**Class – 12, Physics, Module – 02, **

**Current Electricity**

**Chapter – 01, Electric Current**

The word ” **electric** ” is derived from the Greek word ” **elecktron** ” and the word ” **Current** ” is derived from the Latin word ” **currere** ” which means ” run”. Hence, we can conclude that the ” electric current ” means the running of an electron.

The phenomenon related to moving of electrons in a particular direction is termed as the **current electricity**.

**Electric Current:** The rate of flow of electric charges is called the electric current. It is represented by i. It is a scalar quantity though it has direction. As it does not follow the laws of vector, it is a scalar quantity.

**Unit of Electric Current: **

SI Unit: Ampere ( represented by A )

CGS unit: Stat ampere ( represented by stat- A )

**Expression of the electric current:**

Let us assume that the electric charges pass through the electric conductor in ‘ t ‘ sec is q.

The amount of electric charges flowing in t sec = q

The amount of electric charges flowing in 1 sec = q / t

∴ **Electric current ( i ) = q / t **

In differential form,

**i = dq / dt **

if number of electron ( electronic charge ‘ e ‘ ) is n then

q = ne

∴ **i = ne / t **

**Current carrier :** The particles that carry the electric current thrtough which the electric current will flow, is called as current carriers.

Here is the table through which we may understand the current carrier in different substances.

Serial No | Types of conductor | examples | current carriers |
---|---|---|---|

01 | Solid conductor | all metals | free electrons |

02 | Liquid Conductor( electrolyte ) | the solution of CuSO_{4} etc |
ions ( positive and negative ) |

03 | Gas ( low pressure and high potential difference ) | gasses | free electrons and ions |

**Types of electric current :**

**Electric Current**

** ↓**

**↓————————————-↓———————————↓————————————-↓**

**Steady current Varying current Direct current Altering current **

magnitude = magnitude ≠ direction = direction

constant constant constant changes ( w.r.t time ) periodically

**The direction of electric current:** The direction of electric current is

i ) the opposite of the direction of flowing of the electron.

ii ) the direction of flowing of a positive charge.

iii ) form a higher potential region to a lower potential region.

iv ) form the positive terminal to the negative terminal of a battery.

**Electric Potential:** We have already learned about the electric potential in the module – 01 ( Electrostatics ). It is the work done required to bring a unit positive charge from infinity to a particular point in an electric field.

**Potential Difference:** It is defined as the amount of work done required to move a unit positive charge from one point to another point in the electric field when the electric circuit is closed. In other words, it is the difference in the electric potentials of two terminals of a closed electric circuit.

SI unit → Volt

CGS unit → stat – volt

**Ohm’s Law:**

**Statement:** It states that ” the electric current between the two ends of a conductor or an electric circuit is proportional to the potential difference between the two ends provided the other physical condition like temperature etc should be kept constant”.

**Mathematical Form:**

Following the figure,

V = potential difference between two terminals

I = Electric current flowing through the electric circuit.

According to Ohm’s law,

I ∝ V

I = V / R

Where 1 / R is the proportionality constant and R is termed as the electrical resistance.

**Graphical Representation:**

In the above graphical representation, we can see that the resistance is the slope of the I – V graph and remains constant whatever be the current and potential difference.

The properties of the graph are given below :

i ) The graph is a linear or straight line passing through the origin.

ii ) The slope of the graph is constant.

iii ) Resistance is independent of the electric current as well as the potential difference.

**Types of the electric conductor on the basis of Ohm’s law:**

There are two types of electric conductors.

**1 ) Ohmic conductor: **The conductor that follows the ohm’s law and has a linear I – V graph is termed as the ohmic conductor.

Examples: All metals and a resistor etc.

**2 ) Non – ohmic conductor: ** The conductor that does not follow the ohm’s law and has a non – linear I – V graph is termed as the non – ohmic conductor.

Example: semiconductor, thermistor, transistor, and diode etc.

**Resistance:** The resistance is the property of a substance by virtue of which it resists to flow the electric current.

According to Ohm’s law, the resistance is defined as the ratio of the voltage to the electric current.

**∴ R = V / I **

Units:

Its SI unit is Ohm represented by Ω and its CGS unit is stat – ohm i.e stat – Ω.

**Factors on which the resistance depends:**

**1 ) Length of the conductor:**

If the cross-sectional area of an electric conductor and other physical conditions like temperature etc are kept constant, the resistance is directly proportional to its length.

i.e R ∝ l ———————————————– ( 1 )

**2 ) The cross-sectional area of the conductor :**

If the length of an electric conductor and other physical conditions like temperature etc are kept constant, the resistance is inversely proportional to its cross-sectional area.

i.e R ∝ 1 / A ———————————————– ( 2 )

**3 ) Temperature:**

Resistance depends on the temperature of the substance.

For the conductor, it increases linearly with a small increase in temperature.

**Specific Resistance ( ****ρ ) :**** **

We have seen that the resistance ‘ R ‘ is directly proportional to the length ( l ) of the resistance provided the cross-sectional area ( A ) of the conductor kept constant while it is inversely proportional to the cross-sectional area of the conductor provided its length should be kept constant. That is,

R ∝ l ——————– ( 1 )

R ∝ 1 / A —————————— ( 2 )

On combining the above two equations, we get

R ∝ l / A

R = ρl / A

where ρ is the proportionality constant, and it is known as specific resistance.

**Definition:** The resistance of an electric conductor of unit length and unit cross-sectional area is called as the specific resistance.

**ρ = RA / l**

It does not depend on the cross-sectional area and the length of the conductor. It depends on the nature of the material of the conductor and its temperature.

**The expression for the Resistance:**

1 ) R = V / I

2 ) R = ρl / A

3 ) R = ρl / πr^{2}

4 ) R = 4ρl / πd^{2}

5 ) If ρ = constant and l = constant then R ∝ r^{2 }and R ∝ d^{2}

**The expression for the Specific Resistance:**

1 ) ρ = RA / l

2 ) ρ = πr^{2}R / l

3 ) ρ = πd^{2}R / 4l

**Temperature Coefficient:** We know that the resistance of an electronic substance depends on its temperature. If there is a change in temperature by unity, it affects the resistance of the electronic substance. This degree of measure by which the resistance changes with a unit change in temperature is called the temperature coefficient.

R = R_{0} ( 1 + ∝△t )

where R = resistance after changing the temperature

R_{0} = resistance before changing the temperature

∝ = temperature coefficient of the electronic substance

△t = change in temperature

**Internal Resistance :**

The resistance offered by the electrolytes of the cell is called the internal resistance.

It depends on the following factors :

- distance between two plates
- nature of electrolytes
- nature of electrodes
- area of plates
- the concentration of the electrolytes

**Distance between two plates:** As the distance between the two plates increases the internal resistance also increases and vice versa.

**Nature of electrolyte:** The value of internal resistance depends on the nature of the electrolyte. As we know that the internal resistance is nothing but the resistance offered by the electrolyte if we change the electrolyte of the cell its internal resistance also changes.

**Nature of electrodes:** The value of internal resistance also depends on the nature of the electrodes. Its value is different for the different electrodes used in the cell.

**Area of plates:** The value of internal resistance increases with the decrease in the area of plates and vice versa that is the internal resistance is inversely proportional to the area of plates.

**The concentration of the electrolytes:** The internal resistance depends on the concentration of the electrolyte used in the cell. It is directly proportional to the concentration of the electrolytes.

**The expression for the internal resistance of the cell:**

Following the circuit,

r = internal resistance

R = external resistance

E = E.M.F of the cell

V = voltage drop across R = iR

I = current flowing throgh the circuit = emf of the battery / total resistance of the ciruit

I = E / ( R + r )

⇒ iR + ir = E

⇒ V + Ir = E

**⇒ Ir = E – V ——————————————— ( A )**

⇒ **r = ( E – V ) / I ———————————– ( 1 ) **

∵I = V / R, putting this value of i in the above eqaution, we get

r = ( E – V ) / ( V / R )

⇒ r = ( E / V – V / V ) R

**⇒ r = ( E / V – 1 ) R ————————————- ( 2 ) **

**Special Cases :**

1 ) If the circuit is opened that is there is no current flowing through the circuit, i = 0

then from equation ( A ), we get

ir = ( E – V )

⇒ 0 = E – V

**E = V **

Hence, the potential difference across the external resistance R is equal to the emf of the cell.

2 )** Discharging: **

ir = E – V

3 )** Charging:**

ir = E + V

**Combination of the resistors:**

1 ) Parallel combination

2 ) Series Combination

**1 ) Parallel Combination of the resistors:**

If the ends of the resistors are connected to each other, then the combination is termed as the parallel combination. In this type of connection, the current is divided among the different arms but the potential difference across each arm remains constant.

Let the equivalent resistance of the parallel combination of the n resistors is R_{p}

I = I_{1} + I_{2} + I_{3} + ………………….. + I_{n}

V / R_{p} = V / R_{1} + V / R_{2} + V / R_{3} + ……………………… + V / R_{p}

where V is the potential difference across each arm.

1 / R_{p} = 1 / R_{1} + 1 / R_{2} + 1 / R_{3} + ……………………… + 1 / R_{p}

**Different case:**

1 ) For two differents resistors,

1 / R_{p} = 1 / R_{1} + 1 / R_{2}

R_{p} = R_{1}R_{2} / ( R_{1} + R_{2} )

2 ) For two equal resistors, that is R_{1} = R_{2} = R

R_{p} = R*R / ( R + R )

R_{p} = R / 2

Hence, we can see that the equivalent resitance of the parallel combination of the resistors is always less than the individual one.

**2 ) Series Combination of resistors:**

Let the equivalent resistance of the parallel combination of the 3 resistors is R_{s}

V = V_{1} + V_{2} + V_{3}

IR_{s} = IR_{1} + IR_{2} +IR_{3}

where I is the electric current flowing through each arm.

**R _{s} = R_{1} + R_{2} +R_{3} **

**Different case:**

1 ) For two differents resistors,

R_{s} = R_{1} + R_{2}

2 ) For two equal resistors, that is R_{1} = R_{2} = R

R_{s} = R + R

R_{s} = 2R

Hence, we can observe that the equivalent resistance of series combination of two equal resistors is twice of individual.

**Drift Velocity:**

When an external electric field is applied to a conductor, all free electrons that were moving randomly, now arranged in a particular direction and move with a specific velocity. The average velocity with which the electorns move under the influence of exterenal applied electric field, is termed as the **drift velocity**.

**Expression for the Drift velocity ( v _{d} ):**

Let v_{1} , v_{2} , v_{3} …………… are the final velocity of the electrons movign under the influence of applied electirc field E. τ is the relaxation time between two succesive colission of electrons when electric current flows ‘n’ ihe electron density.

*v _{d}* = ( v

_{1}+ v

_{2}+ v

_{3}+ ………………… ) / n

*v _{d}* = { ( u

_{1}+ aτ

_{1}) + ( u

_{1}+ aτ

_{2}) + ………………………. } / n

*v _{d }*= { ( u

_{1}+ u

_{2}+ …………….. ) / n } + { a ( τ

_{1}+ τ

_{2}+ ……………….. ) / n } —————- ( 1 )

Before applying electirc field, the average velocity is zero as all electrons move randomly and effect produced by these electrons are canclled by each other.

therfore,

( u_{1} + u_{2} + …………….. ) / n = 0

and we Know that force experineced by each electron is given by

F = e E

ma = eE

a = eE / m

and ( τ_{1} + τ_{2} + ……………….. ) / n = τ = average relaxation time

where, a is the acceleration produced by each electron, m is the mass of the electorn, E is the applied electric field.

Now, putting the value of ‘a ‘ in equaion ( 1 ), we get

*v _{d}* = 0 + eEτ / m

*v _{d}* = ( e E / m ) τ

**Relation between the electric current ( I ) and Drift velocity ( v _{d} ):**

let

*l* = length of the conductor

A = cross sectional area of the conductor

V = potential differnce across the two ends of the conductor.

n = number of the free electrons per unit volume

N = total number of free electrons = n * volume of the conductor = nA*l*

All free electrons move randomly before applying any electric field. When the electric field is applied to the conductor , all the electrons arrnaged and move in a specific direction and constitutues electric current flowing in opposite direction of the electorn. This electric current is nothing but the drift velocity.

Let I = electric current flowing through the conductor

I = total charge / t

I = Ne / t

I = nA*le / t*

I = neA ( *l* / t )

**I = neA v_{d}**

where *v _{d} = l / t *

**Relation Between drift velocity and potential difference :**

*v _{d}* = ( e E / m ) τ ………………………………. ( 1 )

where

e → electronic charge

E → strength of applied electric field

m → mass of each electron

τ → average relaxation time.

*l* → length of the conductor

we know that potential differnec across the ends of the conducor is given by

V = E*l*

E = V*l*

Putting the value of E in equation ( 1 ), we get

*v _{d}* = ( e V / ml ) τ

This is the required relation.

**Carbon Resistors**

In an electric circuit, some specific value of resistance is required. To fullfil this requirement, a component, having some color code is used. This component is called Carbon Resistors.

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