Direct Current Circuit (D.C. Circuit) Class 12 Physics | Notes

Current Electricity:

The branch of physics which deals with the study of motion of charge through a conductor, semiconductor or electrolyte is known as current electricity.

Electric Current (I):

It is defined as the flow of charge per unit time. It is denoted by ‘I’. It is a scalar quantity.

Mathematically,

Electric current (I) = $\frac{charge\text{ }(q)}{time\text{ }(t)}$ …. (1)

Its unit is CS^-1 or Ampere (A)

From quantization of charge, q = ne, then equation (1) becomes

I = $\frac{ne}{t}$

For I = 1A & t =1 sec, we have

    n =  $\frac{1}{e}$

Or, n = $\frac{1}{1.6\times {{10}^{-19}}}$ [ $\because $ e = $1.6\times {{10}^{-19}}$C ]

$\therefore $ n = 6.25×10¹⁸

Thus, current is said to be of one ampere if 6.25×10¹⁸ number of electrons flow through a conductor in 1 second.

Current carriers in different materials:

Conductor: Free electrons

Electrolyte: +ve charge (cation) and -ve charge (anion) 

Gases: +ve charge and free electrons

Semiconductor: Free electrons and holes

Direction of current

Conventional flow of current:

The flow of current from positive terminal to negative terminal of a battery in a circuit is known as conventional flow. 

Electron flow:

The actual flow of current in conductors is due to the flow of electrons which from negative terminal to positive terminal of a battery in a circuit is known as electron flow (directional flow). 

Types of current

i. Direct current (D.C.)

Current is said to be direct current if its magnitude and direction do not change with respect to time. 

ii. Alternating current (A.C.) 

Current is said to be alternating current if the magnitude of current changes with the time and direction reverses periodically. 

Conductor:

Those substances which have a large number of free electrons even at normal temperature are known as conductors. For example: copper, iron, brass etc.

Insulators:

Those substances which do not have any free electron at normal temperature are known as insulators. For example: wood, paper, plastic etc.

Semiconductor:

Those substances whose conductivity lies between conductors and insulators are known as semiconductors. For example Silicon, Germanium etc.

Mechanism of metallic conduction:

Since the conductor contains a large number of free electrons even at normal temperature. The free electrons are moving with very high velocity in random direction due to which the average velocity is zero. So, there is no electric current although the electrons are moving with high velocity without application of electric field. 

Fig: Electrons passing through the conductor in an applied electric field. 

Let us consider a conductor XY of length ‘l ‘ and area of cross section ‘A’. Let ‘n’ be the number of free electrons per unit volume and ‘e’ be the charge on each electron. The conductor XY is connected with the source (battery) in such a way that the end X is connected with the negative terminal and Y is connected with the positive terminal. When the conductor is connected with the source, the free electrons experience columbic repulsive force and they get accelerated. When free electrons get accelerated, a collision takes place between them and loses their velocity. By this process, the free electrons attain a certain average velocity on moving from X to Y. The average velocity gained by electrons in a specific direction on the application of electric field is known as drift velocity.

Now, let us consider ‘n’ be the number of free electrons per unit volume i.e. n = $\frac{N}{V}$

Volume of conductor, V = Al

In volume V, total number of free electrons (N) = nV = n Al

$\therefore $Total charge (q) = Ne

Or, q = nAle

Since electric current is defined as the flow of charge per unit time.

$\therefore $ Electric current (I) = $\frac{q}{t}$  =  $\frac{nAle}{t}$      

Or, I = n A V_d e             ($\because $ Drift velocity,  V_d = $\frac{l}{t}$)

$\therefore $ I = V_d e n A (or simply I = v e n A) 

This is the required relation between current and drift velocity of electrons.

Current density (J):

The current density at any point in the conductor is defined as the current flowing per unit cross-sectional area perpendicular to the direction of the flow. It is denoted by J. It is a vector quantity.

From the expression of drift velocity,

               I = V_d e n A

         Or, $\frac{I}{A}$ = V_d e n

Since, current density, J = $\frac{I}{A}$ 

         $\therefore $  J = V_d e n         This is the required expression for current density.

Ohm’s law: (i.e. V=IR)

Ohm’s law states that,” the current flowing through a conductor is directly proportional to potential difference across its two ends.” Keeping physical conditions (temperature and mechanical strain) constant. 

Let ‘V’ the potential difference between two ends of a conductor and current flowing through it is ‘I’. Then, according to Ohm’s law, we can write

I $\propto $ V

Conversely, V $\propto $  I

V = RI

Or, R = $\frac{V}{I}$

i.e. V = IR

Where R is proportionality constant and known as resistance of conductor. Its unit is VA^-1 or ohm ($\Omega $). 

Experimental verification of Ohm’s Law: 

Fig: Experimental Verification of Ohm’s law

The experimental setup for verification of Ohm’s law is shown in figure above. In the figure, a resistor of resistance ‘R’ is connected with a battery through ammeter (A), variable resistance (R_h) and a key (K) as shown in a figure. A voltmeter (V) is connected across the resistor (i.e. in parallel with resistor). When the circuit is open, no current flows through it and the ammeter and voltmeter do not show any reading. When the circuit is closed, an amount of current I₁ flows through the circuit measured by ammeter and corresponding potential is V₁ measured by voltmeter. Now, the value of current in the circuit is changed with the help of variable resistance (rheostat, R_h). Let I₂ be the current flowing through the circuit and corresponding potential is V₂. Let I₃, I₄, I_{5 …….} be the current flowing through the circuit for different values of R_h and corresponding potential are V₃,V₄,V₅, . . . respectively. Experimentally, it has been found that 

$\frac{{{V}_{1}}}{{{I}_{1}}}$= $\frac{{{V}_{2}}}{{{I}_{2}}}$= $\frac{{{V}_{3}}}{{{I}_{3}}}$= $\frac{{{V}_{4}}}{{{I}_{4}}}$= $\frac{{{V}_{5}}}{{{I}_{5}}}$……….

This shows $\frac{V}{I}$= constant i.e. V$\propto $ I

If a graph is plotted between current and voltage, straight line passing through the origin is obtained. 

Fig: Current Vs voltage graph

The above experiment shows that current flowing through a conductor is directly proportional to potential difference across its two ends, which is Ohm’s law.                                       

Ohmic conductor: 

A conductor which obeys Ohm’s law is called ohmic conductor. For example: metals (copper, silver, iron etc)

When a graph is plotted between current and voltage, a straight line passing through the origin will be obtained. I-V graph of an ohmic conductor is shown in figure.                   

Non-ohmic conductor:

A conductor which does not obey Ohm’s law is known as a non-ohmic conductor. For example: electrolyte, junction diode etc.

If a graph is plotted between current and voltage for a non-ohmic conductor, a straight line passing through origin will not be obtained. The I-V graph of a diode (a non- ohmic conductor) is shown in figure. 

Resistance:

The property of conductor by virtue of which it opposes the flow of current through it is known as resistance. And the device is called a resistor. It is denoted by R and the unit is Ohm ($\Omega $).

Law of resistance:

Let us consider a conductor of length (l ) and area of cross section (A). Let R be the resistance of the conductor. Experimentally it is found that resistance of conductor is 

(i) directly proportional to length of conductor

i. e. R $\propto $ l……………………….(i)

 (ii) inversely proportional to area of cross section of the conductor.

i.e. R $\propto $$\frac{1}{A}$………………………..(ii)

On combining (i) and (ii)

R $\propto $$\frac{l}{A}$

Or, R = $\rho $$\frac{l}{A}$              

where ‘ρ’ is a proportionality constant called resistivity of a conductor.

Or, $\rho $ = $\frac{RA}{l}$

If A = 1 m² and l =1m, then $\rho $ = R

Thus, resistivity of a conductor is defined as the resistance of a conductor of unit area of cross sectional area per unit length. It’s unit is $\Omega $m.

Its dimensional Formula is [ML³T^-3A^-2]

It is the property of a material. So, independent to the dimension of conductor.

Conductivity:

It is defined as the reciprocal of resistivity. It is denoted by ($\sigma $).

Mathematically,

Conductivity, $\sigma $ = $\frac{1}{\rho }$

Or, $\sigma $ =   $\frac{l}{RA}$ 

Its unit is $\Omega $^-1m^-1

Conductance:

It is defined as the reciprocal of resistance. 

It is denoted by ‘C’

Mathematically,

Conductance, C = $\frac{1}{R}$

Its unit is   $\Omega $^-1 or ohm^–1 or (mho) or Sieman (S).

Variation of resistance with temperature:

Let us consider a conductor of resistance R₀ at 0^oC and ${{R}_{\theta }}$  at $\theta $^oC.

Experimentally, it has been found that the increase in resistance (${{R}_{\theta }}$ – R₀) is 

(i) directly proportional to original resistance i.e. Resistance at 0^oC

i.e. (${{R}_{\theta }}$ – R₀) $\propto $ R₀…………………….(i)

(ii) directly proportional to increase in temperature(θ-0)

i.e. (${{R}_{\theta }}$ – R₀) $\propto $ $\theta $…………………(ii)

Combining (i) and (ii) we get, 

(${{R}_{\theta }}$ – R₀) $\propto $ R_{0$\theta $}

Or, (${{R}_{\theta }}$ – R₀) = $\alpha $R_{0$\theta $}…………(a) where ‘α’is proportionality constant and known as temperature coefficient of resistance.

Or, $\alpha $= $\frac{({{R}_{\theta }}-{{R}_{0}})}{{{R}_{0}}\theta }$

Thus, temperature coefficient of resistance of a conductor is defined as the increase in resistance per unit original resistance i.e. resistance at 0^oC per degree rise in temperature.

Its unit is ^oC^-1 or K^-1. 

The value of temperature coefficient is positive for conductor (metals), negative for semiconductors and almost zero for standard resistor (constantan and manganin).

Also, from equation (a)

${{R}_{\theta }}$ = R₀ + $\alpha $R_{0$\theta $}

${{R}_{\theta }}$ = R₀ (1+ $\alpha $$\theta $)   

This is the required expression for variation of resistance with temperature.

Grouping of resistance:

Resistors are to be grouped in circuit to decrease or increase the equivalent resistance of the circuit.

(i) Series Combination:

The combination in which one end of resistance is connected to the one end of another resistance and so on, so that the same current flows through all the resistors then this type of combination is called series combination of resistances.

Fig: Series Combination of resistors and Series equivalent resistance

Let us consider three resistors of resistance R₁, R₂ and R₃ are connected in series with a battery of potential V. Let ‘I’ be the current supplied by the battery. Since resistors are in series combination, so the same current ‘I’ flows through each resistor but potential difference is different depending upon the resistance of the resistor. Let V₁, V₂ and V₃ be the potential across R₁, R₂ and R₃ respectively. Let equivalent resistance of the circuit is ‘R_s’. 

Then

p.d. across R₁, V₁ = IR₁

p.d. across R₂ , V₂ = IR₂

p.d. across R₃ , V₃ = IR₃

Now,

Total potential, V = V₁ + V₂ + V₃

Or, V = IR₁+ IR₂+ IR₃

Or, V = I (R₁+ R₂+ R₃)…………. (i)

If R_s is the equivalent resistance of the circuit, then we can write

V = IR_s……………………. (ii) 

From (i) and (ii), 

IR_s =  I (R₁+ R₂+ R₃)

$\therefore $ R_s = (R₁+ R₂+ R₃)

This is the required expression for equivalent resistance in series combination.

Properties of series combination:

(i) Same amount of current flows through all resistors.

(ii) The Potential is divided and potential difference is different depending upon resistance of resistor.

(iii) The value of equivalent resistance is equal to the sum of individual resistance.

(iv) The value of equivalent resistance is greater than that of even the greatest resistance.

(ii) Parallel Combination:

The combination in which one end of all resistors are connected with positive terminal and another end of all resistors is connected with negative terminal of a battery so that voltage drop across each resistance remains same is known as parallel combination of resistors.

Fig: Parallel Combination of resistors and Parallel equivalent resistance

Let us consider three resistors of resistance R₁, R₂ and R₃ are connected in parallel combination with a battery of potential ‘V’. Since resistors are in parallel combination, so potential difference across each resistor is same (i.e. V).  Let I be the total current supplied by the battery. From point A, the current is divided into I₁, I₂ and I₃ and flows through R₁, R₂ and R₃ respectively. Let R_P be the equivalent resistance of the circuit.

Then, Current through R₁, I₁= $\frac{V}{{{R}_{1}}}$

Current through R₂, I₂= $\frac{V}{{{R}_{2}}}$

Current through R₃, I₃= $\frac{V}{{{R}_{3}}}$

Now,

Total current, I = I₁+I₂+I₃

Or, I = $\frac{V}{{{R}_{1}}}$+ $\frac{V}{{{R}_{2}}}$+$\frac{V}{{{R}_{3}}}$

Or, I = V$\left( \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}} \right)$

The equivalent resistance of the circuit is R_p. Then, we can write,

I = $\frac{V}{{{R}_{p}}}$…………………… (ii)

From (i) and (ii)

$\frac{V}{{{R}_{p}}}$= V$\left( \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}} \right)$

$\therefore $     $\frac{1}{{{R}_{p}}}$= $\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}$

Properties of parallel combination:

(i) The potential difference is the same across each resistor.

(ii) The current is divided and the value of current different depending upon resistance of resistor.

(iii) The reciprocal of equivalent resistance is equal to the sum of the reciprocal of individual resistance.

(iv) The value equivalent resistance is less than that of even the smallest resistance. 

Note: If there are only two resistors in parallel then to find equivalent resistance, it is better to use the formula 

R_eq  or R_p =  $\frac{{{R}_{{1}}}\times {{R}_{{2}}}}{{{R}_{\mathbf{1}}}+{{R}_{{2}}}}$ instead of $\frac{1}{{{R}_{p}}}$= $\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}$

Galvanometer:

It is an electrical instrument which is used to detect the flow of current in the circuit. It also gives the direction of current and measures a small amount of current and potential difference. The resistance of the galvanometer is low. It’s symbol is  

Ammeter:

An ammeter is an electrical instrument used to measure the current passing through it. The resistance of the ammeter is very low. It is always connected in the series in the circuit. It’s symbol is 

Voltmeter:

Voltmeter is an electrical instrument used to measure the potential difference across it. The resistance of the voltmeter is very high.  It is always connected in the parallel with the load resistance in the circuit. It’s symbol is

Shunt:

 It is a low value resistance which is connected in parallel with the galvanometer to convert the galvanometer into an ammeter.

Conversion of galvanometer into an ammeter:

Galvanometer is an electrical instrument which is used to detect the flow of current in the circuit. The resistance of the galvanometer is low.

Ammeter is an instrument used to measure the current passing through it. The resistance of the ammeter is very low. It is always connected in the series in the circuit.

Galvanometer is converted into an ammeter by using suitable shunt (a low value resistance) in parallel with it.  

Fig: Conversion of galvanometer into an ammeter

Let us consider a galvanometer of resistance ‘G’ and shows full scale deflection when an amount of current I_g flows through it. We have to convert the galvanometer into an ammeter which can measure current upto ‘I’ ampere. For this, let us connect a shunt ‘S’ (a low value resistance) in parallel with the galvanometer.

The value of shunt is chosen in such a way that only a desirable amount of current I_g flows through the galvanometer and remaining current (I–I_g) flows through the shunt. 

Since, shunt and galvanometer are in parallel combination, 

$\therefore $ p.d. across shunt (S) = p.d. across galvanometer (G) 

Or, (I–I_g).S = I_gG

Or, S = $\frac{I{}_{g}}{I-{{I}_{g}}}$.G

This is the required value of shunt to convert a galvanometer into an ammeter to measure current upto (I) ampere.

Now the resistance of ammeter is the equivalent resistance of parallel combination of shunt and galvanometer

So,

R_A= $\frac{SG}{S+G}$

So, the resistance of ammeter (R_A) is very small (even smaller than G & S). This is the necessary criterion to be an ammeter.

Conversion of galvanometer into voltmeter:

Galvanometer is an electrical instrument which is used to detect the flow of current in the circuit. The resistance of the galvanometer is low.

Voltmeter is an instrument used to measure the potential difference across it. The resistance of the voltmeter is very high. It is always connected in the parallel with the load resistance in the circuit. 

Fig: Conversion of galvanometer into voltmeter

Let us consider a galvanometer of resistance ‘G’ and show full scale deflection when an amount of current I_g flows through it. We have to convert the galvanometer into voltmeter measure to measure potential difference upto V volt. For this, let us connect a high value resistance ‘R’ in series with a galvanometer. Since galvanometer and high value resistance(R) are in series combination, so total resistance becomes R+G

Now,

Potential (V) = I_g(R+G)

Or, $\frac{V}{{{I}_{g}}}$= R+G

Or, R = $\frac{V}{{{I}_{g}}}$– G

Where R is required value of high value resistance to convert a galvanometer into voltmeter to measure potential difference upto V volt.

The resistance of voltmeter R_v = G + R. 

As the value of R is high, so R_v is also high. This is the necessary criterion to be a voltmeter.

Also Read: Electrical Circuit Class 12 Physics | Notes