CLASS 12 CURRENT ELECTRICITY NOTES

3.1 INTRODUCTION
• Charges in motion constitute an electric current
• Lightning - charges flow from the clouds to the earth through the atmosphere
3.2 ELECTRIC CURRENT

• (steady current)
• Current is the rate of flow of electric charges through any conducting path
• Let ΔQ be the net charge flowing across a cross-section of a conductor during the
time interval Δt. Then

• SI Unit – Ampere (C/s)

3.3 ELECTRIC CURRENTS IN CONDUCTORS
• In nature, free-charged particles exist in the upper strata of the atmosphere - the
ionosphere
• Conductors – the movement of free electrons constitutes the current
• Electrolytic solutions – the movement of both positive and negative charges
constitutes the current
• Semiconductor – movement of free electrons and holes
• Discharge tubes – ionized gas
• When E=0
➢ The electrons move due to thermal motion, colliding with the fixed ions.
➢ An electron colliding with an ion retains its speed but changes direction
randomly after the collision
➢ The number of electrons traveling in any direction equals the number traveling
in the opposite direction, resulting in no net electric current
• When E ≠ 0

➢ The electric field will accelerate the electrons towards +Q, causing them to
neutralize the charges and create an electric current
➢ When connected to cells or batteries, a steady electric field in the conductor
results in a continuous current.

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3.4 OHM’S LAW
• Imagine a conductor with current I and potential difference V between its ends. Ohm’s
law states that

where R – proportionality constant – resistance

• SI Unit of R – ohm(Ω)
• R depends on
➢ Material of conductor
➢ Dimension of conductor
• Ohm’s law states that the voltage across a conductor is directly proportional to the
current flowing through it, provided all physical conditions and temperatures remain
constant.

V = IR

I remain same

V is doubled (V+V)

R = 2V/I = 2R (Doubled) ->

I is halved

V remains same

R = V/(I/2) = 2R ->

• So,

• Here ρ – resistivity (depends on material )

• Current per unit area j = I/A - current density- SI units - A/m2
• if E is the magnitude of the uniform electric field in the conductor whose length is l,
then the potential difference V across its ends is El

• The current density is also directed along E and is also a vector

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 Equivalent form of Ohm’s Law
Where σ = 1/ ρ – conductivity
• Resistivity - measures the electrical resistance
• Conductivity - material's ability to conduct electric current
• Conductance is an expression of the ease with which electric current flows through
materials – G – inverse of R
3.5 DRIFT OF ELECTRONS AND THE ORIGIN OF RESISTIVITY
Case I –> E = 0
When electrons collide with fixed ions, they emerge in random directions at the same speed,
resulting in an average velocity of zero due to their random directions. Iif there are N
electrons and the velocity of the ith electron (i = 1, 2, 3, ... N ) at a given time is vi

Case II -> E ≠ 0
Electrons collide with other particles, get accelerated, and gain drift velocity.
F = ma
a = F/m = Eq /m = -eE /m
Consider the i th electron at a given time t and let t i be the time elapsed after its previous
collision. If vi was its velocity immediately after that previous collision, then its velocity Vi at
time t is

since v = u +at

Let τ - the average time between successive collisions(relaxation time)

The average velocity of the electrons at time t is the average of all the Vi ’s , so averaging
over the N-electrons at any given time t gives us for the average velocity vd

This tells us that the electrons move with an average velocity which is independent of time,
although electrons are accelerated. This is the phenomenon of drift and the velocity vd is
called the drift velocity.
Because of the drift, there will be net transport of charges across any area perpendicular to E

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Consider a planar area A, inside the conductor whose normal is parallel to E.
In time Δt, all electrons to the left of the area would cross distance = |vd|Δt (since velocity =
disp/time)
n - number of free electrons per unit volume in the metal
Total no. of electrons = n Δt |vd|A
Total charges transported = –ne A|vd|Δt (opp to E)
Total charges transported in the direction of E = ne A|vd|Δt
So, I Δt = ne A|vd|Δt
I Δt = ne (-eE τ/m )A Δt

From the formula of current density, I = |j|A

(or)

Here
So, we get j = σE which the equivalent form of Ohm’s Law.
3.5.1 Mobility
• mobility μ is defined as the magnitude of the drift velocity per unit electric field

(or)
• SI unit of mobility is m2 /Vs
3.6 LIMITATIONS OF OHM’S LAW
a) V ceases to be proportional to I

(b) The relation between V and I depends on the sign of V i.e, if I is the current for a certain
V, then reversing the direction of V keeping its magnitude fixed, does not produce a current
of the same magnitude as I in the opposite direction. Ex: diode

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 c) The relation between V and I is not unique, i.e., there is more than one value of V for the
same current I. Ex: GaAs

3.7 RESISTIVITY OF VARIOUS MATERIALS

1. Conductors – Metals - 10–8 Ωm to 10–6 Ωm
2. Semiconductors - resistivities decrease with a rise in temperature or by adding a small
amount of suitable impurities
3. Insulators - 1018 times greater than metals or more
3.8 TEMPERATURE DEPENDENCE OF RESISTIVITY
The resistivity of a metallic conductor is
temperature co-efficient of resistivity

At T At To
• Dimension of α = (temperature)-1 used in wire-bound standard resistors as R changes
• α - +ve for metals very little with temp. Ex: manganin, constantan

•
• So, ρ depends inversely both on the number n of free electrons per unit volume and on
the average time t between collisions.
• Temp ↑, Vthermal ↑, collision ↑, τ↓, ρ↑
• Metals - Temp ↑, n-constant, τ↓, ρ↑
• Semiconductors & Insulators - Temp ↑, n↑ > τ↓, ρ↓

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3.9 ELECTRICAL ENERGY, POWER
• Consider a conductor with endpoints A and B, in which a current I is flowing from A
to B. The electric potential at A and B is denoted by V(A) and V(B) respectively.
Since the current is flowing from A to B, V(A) > V(B) and the potential difference
across AB is V = V(A) – V(B) > 0.
• In a time interval Δt, charge ΔQ = I Δt travels from A to B.
• The potential energy of the charge at A = Q V(A) and similarly at B = Q V(B)
ΔUpot = Final potential energy – Initial potential energy
= ΔQ[(V (B) – V (A)] = –ΔQ V
= –I VΔt < 0
By , Conservation of total energy ΔK = –ΔUpot = I VΔt > 0
• When charges move freely through a conductor due to an electric field, their kinetic
energy increases. This energy is then transferred to the atoms during collisions,
causing the atoms to vibrate more vigorously, which in turn heats up the conductor.
amount of energy dissipated as heat in the conductor during the time interval Δt = ΔW = I
VΔt
The energy dissipated per unit time = power dissipated P = ΔW/Δt
And P = I V , by Ohm’s law

This represents Ohmic loss in a conductor

• Power comes from any external source or from the chemical energy of the electrolytic
cell.
• Power depends on V & I
• Electrical power is transmitted from power stations to homes and factories, which
may be hundreds of miles away, via transmission cables.
• To minimize power loss:
Consider a device R, to which a power P is to be delivered via transmission cables
having a resistance Rc to be dissipated by it finally. If V is the voltage across R and I
the current through it, then P = V I
The connecting wires from the power station to the device have a finite resistance Rc.
The power dissipated in the connecting wires, which is wasted is Pc with

the power wasted in the connecting wires is inversely proportional to V2

• To reduce Pc, these wires carry current at enormous values of V and this is the reason
for the high voltage danger signs on transmission lines
• Electricity at high voltages is unsafe, so a transformer lowers the voltage for safe use.

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3.10 CELLS, EMF, INTERNAL RESISTANCE
• an electrolytic cell has two electrodes - the positive (P) and the
negative (N) immersed in electrolytic solution
• The positive electrode has a potential difference V+ (V+ > 0) between
itself and the electrolyte solution immediately adjacent to it marked A
& the negative electrode develops a negative potential – (V– ) (V– ≥ 0)
relative to the electrolyte adjacent to it marked B
• When I = 0 the electrolyte has the same potential & the potential
difference between P and N is V+ – (–V– ) = V+ + V– = ϵ
• ϵ - electromotive force of cell(emf)
• emf is the potential difference between the positive and negative electrodes in an open
circuit, i.e., when no current is flowing through the cell
• consider a resistor R connected across the cell and suppose it is infinite
• A current I flow across R from C to D
• I = V/R = 0
• V = ϵ = Potential difference between P and A + Potential difference between A and B
+ Potential difference between B and N
• When R is finite, I is not zero and

• Here, the electrolyte through which a current flow has a finite resistance r, called the
internal resistance
• The internal resistance of dry cells, however, is much higher than the common
electrolytic cells.
• internal resistances of cells in the circuit may be neglected when ϵ >> I r
• Using Ohm’s Law

• The maximum current that can be drawn from a cell is for R = 0 and it is Imax = ϵ/r
3.11 CELLS IN SERIES AND IN PARALLEL
A) Series

one terminal of the two cells is joined together leaving the other terminal in either cell-free
V (A) – V (B) is the potential difference between the positive and negative terminals of the
first cell
VAB = V(A) – V(B)= ϵ1 - I r1
2 rules
VBC = V(B) – V(C)= ϵ2 - I r2
(i) The equivalent emf of a series combination of n
VAC = V(A) – V(C)= V(A) – V(C) + V(B) - V(B) cells is just the sum of their individual emf’s, and
(ii) The equivalent internal resistance of a series
= V(A) – V(B) + V(B) - V(C)
combination of n cells is just the sum of their
internal resistances.

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 = ϵ1 - I r1 + ϵ2 - I r2
= ϵ1 + ϵ2 - I (r1 + r2) = ϵeq – I req
Where ϵeq = ϵ1 + ϵ2 & req = r1 + r2
B) Parallel

Let V (B1 ) and V (B2 ) be the potentials at B1 and B2 , respectively.

Now, I = I1 + I2
Potential difference across cell 1 = V (B1 ) – V (B2 ) = ϵ1 – I1 r1
The potential difference across cell 1 = V (B1 ) – V (B2 ) = ϵ2 – I2 r2

Hence V = ϵeq – I req where,

Rules:

If there are n cells of emf ϵ1, . . . ϵn and of internal

resistances r1 ,... rn respectively, connected in parallel,
the combination is equivalent to a single cell of emf
In simple terms ϵeq and internal resistance req, such that

3.12 KIRCHHOFF’S RULES

(a) Junction rule: At any junction, the sum of the currents entering the junction is equal to the
sum of currents leaving the junction
Explanation:

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"When currents are steady, there is no accumulation of charges at any junction or at any point
in a line. Thus, the total current flowing in must equal the total current flowing out."
(b) Loop rule: The algebraic sum of changes in potential around any closed loop involving
resistors and cells in the loop is zero
Explanation:
Electric potential depends on the location of the point. Therefore, the total change when
returning to the same point must be zero.
Labelling:
• Cell – negative to positive - +
• Resistor – current & path same direction – negative
3.13 WHEATSTONE BRIDGE
• The bridge has four resistors R1 , R2 , R3 and R4
• Across one pair of diagonally opposite points (A and C )
source is connected. -- battery arm
• Between the other two vertices, B and D, a galvanometer
G (which is a device to detect currents) is connected - the
galvanometer arm
• the cell has no internal resistance
• balanced bridge – current through galvanometer = 0
• for determination of an unknown resistance
• A practical device using this principle is called the meter
bridge
• By applying Kirchhoff’s junction rule in junctions B & D
I1 = I3 and I2 = I4
Next, we apply Kirchhoff’s loop rule to closed loops ADBA and CBDC

Using I1 = I3 and I2 = I4 in the second loop

Using these equations:

So, we get

• To compute the unknown resistor, we keep it as R4, and keeping known resistances R1
and R2 in the first and second arm of the bridge, we go on varying R3 till the
galvanometer shows a null deflection. The bridge then is balanced, and from the
balance condition the value of the unknown resistance R4 is given by,

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