Current Electricity Class 12 notes Physics Chapter 3

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/m².

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.

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_dt)

So, moving charge in t second Q = (nAv_dt).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

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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 m²V^-1s^-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).

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 = i²Rt

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 = i²R = `\frac{V^2}{R}`

Resistor Colour Codes

Current Electricity Class 12 Notes Chapter-3

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 = R₁ + R₂ + R₃

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 R₁, R₂, R₃ and R₄ 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

I₂ = I₄ and I₁ = I₃

Applying loop rule to loop ABDA

I₂R₂ + 0 – I₁R₁ = 0

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

Applying loop rule to loop BCDB

I₄R₄ – I₃R₃ + 0 = 0

I₂R₄ – I₁R₃ = 0 (Using I₄ = I₂ and I₃ = I₁)

`\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 K₁. 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 l₁. Resistance of the portion AD is R_AD = R_ml₁ and resistance of the portion DC is R_DC = R_m(100 – l₁), 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 = l₁. 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., l₁ 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.