( ) Tutorial:
The current density at any point inside a conductor is defined as a vector quantity whose magnitude is equal to current per unit infinitesimal area at that point, the area held normal to the direction of flow of current.
Current density  points along the direction of current flow. If Anis small area normal to Current I, then current density  = 
The unit of current density is Am-2 from above, we have I = JA cos  =  Drift Velocity :
When no potential difference is applied across a conductor, the free electron are in random motion and hence the average velocity of free electrons is zero and consequently this motion does n't constitute a net transport of charge across any section of conductor and so no current flows in the conductor.
On application of potential difference across a conductor, the electrons gain some average velocity in the direction of positive potential. This average velocity in addition to random velocity of electrons is called Drift Velocity (vd).
Drift Speed of electrons in Conductor:
The drift speed of electrons in a conductor through which a current 'I' flows is given by  where ‘n’ is the number of electrons per unit volume of conductor, ‘e’ is electronic charge and ‘A’ is the cross sectional area of conductor.
Electric Cell:
a.) Cell is a source of E.M.F. (Electro Motive Force)
b) When connected between the ends of a conductor, it maintains potential difference between the ends of a conductor.
c) It is a device which converts chemical energy into electrical energy.
Primary cells :
a.) Voltaic, Leclanche, Daniel and Dry cells are primary cells.
b.) They convert chemical energy into electrical energy. They can't be recharged.
c.) They supply small currents.
Secondary cells (Storage cells) :
a.) In these cells, the electrical energy is first converted into chemical energy and then stored chemical energy is converted into electrical energy.
b.) They can be recharged.
c.) Internal resistance of primary cells is large and secondary cells is low.
Eg : Lead accumulator, Edison cell.
EMF of a cell :
The work done in carrying a unit positive charge once in the circuit including the cell is defined as the electro motive force. E = . 
a.) It is the difference of potential between the terminals of a cell in open circuit and is responsible for the flow of current in a circuit.
b.) It is equal to the work done in moving unit charge through a circuit.
c.) It is a fixed quantity and it depends only on the nature of electrolyte used in the cells. It depends on 1) nature of electrolyte concentration 2) nature of electrodes.
d.) EMF does not depends on 1) area of electrodes 2) distance between the electrodes, 3) quantity of electrolyte 4) size of the cell.
e.) direction of current inside the cell is from (-)ve pole to (+)ve pole.
f.) Unit : J/C or volt
Internal resistance of a cell :
a) It is the resistance offered by the electrolyte of the cell.
b) It depends on (i) r  a  area (ii) r a d, d - distance (iii) nature of electrolyte (iv) area of cross section of the electrolyte through which the current flows. (v) concentration (r c) (vi) temp of electrolyte (r a  ) (vii ) r =  .
c) Internal resistance of ideal cell is zero.
d) Power transferred to the load is maximum when external resistance becomes equal to internal resistance.
13. Terminal Pd of cell :
a) It is P.D. between the terminals of cell when it is in closed circuit.
b) Terminal P.D. of cell when cell being discharged is V = E - ir
PD = iR =
i = circuit current
r = Internal resistance of cell
R = External resistance
c) Terminal P.D. of cell during charging is V = E + ir
d) Terminal P.D. of cell is not fixed quantity. It depends on the external resistance.
–– 
fig
e) P.D. is less than emf during discharge and greater than emf during charging.
f) when no current is drown from the cell V = E
Lost Volts :
a) It is the difference between E.M.F. and P.D. of a cell.
b) It is used in driving the current between the terminals of a cell.
Current in a circuit : i = 
Combination of Cells :
There are 3 possible arrangements of grouping of cells.
Series Arrangement :
In this arrangement, the positive terminal of one cell is connected to negative terminal of other in succession. Figure represents (n) cells, each of e.m.f (E) and internal resistance (r) connected in series with an external resistance (R)
(a) Since all the cells are in series, net e.m.f = nE
fig
(b) similarly all the resistances are in series, so net internal resistance = nr
(c) Total resistance of circuit = R + nr
(d) Current I = 
(e) If R &g.&g. nr, then
 current due one cell
(f) If R &l.&l. nr, then
 current due to one cell.
(g) When net external resistance &g.&g. net internal resistance, then to get maximum current, cells must be connected in series .
Parallel Arrangement :
In this arrangement, the positive terminals of all cells are connected to one point and negative terminals to one point. Figure represents (m) cells, each of e.m.f (E) and internal resistance (r), connected in parallel and an external resistance R is connected across the combination.
fig
(a) Since all the cells are in parallel, the net e.m.f equals to e.m.f due to a single cell.
Net e.m.f = E
(b) similarly all resistances are in parallel, so
Net internal resistance = Rint = 
(c) Total resistance of circuit = 
(d) Current I = 
(e) If R &l.&l.  , then
 current due to one cell
(f) If R &g.&g.  , then
 current due to one cell.
(g) when net internal resistance &g.&g. net external resistance, then to get maximum current, the cells must be connected in parallel.
MIXED GROUPING :
In this arrangement, cells are conneted both in series &am. parallel.
This arrangement comprises of 'm' rows of cell in parallel, each row containing 'n' cells in series. Total no. of cells = mn.
fig
(a) The e.m.f of each cell is (E) and internal resistance is (r). The combination is connected to external resistance R. Net e.m.f = nE
(b) Net internal resistance = Rint = 
(c) Net resistance of circuit = R + 
(d) Current I = 
&THOR. 
(e) Thus for maximum current, cells should be connected in mixed grouping such that external resistance R = net internal resistance i.e.
Rext= Rint &THOR. R = 
and
Imax = 
Mixed grouping is preferred when more power is needed.
Imp
a) Of two cells connected current flows in the direction of higher emf. If E1&g. E2, current flows in anticlock wise direction.
i = 
  fig
b) If two cells are connected in parallel current i1=  . i2= 
i = i1+ i2=  +  = 
Effective internal resistance = 
EMF of equivalent source or effective P.d
E = ir = 
E = 
Fig.
i = 
If cells are connected as shown
T.P.d(v) = 
Wrongly connected cells in series :
N cells each of emf 'E' are to be connected in series. If n cells of them are wrongly connected, the resultant emf = (N - 2n) E
resultant internal resistance = Nr
Ohm's law :
a) At constant temperature, the current 'i' through a conductor is proportional to P.D. 'V' across its ends V = iR (or) R = 
b) Ohm's law is obeyed by metals and alloys. They are called ohmic conductors. Graph drawn between i-V for ohmic conductors is straight line.
Fig.
c) Ohms law is not obeyed by vacuum tubes, transistors, discharge tubes, Electrolyte, Thermistor. They are called non ohmic conductors. Graph drawn between i-V for non ohmic conductors is a curve.
Fig.
. Resistance : R
a) It is the property by virtue of which a conductor opposes the flow of charge in it.
b) It depends on length, area of cross section, nature of material and temperature.
c) It does not depend upon current and potential difference.
d) S.I. unit of resistance = ohm
I ohm = 
e) Reciprocal of resistance is conductance. G = 
S.I. Unit = siemen (S)
Specific resistance :
a) R  or R =  or s = 
b) It is equal to resistance of the conductor of unit length and unit area of cross section.
c) S.I. Unit : ohm - metre.
d) It depends only on the material of the conductor, and temperature.
e) It is independent of dimensions of the conductor.
f) For silver and copper specific resistance is less.
g) For Nichrome, constantan, Manganin it is more.
Conductivity : (or)
specific conductance : (s)
a) It is reciprocal of resistivity. s = 
b) S.I. Unit : siemen / m
c) For Insulators s = 0
d) For perfect conductors,  = Infinity
Variation of resistance with temperature :
a) Rt= Ro(1 + t) or
a = 
Rt= Resistance at t & or d.C
R0= Resistance at 0 & or d.C
a = Temperature coefficient of resistance.
b) is positive for metals. Hence resistance of metals increases with increase of temperature.
c) is negative for carbon, mica, India rubber, Thermistor, electrolytes, semi conductors and insulators. Their resistance decreases with increase of temperature.
Comparision of resistances :
a) If 2 wires are made of different materials.
b) If 2 wires are made of same material
c) If 2 wires made of same material have lengths l1, l2and masses m1, m2
d) If 2 wires made of same material have equal masses (or) when a wire is stretched from length l1to l2
e) If wire of resistance R is stretched to 'n' times its original length, resistance becomes n2R.
f) If wire of resistance R is stretched until radius becomes  th original radius, resistance becomes n4.R.
g) When wire is stretched to increases its length by x% (where x is a small) resistance increases by 2x%.
h) When wire is stretched to reduce its radius by x% (where x is small), resistance increases by 4x%.
Resistances in series :
fig
a) R = R1+ R2+ R3
b) If n wires each having resistance R1 are connected in series, equivalent resistance R = nR1
c) Resistance of combination is more than highest of components.
d) Current 'i' is same through all the resistances.
e) P.D. 'V' R and V1: V2: V3= R1: R2: R3
Resistances in Parallel :
a) 
b) If 2 wires are connected in parallel, R = 
c) If n wires each having resistance R1are connected in Parallel, R =  .
d) Resistance of combination is less than lowest component.
e) Potential difference 'V' is same across the ends of any wire.
f) Current is different in different wires and i. 
i1: i2: i3= 
g) If 2 wires of resistance R1, R2are connected in parallel, the total current splits into i1, i2such that i = i1+ i2
Current through R1is i1= 
Current through R2is i2= 
h) More current passes through less resistance and less current passes through higher resistance.
i) Used to decrease the resistance in a circuit.
. KIRCHHOFF'S RULE :-
All electrical networks can't be reduced to simple series parallel combinations. Kirchhoff gave two simple and general rules which can be applied to find current flowing through (or) voltage drops across resistances in such complex networks.
First Rule : (Junction rule) :-
a) The algebraic sum of currents at a junction is zero.
b) This rule follows from conservation of charge, since no charge can accumulate at a junction.
c) while applying this rule, we (arbitrarily) take the currents entering into a junction as positive and those leaving it as negative.
Fig.
Second Rule : (loop Rule)
a) The algebraic sum of changes of potential (potential rise and drop) across the elements of a circuit in a closed loop is zero.
b) This rule follows from law of conservation of energy.
Following simple procedure should be adopted while applying rules to some network.
a) Assign direction to each unknown current. It is advisable to check that all currents must not leave a junction (or) enter a junction. If the actual direction of a particular current is opposite to assumed direction, the value of that current will emerge from solution with a negative sign attached to it.
b) Apply the junction rule to all junctions remembering that a bend in a single wire can't be taken as a junction.
c) Select any loop in network. While taking a loop you must start and end at same point.
d) Starting from any point go around the loop in designated direction, adding all e.m.f's and potential drops.
(e) Take an e.m.f as positive if a cell is traversed from (-) to (+).
(f) Take an IR drop as positive if resistor is traversed opp. to assigned directions of current i.e.
Similarly if we move in direction of current, then there is a potential drop of IR and hence
 .
If necessary, select another loop and repeat steps (e) and (f) till you get as many equations as number of unknowns.
Example : Consider the following circuit.
Fig.
Let I1, I2and I3be currents through R1, R2 and R3respertively. The assumed directions of current are shown in figure. Applying junction rule to point 2 gives
I1+ I3 = I2 ................ (1)
Applying loop rule to loops 12561 and 23452, we get
E1- I2R2 - I1R1 = 0 .................... (2)
- E2 + I3R3+ I2R2= 0 ....................(3)
solving these three equation I1, I2and I3can be obtained.
. Wheatstone's Bridge :
a) Bridge is balanced when galvanometer current is zero.
b) condition for balance is  (or) PS = QR.
c) The two ends of Galvanometer remain at same potential.
d) The balance is not effected when positions of battery and Galvanometer are inter charged.
e) The bridge is most sensitive if P = Q = R = S.
f) equivalent resistance of balanced bridge across the ends of battery = 
g) If 
and R = 
Metre bridge :
By any reason, if resistance in left gap increases (or) resistance in right gap decreases, balancing point shifts towards right side. By any reason, if resistance in left gap decreases (or) resistance in right gap increases, balancing point shift towards left.
a) used to find resistance of unknown wire, specific resistance of wire and to compare resistances.
b) In balanced bridge
Where l = balancing length from left end.
c) A high resistance is connected in series to galvanometer to protect it from high currents.
. Potentiometer :
1. It is a device used to measure :
a) P.d between two points in a circuit.
b) Compare e.m.fs of cells.
c) Find internal resistance of a cell.
d) To calibrate an ammeter.
e) To calibrate a voltmeter.
2. It works based on principle that potential in secondary is compared with known variable p.d across potentiometer wire.
3. As it is a null deflection method it gives accurate measurement.
Fig.
ABCD - Primary circuit.
AJGFA - Secondary circuit.
Conditions to be fulfilled
a) E &g. E1
b) higher potential points of primary and secondary are to be connected together at A.
4. In balanced condition potential difference in secondary Ei= Potential gradient (X) x balancing length (l)
5. Potential gradient (X) :
It is the fall of potential per unit length
X =  X = 
X = 
Here
E - e.m.f of cell in primary
R - series resistance
r - internal resistance of cell in primary
Rp- Resistance of potentiometer wire
L - Length of potentiometer wire
- resistivity of potentiometer wire
A - Area of cross section of potentiometer wire
6. Comparison of e.m.f. is using potentiometer
a) If two cells of e.m.f.s E1and E2used in secondary gives balancing length l1and l2respectively then 
b) If the above cells are used at a time first in such a way that they support each other and later in such a way that they oppose each other and corresponding balancing lengths are l1and l2then
7. For finding internal resistance of cell
If l1and l2are balancing lengths respectively in secondary circuit if cell alone is used and when a parallel resistance R is included to it, then r = ( ) R.
8. a) A potentiometer is said to be more sensitive if it measures a small potential difference more accurately.
b) The smaller the potential gradient, the more is the sensitivity of potentiometer.
c) Sensitivity of potentiometer can be increased by
i) increasing resistance in primary circuit
ii) decreasing current in primary circuit
iii) increasing length of potentiometer.
Equivalent resistance in some special cases
(a) 12 wires each of resistance 'r' are connected in the form of 12 sides of a cube. Effective resistance across
i) Diagonally opposite corners = 
ii) Face diagonal = 
iii) 2 adjacent corners = 
(b) A wire of resistance 'R' is cut into n equal parts and all of them are connected in parallel the effective resistance is R/n2.
(c) If n wires each of resistance 'r' are connected to form a closed polygon, equivalent resistance across two adjacent corners is R =  r.
(d) Three equal resistances of each 'r' are connected to form a triangle. Equivalent resistance across any two verticles is  .
(e) Four equal resistances of each 'r' are connected to form a square. Equivalent resistance between
i) Adjacent corners is 
ii) Ends of diagonal is r
(f) A wire of resistance 'r' is bent into a circle. The effective resistance across any two vertices is  .
(g) A wire of resistance 'r' is bent into a triangle. Equivalent resistance across any two vertices is 2r/9.
(h) If n wires of equal resistances are given, the number of combinations they can be made to give different resistances is 2n - 1.
(i) If n wires of unequal resistances are given, the number of combinations they can be made to give different resistances is 2n(if n &g. 2).
(j) If 2 wires of resistivities s1, s2lengths l1, l2and of same area of cross section are connected in series.
equivalent resistivity s = 
If l1= l2, s = 
If l1= l2conductivity 
(k) If 2 wires of resistance s1, s2area of cross sections A1, A2and of same length are connected in parallel.
Equivalent resistivity s = 
If A1= A2, s = 
 .
THERMAL &am. CHEMICAL EFFECTS OF CURRENT
Heating effects of current
Imp
a. The P.D. established across the ends of a conductor will develop an electric field along its length.
b. The electrons accelerating by the applied field lose their excess energy continuously by the collisions with the lattice atoms.
c. This energy is converted into heat and increases the lattice vibrations.
d. If a P.D. 'V' is applied across a resistor of resistance R and a current 'i' is passing through it, then the energy dissipated or heat produced is given by H = Vit = i2Rt = .t joule since V=Ri. or  =  calories.
e. This was first observed and experimentally proved by Joule and so it is called 'Joule heating effect'.
. Joule's laws of heating effect :
a) The heat produced in a given time in a given resistor is directly proportional to the square of the current (Hai2).
b) The heat produced due to a given current in a given resistor directly proportional the time of flow of current (Hat)
c) The heat produced in a given time due to a given current is directly proportional to the resistance of the conductor (H a R)
Imp.
i. This effect is irreversible.
ii. As H a i2heating effect is common to both DC and AC
iii. As H a i2, filament bulb, heater, geyser etc work on both DC and AC
. Electric power :-
The electric power of a device
P = = Vi = i2R  = watt.
Consumption of electric power :
a) If a bulb is marked as 100W - 220V, then its power will be 100W when connected to 220V mains only.
b) If the applied voltage changes, its electric power also changes.
c) Among the bulbs of 100W - 220 V and 40W - 220V, the bulb of low wattage will have because more resistance as P a  as V is same. [40W bulb will have more resistance than 100W bulb).
d) If a bulb of 100W - 220 V is connected to the mains of a different voltage, resistance of the bulb remains same but power consumption changes.
Resistances in series :-
a. As the current i is same, the power (Heat) developed is proportional to the resistance  or  greater the resistance, larger will be the power consumed.
b. An electric bulb of low wattage will glow more in series because its resistance is more than a high wattage bulb.
c. If any one of the bulbs in series is blown off or fused out, then all the bulbs will not glow in that row and an infinite resistance is developed.
d. If n equal resistances each of R and power P are connected in series, then Ps=  , the power consumption or heat developed or the intensity of light produced will decrease by n times. (Rs= R1+ R2+ ........)
Resistances in Parallel :
a. When resistances are connceted in parallel, V is same  , ,ie., power (Heat) is inversely proportional to the resistance ie, P1R1= P2R2or H1R1= H2R2.
b. More power is consumed in smaller resistance of the Combination.
c. If n equal resistances each R and power P are connected in parallel, then
PP --=  = np.  . the power consumption or heat developed or intensity of light by the lamps will increase by n times.
d. When two bulbs of 60W - 220V and 100W - 220V are connected in parallel, the 100W bulb glows more because its resistance is low. ( P  in parallel) ie., in parallel combination the bulb of high waltage glows more.
e. As V is to be same, the electrical appliances in houseare connected in parallel.
f. Ps = and PP= n.  = n2.
i. power consumed by n equal resistances in parallel is n2times the power consumed in series when V is same for both combinations.
g. An electrical appliance consumes the specified power P only when it works on the specified voltage Vs.
h. If the applied voltage VAis different from the marked or specified voltage Vs, then the power consumption will be
P1=  .
i. When VA&g.VS, the appliance will get damaged because the current exceeds the limit (VS/R).
j. If VA&l.VS, power consumption will be lesser than specified power and the bulb gives less brightness.
Fuse :-
a. It is a metallic conducting wire with low melting point and high resistance.
b. It is placed in series with the appliance.
c. When the current in the circuit exceeds the specified value, the fuse is damaged by melting and breaks the circuit and the device is saved.
d. The current capacity of a fuse is independent of its length and varies wih the radius.
Unit of electrical energy consumed :
a) Board of Trade Unit (B.T.U.)
b) B.T.U. = It is the electrical energy consumed at the rate of 1000 joules per second in one hour.
c) B.T.U. = 1 kWH = 36 x 105joules
d) Units of electrical energy consumed by an electrical appliance
 = Units. (kWH).
Time taken in heating of water
A geyser of power P Watt is used for t seconds to rise the temperature of m gm mass of water through q0c, then  = msq, where S = specific heat of water (cal/gm/0C)
To boil certain mass of water a coil will take, t1time and another coil will take t2time. If they are connected.
a) in series : H =  .t
As H and V and J are constant, R a t
In serie. R = R1+ R2
ts= t1+ t2
b) in parallel,
As R =  ,
tp= 
c)  = 
. Thermocouple
a. The electrical conductivity in a metal is due to motion of electrons.
b. Different metals have different free electron densities.
c. When two dissimilar metals are joined at the junction a potential difference is developed due to diffusion of electrons from one metal to the others.
d. The rate of diffusion depends upon temperature of the junction.
e. When the two junctions of loop are at the same temperature, the two junctions will be at the same potential and no current passes through it.
f. If the two junctions are kept at different temperatures, their potentials will be different and a current passes through it
g. The electricity developed is called thermo-electricity.
The emf is called Thermo-emf.
The current is called thermoelectric current.
The phenomenon is called Seebeck effect.
The arrangement is called thermo-couple.
h. The heat energy is converted into electrical energy.
i. Two different metals forming a closed circuit with two junctions is called thermo couple.
j. Seebeck effect is reversible.
k. If the junctions are interchanged, direction of current is reversed.
l. In the case of iron-copper thermo-couple, current flows from iron to copper at the cold junction.
Thermoelectric series :-
a. Seebeck arranged the metals in an order that can form a thermocouple. This order is called thermoelectric series.
b. Antimony, nichrome, iron, Zinc, copper, gold, silver, lead, aluminium, mercury, platinum. Rhodium, platinum, nickel, constantan and Bismuth are some members in the series.
c. The elements before lead are thermoelectrically positive and after lead are thermoelectrically negative.
d. If two metals in the series form a thermocouple, current flows at the cold junction from the metal that comes earlier in the series to the metal later in the series.
e. For a given temperature difference, farther the metals selected in the series, larger will be the emf.
f. For a given temperature difference, thermo emf is maximum for antimony-bismuth thermocouple.
g. In the case of antimony-bismuth thermocouple, at the cold junction, current flows from antimony to bismuth.
Variation of emf with temperature :
a. When the cold junction is at 0oC and hot junction is at toC, the thermo emf developed is given by
e = at-bt2when a & b are both positive
e = at+bt2where a is positive but b is negative
The values of '; a & b '; are depend on the metals in thermo couple.
b. The equation shows that the graph between temperature difference and thermo emf is a parabola.
c. The emf increases rapidly first and then slowly, reaches a maximum and then decreases to zero and then reverse its sign with the increase of temperature of hot junction.
Neutral temperature :-
a. It is the temperature of hot junction at which thermo emf is maximum.
b. It depends upon the nature of the pair of metals.
c. It is a constant for a given thermo-couple.
d. It does not depend upon temperature of cold junction.
e. tn =  where a is positive but b is negative.
Fig.
Temperature of inversion :-
a. It is the temperature of hot junction at which thermo emf becomes zero or its sign is just reversed.
b. It depends on the nature of metals and the temperature of cold junction.
c. As temperature of cold junction increases, temperature of inversion decreases.
d. Temperature difference between neutral temperture and cold junction is equal to the temperature difference between neutral temperature and inversion temperature.
ie. Ti- Tn= Tn- Tc
e. For iron - copper thermocouple, Tn= 270oC
Thermo-electric power :-(Seebeck coefficient)
f. The rate of change of thermo emf with temperature is known as thermoelectric power (P) or Seebeck Coefficient (S).
g. P =  (at + bt2) = a + 2bt = s
h. S.I. Unit of P is VK-1. Dimensional formula is ML2T-3I-1K-1
i. Thermoelectric power is zero at neutral temperature At t = Tn, P = 0 and Tn= -  .
j. At inversion temperature, P is maximum.
At t = T. emf = 0 and Ti = -  (Tc= 0oC)
Thermoelectric diagram : -
a. It is a graph drawn between the temperature and thermoelectric power.
b. It is a straight line and called thermoelectric power line.
c. For metals of positive Thomson coefficient, it is a straight line with slope upwards.
d. For metals of negative Thomson coefficient, it is a straight line with slope downwards.
e.; The point of intersection of the above two power lines gives neutral temperature when the two metals form the thermocouple.
f.; Thermoelectric power line is drawn for a thermocouple formed with one given metal (positive or negative) and the other lead.
Fig.
Peltier effect:-
aWhen current is allowed to pass through a thermocouple, heat is evolved at one junction and heated up, it is Heat is absorbed at the other junction and it is cooled. This phenomenon is called 'Peltier effect'.
b.If the direction of flow of current is reversed, the hot and cold juction are also interchanged. Peltier effect is reversible.
c.Peltier effect is an converse of Seebeck effect.
d.The junction which was heated in Seebeck will now become cold in Peltier effect.
eThe heat evolved or absorbed at a junction is proportional to the charge passed through it.
Peltier Coefficient :- (p)
a.H  q or H = p q = p it. Where p is a constant known as Peltier coefficient.
b.Peltier Coefficient is defined as numerically equal to the amount of heat evolved or obsorbed at a junction when 1 A of current is passed through it in 1s (or) 1 coulomb of charge passes through the junction.
c.S.I. unit : Joule/ Coulomb.
Dimensional formula is M1L2T-3I-1
d.; depends upon nature of metals forming the junction and its temperature.
e.If T and T+  T are the temperature of the two junctions of a thermocouple and dE is thermo emf produced, then p = T.  , (T= temperature of cold junction).
f. According to Peltier effect, thermal emf is proportional to the temperature and the graph between them must be a straight line. But, in practice it is a parabola. This shows that Peltier effect alone can not explain seebeck effect. This discrepancy lead to the concept of Thomson effect.
.Thosmson effect :-
a.The phenomenon of absorption or evolution of heat due to passage of current in a non uniformly heated conductor is known as Thomson effect.
b.It is a reversible process.
c.; In the case of copper, points of higher temperature will be at higher potential.
When current flows from colder to hotter part, energy is required and heat is absorbed and vice versa.
d.For metals like copper, silver, zinc, cadmium, antimony etc., Thomson effect is positive.
e.In the case of iron, points of higher temperature will be at lower potential.
f.When current flows from colder to hotter part of iron heat is evolved
g.Metals like iron, platinum, nickel, Cobalt, bismuth etc. Thomson effect is negative.
h.In lead Thomson effect is zero.
Thomson Coefficient : (s)
a.It is numerically equal to the amount of heat evolved or absorbed when 1A of current passes through a conductor in 1s with a temperature difference of 1oC across its ends.
b.It is also equal to the potential difference between two parts having a temperature difference of 1oC or 1 K.
c.If a current i is passed through a conductor for t seconds having a temperature difference , the heat evolved or absorbed is given by H = s it dT.
d.s =  where q = i.t, the quantity of electric charge passed through the conductor.
e.s =  where dV is P.D. between two points, of the conductor with a temperature difference dT.
f. S.I. unit : VK-1Dimensional formula is ML2T-3I-1K-1.
g. s =  -T.
Relations between S, p and s:-
a.p = T.  and s = -T  p = T(S)
b. If T1and T2are the absolute temperatures of cold and hot junctions respectively, the total emf developed in the thermocouple is given by
E = 
Where p1and p2Peltier coefficients of the junctions and 1, 2are Thomson Coefficients of the metals.
c.Thomson emf is given by E = 
d.Seebeck emf is the algebraic sum of two Peltier emfs and two Thomson emfs. [ Two metals are involved]
Law of Intermediate Temperatures :-
For the given thermocouple, if thermoemfs when the junctions are at temperatures (T1, T2), 
Law of Intermediate metals :-
For the given temperature difference of the two junctions, if are the thermo emfs developed for thermo couples made with metals (A, B) (B, C) an (A, C)
Then  (Note  )
CHEMICAL EFFECTS OF CURRENT
Definitions of various terms -
1 Electrolysis - The process of splitting up or decomposing a liquid by passing an electric current through it, is defined as electrolysis.
2. Electrolyte - The compound, whether fused or in solution, which undergoes decomposition by an electric current, is defined as electrolyte.
3. Anions and Cations -The decomposed substance appears in the form of ions. The ions appearing at the anode are known as anions and those at the cathode are known as cations.
4. Ionisation - The phenomenon of separation of a molecule into oppositely charged ions is known as ionisation.
5 Equivalent weight - The ratio of the atomic weight of an element to its valency is defined as its equivalent weight.
6.Atomic weight - The ratio of the average mass of an atom of an element to the mass of an atom of hydrogen, taken as 1.008, is defined as the atomic weight of that element.
7.Valency - The valency of an element is the number of atoms of hydrogen or chlorine which combines with or is displaced by one atom of the element.
8. Molecular weight - The molecular weight of a substance is the ratio of the mass of one molecule of the substance to the mass of an atom of oxygen which is taken as 16.
9.Gram-equivalent - It is the weight in grams of an element which will combine with or replace 1 gm of hydrogen.
Gram-equivalent = 
10. Gram-molecule - The molecular weight of any substance, expressed in grams is defined as gram-molecule of that substance.
11. Avogadro number - The number of molecules of a substance in its one-gram-molecule is known as Avogadro number.
12. Normal solution - A solution containing one gram-equivalent of the solute per litre is called a normal solution.
Faraday';s laws of Electrolysis -
1.First law - The total mass of ions liberated at an electrode, during electrolysis, is proportional to the quantity of electricity which passes through the electrolyte.
i.e. m  Q But Q = it
Hence the first law may also be stated as follows -
The mass of ions liberated at an electrode during electrolysis is proportional to
(i) the current flowing through the electrolyte, and
(ii) the time for which the current flows
2.Second law - If same quantity of electricity is passed through different electrolytes, the masses of the substances (ions) deposited at the respective cathodes are directly proportional to their chemical equivalents (equivalent weights).
i.e. m E (chemical equivalent)
Electro-chemical equivalent (E.C.E.) -
(i) The electro-chemical equivalent of an element is its mass in grams deposited on the electrode by the passage of 1 coulomb of charge through it i.e. by the passage of 1 ampere current for 1 second.
(ii) According to Faraday';s first law -
m i.t
or m = Z .i.t
Where Z is the constant of proportionality known as the electro- chemical equivalent of the substance. It is numerically equal to the mass in grams of the element deposited when a unit current flows in unit time.
(iii) According to Faraday';s first law
m = ZQ
If same charge is passed in two electrolytes, then 
According to Faraday';s second law
or 
(iv) E.C.E. of any substance = E.C.E. of hydrogen x chemical equivalent of the substance.
(v) If silver is taken as the standard substance for which E.C.E. is 0.001118 gm per coulomb, then E.C.E. of any substance = 0.001118 × 
Faraday -
(i) The quantity of electricity (i.e. charge required to liberate a gram equivalent of a substance during electrolysis.)
(ii) As derived above
 or  = constant = F
The constant F is known as one Farady
For example, for copper E = 31.5 gm
and Z = 0.000329 gmC-1
 coulomb
(iii) According to Faraday's first law
m = Z.i.t. = ZQ
If p is the valency of the element, then electrons has to flow through the solution to deposit one atom.
charge required to deposit 1 mol of the substance = Npe, where N = Avogadro number and m = M
M = ZNpe
Where F = Ne = Faraday constant
or F = 6.0229 × 1023× 1.602 × 10-19
= 96487 C  96500 C.
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