Current electricity

The study of electric charges in motion is called current electricity.

1. Electric Current

The rate of flow of electric charges through a conductor is called electric current.

Current is defined as the rate of flow of electric charge.

I=qt

or Instantaneous current I=dqdt

Conventionally, the direction of current is taken along the direction of flow of positive charge and opposite to the direction of flow of negative charge (electron).

Current is a scalar quantity. SI unit of electric current is ampere (A).

2. Flow of Electric Charges in a Metallic Conductor

A metallic conductor contains free electrons as charge carriers, while positive ions are fixed in the lattice. When no potential difference is applied, the motion of free electrons is random so there is no net current in any direction. When a potential difference is applied across the conductor the free electrons drift along the direction of positive potential so a current begins to flow in the conductor, the direction of current is opposite to the direction of the net electron flow.

3. Drift Velocity and Mobility

Drift velocity is defined as the average velocity with which the free electrons get drifted towards the positive end of the conductor under the influence of an external electric field applied. It is given by the relation

vd−→=–eE⃗ mτ

where m = mass of electron, e = charge of electron

E = electric field applied

τ=relaxationtime=meanfreepathrootmeansquarevelocityofelectrons

Mobility of an ion is defined as the drift velocity per unit electric field i.e.,

μ=vdE=eτm

Its unit is m2/Vs.

4. Relation between Drift Velocity and Mobility with Electric Current

Current, in terms of drift velocity i=neAvd,

Current, in terms of mobility i=neAµE,

where, n = number of free electrons per metre3

A = cross-sectional area of conductor.

5. Ohm’s Law

It states that the current flowing in a conductor is directly proportional to the potential difference applied across the conductor provided physical conditions, e.g., temperature, pressure, etc. remain the same.

I∝VorV∝IorV=RI

where R is called electrical resistance. Its unit is volt/metre or ohm.

Ohm’s law is not applicable to all types of conductor. It is applicable only for those conducting materials for which V-I graph is non linear.

6. Electrical Resistance

The hindrance offered by a conductor to the flow of current is called the electrical resistance of the conductor. The electrical resistance of a conductor depends on its length l, cross-sectional area A and nature of material and is given by

R=ρlA

where ρ is the resistivity of the material and is given by

ρ=mne2τ ∴R=mlne2Aτ

7. V-I Characteristics: Linear and Non-linear — Ohmic and Non-ohmic Conductors

The conductors or circuit elements for which V-I graph is linear are called ohmic conductors. The examples are metallic conductors.

On the other hand, the circuit elements for which V-I graph is non-linear are called non-ohmic conductors. The examples are junction diodes and transistors.

Electrical Energy and Power

8. Joule’s Law of Heating

The heat which is produced (or consumed) due to the flow of current in a conductor, is expressed in joules.

Mathematically, amount of heat produced (consumed) is proportional to square of amount of current flowing through conductor, electrical resistance of wire and the time of current flow through it.

So, Hαi2RT

⇒ H=i2RTJ

where J is a joule constant. 1 joule constant is 4.18×103J/kcal

⇒ H=i2RTJ=VitJ=V2JRt

Where V is the potential difference across wire.

9. Power

Rate of energy dissipation in a resistor is called the power i.e.,

Power P=Wt=Vi=i2R=V2R

The unit of power is watt.

10. Fuse

It is a safety device used in electrical circuits. It is made of iron-lead alloy. The characteristics of fuse are high resistivity and low melting point.

When high current (more than fuse-rated value) flows through a circuit, the fuse wire melts and causes a break in the circuit.

11. Resistivity (or Specific Resistance)

Resistivity of a substance is defined as the resistance offered by a wire of that substance of 1 metre length and 1 square metre cross-sectional area.

Resistivity depends only on the material and is independent of dimensions at a given temperature. The SI unit of resistivity is ohm × metre (Ωm).

12. Conductance and Conductivity

The reciprocal of resistance is called the conductance (G)

i.e., G=1R

Its SI unit is (ohm)–1 or mho or siemen (S).

The reciprocal of resistivity is called the conductivity (σ).

i.e., σ=1ρ

Its SI unit is ohm–1 metre–1 (or mho m–1) or Sm–1

13. Colour Code for Carbon Resistances

Very high resistances are made of carbon. The value of high resistance is specified by four bands of different colours. The first three bands represent value of resistance while the last band represents tolerance (variance). The first band represents first digit, second band represents second digit and third band represents multiplier in powers of 10. The colour of fourth band tells the tolerance. Absence of fourth band means a tolerance of 20%. The following table gives the colour code for carbon resistances.

First letter of colour	Colour	Figure	Multiplier	% Tolerance
B	Black	0	100=1
B	Brown	1	101
R	Red	2	102
O	Orange	3	103
Y	Yellow	4	104
G	Green	5	105
B	Blue	6	106
V	Violet	7	107
G	Grey	8	108
W	White	9	109
	Gold	—	10–1	5
	Silver	—	10–2	10
	No colour	—	—	20

To memorise these colour codes, the following sentence is of great help.

B.B. ROY (of) Great Britain (has) Very Good Wife.

14. Resistances in Series and Parallel

(i) When resistances are connected in series, the net resistance (Rs) is given by

R = R1+R2+R3+.......+Rn

In series I1 = I2 = I3 = Is (same)

voltage, Vs = V1+V2+V3+.....+Vn

(ii) When resistances are connected in parallel, the net resistance (Rp) is given by

1RP=1R1+1R2+.....+1Rn

In parallel, current IP = I1 + I2 + I3 + ....... + In

voltage VP = V2 = V3 = VP

For two resistances R1 and R2 in parallel

1RP=1R1+1R2⇒RP=R1R2R1+R2

15. Temperature Dependence of Resistance

The resistance of a metallic conductor increases with increase of temperature.

Rt = R0 [1 + a (t – t0)]

where R0 is resistance at 0°C and Rt is resistance at t°C and α is temperature coefficient of resistance. In general if variation of temperature is not too large, then

α=R2–R1R1(t2–t1)peroCorperK

In terms of resistivity

αr=ρ2–ρ1ρ1(t2–t1)peroCorperK

However, the resistance of a semiconductor decreases with rise in temperature.

16. Super Conductors

Some substance lose their resistance when cooled below a certain temperature. These substances are called superconductors and the temperature below which they lose resistance is called transition temperature. The transition temperature of Hg is 4.2 K.

17. Electric Cell

It is a device which converts chemical energy into electrical energy.

EMF of a cell (E) is defined as the maximum potential difference when no current is being drawn from the cell.

Terminal Potential difference (V) is defined as the potential difference when current is being delivered to external load resistance.

Internal Resistance (r) of a cell is the hindrance offered by the electrolyte of cell to the flow of current. Internal resistance of a cell depends on

(i) separation between electrodes.

(ii) area of immersed part of electrodes.

(iii) concentration and nature of electrolyte.

E = V + ir ⇒ V = E – ir

When a current i is passed in cell in opposite direction by external battery, then terminal potential difference V = E + ir

18. Combination of Cells

(i) When n-identical cells are connected in series

NoSuchKeyThe specified key does not exist.0e1358dc-0c53-465a-ab64-4c9bbc46dc3d/mathml/EQ-0930-163102.xmlC7CA542302C1D2E8r5hkCM7JSY16myYetACc9aC+B8FLJTjRUkL//eYA+XIwzQ3Kbp94wkJmtKcy2ruf7gFT6YokAPE=

For useful series combination, the condition is Rext >>Rint

(ii) When m-identical cells are connected in parallel

i=EnetRext+Rint=ER+r/m

Condition of useful parallel combination is R < r/m.

(iii) When N = mn, cells are connected in mixed grouping (m-rows in parallel, each row containing n cells in series)

Current, i=nER+nrm=mnEmR+nr

Condition for useful mixed grouping is Rext = Rint

i.e., R=nrm

(iv) When two cells of different emfs E1 and E2 and different internal resistances r1 and r2 are connected in parallel as shown in fig. then net emf of combination is

E=E1r1+E2r21r1+1r2=E1r2+E2r1r1+r2

Net internal resistance rint

1rint=1r1+1r2⇒rint=r1r2r1+r2

19. Kirchhoff’s Laws

(i) First law (or junction law): The algebraic sum of currents meeting at any junction in an electrical network is zero,

i.e., ∑I = 0

This law is based on conservation of charge.

(ii) Second law (or loop law): The algebraic sum of potential differences of different circuit elements of a closed circuit (or mesh) is zero, i.e.,

∑V = 0

This law is based on conservation of energy.

20. Wheatstone’s Bridge

It is an arrangement of four resistances P, Q, R, and S forming a closed circuit. A potential difference is applied across terminals A and C. A galvanometer is connected across B and D. The condition of null point (no deflection in galvanometer) is

PQ=RS

21. Metre Bridge

Metre bridge is based on the principle of Wheatstone’s bridge. In fact, it is practical application of Wheatstone’s Bridge. It consists of 1 m long resistance wire. The resistance of wire is divided into two resistances P and Q. R is known resistance and S is unknown resistance.

At balance PQ=RS⇒l(100–l)=RS

⇒ Unknown resistance, S= NoSuchKeyThe specified key does not exist.0e1358dc-0c53-465a-ab64-4c9bbc46dc3d/mathml/EQ-0930-165155.xmlEC633D4DAEA304B8Oj6xNA/qWNb9aTtM+Hk9/ImCOWYNT85it7LoO6ZrtZ2cXE2lR8gseSJCZ17YmRJNkqo7LNCBb3Q=

22. Potentiometer

It is a device to measure the potential difference across a circuit element accurately. The circuit containing battery of emf E1 is the main circuit and the circuit containing battery of emf E2 is the secondary circuit. For the working of potentiometer emf E1 > emf E2.

When a steady current is passed through a potentiometer wire AB, there is a fall of potential along the wire from A to B. The fall of potential per unit length along potentiometer wire is called the potential gradient. If L is length of wire AB and V is the potential difference across it then

Potential gradient k=VL

The SI unit of potential gradient is volt/metre.

It is a vector quantity.

If l is the balancing length of cell of emf E, then E = kl.

If l1 and l2 are the balancing lengths for two cells of emfs E1 and E2 for the same potential gradient,

then E1E2=l1l2