## Introduction

We considered all **charges** whether free or bound to be at rest in the previous two chapters. Charges in motion constitute an electric current. Lightning is one of the natural phenomena in which charges flow from clouds to earth through the atmosphere.

In this chapter, we will study some basic laws concerning steady **electric current** and their applications.

## Electric Current

The rate of flow of electric charge through any cross-section of a conductor is known as **electric current**. If ΔQ amount of charge flows through any cross-section of the conductor in the interval t to (t + Δt), then it is defined as

`i=\frac{ΔQ}{Δt}`

The direction of **current** is taken as the direction of motion of positively charged particles and opposite to the direction of negatively charged particles.
SI unit of current is **ampere** (A). It is a scalar quantity.

## Current Density

The **current density** at any point in a conductor is the ratio of the current at that point in the conductor to the area of the cross-section of the conductor. It is a vector quantity and denoted by `\vec{J}`.

`\vec{J}=\frac{Δi}{ΔA}`

The SI unit of current density is A/m^{2}.

## Drift Speed

**Drift Velocity** is defined as the average velocity with which the free electrons move towards the positive end of a conductor under the influence of an external electric field applied. It is denoted by v_{d}.

`v_d=\frac{eE}{m} τ`

Where, τ = relaxation time, E = electric field, m = mass, e = charge of electron.

**Read also:** Moving Charges and Magnetism Class 12 Physics Notes Chapter 4

## Relation between Current Density and Drift Speed

Let, cross-sectional area of any conductor be A, the number of electrons per unit area be n, drift velocity be v_{d}, then the number of total moving electrons in t second will be

N = (nAv_{d}t)

So, moving charge in t second Q = (nAv_{d}t).e

Hence, electric current in t second =`\frac{Q}{t}`

`i=\frac{nAv_{d}te}{t}`

i = neAv_{d}

We know `J=\frac{i}{A}`

Putting i = neAv_{d} in above equation

`\vec{J}`=nev_{d}

## Ohm’s Law

According to this law, "At constant temperature, the potential difference V across the ends of a given metallic wire (conductor) in a circuit (electric) is directly proportional to the current flowing through it". i.e.,

V ∝ i

V = i.R

where, R = resistance of conductor

**Read also:** Electrochemistry Class 12 Chemistry Notes Chapter 3

## Mobility

**Mobility** is defined as the magnitude of the drift velocity per unit of the electric field. It is denoted by μ,

`μ=\frac{v_d}{E}`

Its SI unit is m^{2}V^{-1}s^{-1}.

## Resistance

**Resistance** is the ratio of the potential difference applied across the ends of the conductor to the current flowing through it.

`R = \frac{V}{i}`

The SI unit of R is **ohm** (Ω).

**Read also:** Conceptual Questions for Class 12 Physics Chapter 3 Current Electricity

## Resistivity

**Resistivity** is defined as the ratio of the electric field applied at the conductor to the current density of the conductor. It is denoted by ρ

`ρ=\frac{E}{J}` ......(1)

If the length of the conductor be 'l', the cross-sectional area be 'A', the potential difference at the end of the conductor be 'V' and the electric current be 'i', then `\vec{E}` and `\vec{J}` given by

`\vec{E}=\frac{V}{l}` ......(2)

`\vec{J}=\frac{i}{A}` .......(3)

Putting the value of **E** and **J**, from equations (2) and (3) into (1), we get

`ρ=\frac{\frac{V}{l}}{\frac{i}{A}}`

`ρ=\frac{V}{i}.\frac{A}{l}`

`ρ=R\frac{A}{l}`

The constant of proportionality ρ depends on the material of the conductor but not on its dimensions. ρ is known as **resistivity** or **specific resistance**.

## Conductivity

**Conductivity** is defined as the reciprocal resistivity of a conductor.
It is expressed as,

`σ=\frac{1}{ρ}`

SI unit is mho per metre (Ω^{-1} m^{-1}).

## Superconductivity

The resistivity of certain metals or alloys drops to zero when they are cooled below a certain temperature is called **superconductivity**.

## Electrical Energy

When an electric current is moved in an electric circuit, then the energy of work done by taking a charge from one point to another point is called **electric energy**.

If a charge q at potential difference V is moved from one point to another point, then doing work will be

W = V . q .....(1)

Putting q = i.t in equation (1), we get

W =Vit

Putting V = i.R in equation (1), we get

W = i^{2}Rt

Putting `i=\frac{V}{R}` in equation (1), we get

`W=\frac{V^2}{R}t`

## Power

Electric power is the rate of doing work by electric charge. It is measured in watt and represented by P.

`P=\frac{W}{t}` [∵ 1HP = 746 watt]

Hence, P = Vi = i^{2}R = `\frac{V^2}{R}`

## Resistor Colour Codes

A **carbon resistor** has a set of coaxial coloured rings in them, whose significance is listed in the above table. **First two bands** formed; the First two significant figures of the resistance in ohm. **Third band**; Decimal multiplier as shown in the table.
**Last band**; Tolerance or possible variation in percentage as per the indicated value. For Gold ±5%, for silver ±10% and for No colour ± 20%.

## Combination of Resistors

There are two types of resistance combinations.

### (i) Series Combination

In **Series** Combination, different resistances are connected end to end.
Equivalent resistance can be obtained as the formula,

R = R_{1} + R_{2} + R_{3}

**NOTE:** The total resistance in the series combination is more than the greatest resistance in the circuit.

### (ii) Parallel Combination

In **Parallel** combination, the first end of all the resistances is connected to one point and the last end of all the resistances is connected to another point. Equivalent resistance can be obtained by the formula

`\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}`

**NOTE:** The total resistance in parallel combination is less than the least resistance of the circuit.

## Cells, EMF, Internal Resistance

### (i) Cells

An **electrolytic cell** consisting of two electrodes, called positive (P) and negative (N) immersed in an
electrolytic solution as shown in the figure.

Electrodes exchange charges with the electrolyte. Positive electrode P has a potential difference
V_{+} between itself and electrolyte solution A immediately adjacent to it. Negative electrode N has a potential difference (V_{–}) relative to electrolyte B adjacent to it.

ε = V_{+} - V_{–}

### (ii) EMF

It is the difference in chemical potentials of electrodes used. It is also defined as the difference of potential across the electrodes of the cell when the electrodes are in an open loop.

ε = V_{+} - V_{–}

### (iii) Internal Resistance

It is the opposition offered by the electrolyte of the cell to the flow of current through itself. It is represented by r and given by

`r = \frac{v}{i}`

## Cells in Series and Parallel

## Kirchhoff’s Laws

**Kirchhoff’s** two rules are used for analysing electric circuits consisting of a number of resistors and cells interconnected in a complicated way.

### Kirchhoff’s first rule: Junction rule

At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction.

Σ i = 0

### Kirchhoff’s second rule: the Loop rule

The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.

Σ iR = Σ E

## Wheatstone Bridge

It is an application of Kirchhoff’s rules. The bridge is consisting of four resistances R_{1}, R_{2}, R_{3} and R_{4} as four
sides of a square ABCD as shown in the figure.

Across the diagonally opposite points between A and C, battery E is connected. This is called the battery arm. To the remaining two diagonally opposite points B and D, a galvanometer G is connected to detect current. This line is known as the galvanometer arm.

Currents through all resistances and galvanometer are as shown in figure. In balanced Wheatstone bridge we consider the special case I_{g} = 0. Applying junction rule to junction B and D, we have

I_{2} = I_{4} and I_{1} = I_{3}

Applying loop rule to loop ABDA

I_{2}R_{2} + 0 – I_{1}R_{1} = 0

`\frac{I_1}{I_2}=\frac{R_2}{R_1}` .....(i)

Applying loop rule to loop BCDB

I_{4}R_{4} – I_{3}R_{3} + 0 = 0

I_{2}R_{4} – I_{1}R_{3} = 0 (Using I_{4} = I_{2} and I_{3} = I_{1})

`\frac{I_1}{I_2}=\frac{R_4}{R_3}` .....(ii)

Using equation (i) and (ii), we have

`\frac{R_2}{R_1}=\frac{R_4}{R_3}` .....(iii)

The equation (iii) relating the four resistors is called the balance condition for the galvanometer to give zero or null deflection.

## Meter Bridge

It is the practical application of the **Wheatstone bridge**. A standard wire AC of length one metre and of uniform cross-sectional area is stretched and clamped between two thick metal strips bent at right angles as shown in the figure.

The endpoints, where the wire is clamped are connected to a cell ε through a key K_{1}. The metal strip has
two gaps across which resistors can be connected. One end of the galvanometer is connected to the mid-point of
the metal strip between the gaps. The other end of the galvanometer is connected to a jockey, which can
slide over AC to make electrical connections by its knife edge. R is the **unknown resistance** to be determined.
S is the standard **known resistance** from a resistance box.

Let the jockey be in contact with point D. Length
of portion AD of wire be l_{1}. Resistance of the portion AD is R_{AD} = R_{m}l_{1} and resistance of the portion DC is
R_{DC} = R_{m}(100 – l_{1}), where R_{m} is the resistance per centimetre of the wire. Now R, S, R_{AD} and R_{DC} represent four resistances of the Wheatstone bridge.

`\frac{R}{S}=\frac{R_{AD}}{R_{DC}}=\frac{R_{m}l_1}{R_{m}(100-l_1)}`

Unknown resistance R in terms of standard known resistance S is given by

`R=S(\frac{l_1}{100-l_1})`

When the galvanometer shows zero deflection then length AD = l_{1}. The balance condition gives

The **percentage error** in R can be minimised by adjusting the balance point near the middle of the bridge (i.e.,
l_{1} is closed to 50 cm) by making a suitable choice of S.

## Potentiometer

It is a versatile instrument consisting of a long piece of uniform wire AC across which a standard cell B is connected. more

## Summary

**Electric current:**The rate of flow of charge normally through any cross-section is known as current. It is a scalar quantity and its SI unit is ampere.**Ohm’s law:**When the physical conditions of the conductor remain the same, the current through a conductor is directly proportional to the potential difference across its ends. I ∝ V**Current density**(`\hat{j}`)**:**At a point, it is a vector having a magnitude equal to current for the unit normal area surrounding that point and normal to the direction of charge flow and direction in which current passes through the point.**Drift velocity**(`\vec{v}`_{d})**:**It is the average uniform velocity acquired by free electrons inside a metal by the application of an electric field, which is responsible for current through it.**Mobility (μ):**It is defined as drift velocity per unit of the electric field.**Relaxation time:**Average time interval between two successive collisions of electrons.**Resistance (R):**It is the property of substance by virtue of which it opposes the flow of current through it.**Conductance (G):**It is the reciprocal of resistance G = 1/R.**Resistivity or specific resistance (ρ):**It is the resistance of the conductor per unit length per unit area of cross-section.**Conductivity or specific conductance (σ):**Reciprocal resistivity is called conductivity (σ = 1/ρ).**Electrochemical cell:**It is a device which converts chemical energy into electrical energy to maintain the flow of charge in a circuit.**Electromotive force or emf of a cell (ε):**It is the difference in chemical potentials of electrodes used.**Internal resistance (r):**It is the opposition offered by the electrolyte of the cell to the flow of current through itself.**Kirchhoff’s law****(i) Junction rule:**At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction.**(ii) Loop rule:**The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.**Colour code for resistor value:**Correct sequence can be remembered by B.B. ROY of Great Britain has Very Good Wife. Capital letters correspond to colours in the correct sequence having powers 0 to 9. Tolerance for the gold strip is 5%, Silver is 10% and no colour is 20%.Kirchhoff’s junction rule is based on the conservation of charge. Kirchhoff’s loop rule is based on the conservation of energy.