DC CIRCUITS class 11 Notes

·     Electric circuits
The rate of flow of charge is known as electric circuits. If Q be the charge that flows through any cross-section area in time ‘t’. Then,
           electric current (I)=Q/t

·    
Charge carrier
Solid-free electron
liquid- ions
gas – ions and free electrons
semiconductor – holes and free electrons

·     Types of current
1) Direct current
           The type of current whose magnitude and direction remain constants at all time is called direct current.

2) Alternating current
             Types of current whose magnitude and direction changes with respect to time is called alternating current.

·     

Ohm’s law
       Its states that ‘the current flowing through conductors is directory proportional to the potential difference across its end, the external conditions remain constant.’
  I.e.

          V𝞪I

         V= RI

Where R is the proportionality constant and is called electric resistance.

·    
Drift velocity
    Drift velocity is the average velocity acquired by the free electrons in a conductor subjected to an electric field.
   I=V_denA
            is the relation between drift velocity and electric current
       where,  V_d  - drift velocity
                    e   -  charge of an  electron
                   n   - no of electron per unit volume
                   A   - cross-section area

·    
Current density
      The current density of a conductor is defined as the amount of current that flows per unit area of the conductor. Its units is ampere per square meter.  
       
    

Electrical Resistance
The property of a 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 (Ω). It is measured by the ratio of potential difference across its ends to the current flowing through it. 
R=V/I

 

One Ohm:
The resistance of a conductor is said to be of one ohm if one ampere current flows through it under a potential difference of one volt.

one ohm = 1 volt/1 Amphere

 

Laws of resistance
Consider a conductor of length (l) and area of cross-section (A). Let R be the resistance of the conductor. Experimentally it was found that the resistance of the conductor is:

i. directly proportional to the length of the conductor.
    i.e.   R ∝  L……(1_)

ii. inversely proportional to the area of cross-section of the conductor.
   i.e. ∝1/A……(_2_)

On combining (1) & (2), we get,
R ∝L/A
R =C *L/A

Where, р is the proportionality constant, called the resistivity or the specific resistance of the material of the conductor.

р = RA/L

If L  = 1m and A= 1m², then , р =R

Thus, the resistivity of the material of a conductor is defined as the resistance of the conductor of unit cross-sectional area per unit length. Its unit is ohm meter (Ω m) in SI units.

 

Ohmic and Non-Ohmic Conductors

Ohmic Conductors
Conductors which obey Ohm’s Law, have a constant resistance when the voltage is varied across them or the current through them is increased. These conductors are called ohmic conductors. A graph of the current vs the voltage across these conductors will be a straight line. Some examples of ohmic conductors are tungsten resistors and nichrome wires.

 

Non-Ohmic Conductors
 Conductors who don't obey Ohm’s Law are called nonohmic conductors. The current-voltage graph of a non-ohmic conductor is not a straight line but is a curve. Electrolytes, vacuum tubes, junction diodes, transistors, etc are some examples of non-ohmic conductors.

 

Resistances in Series and Parallel

1)  Series combination of Resistances
Resistances are said to be in series when they are joined end to end, so that same current flows through each of them.

From Ohms law,
V₁ = IR₁
V2= IR₂
V₃ = IR₃

 

If V be the potential difference across the whole combination, then,
V = V₁ + V₂ + V₃
V = IR₁+ IR₂+ IR₃
V= I(R 1+ R₂ +R₃)
V/R= R₁ + R₂ + R₃ ……(1_)

Let, R_S be the effective resistance of the series combination, then,

V =IR_s……. .(_2_)

On comparing equations (1) and (2), we get,

R_s=R₁+R₂+R₃

Thus, when the resistors are connected in series, the effective resistance of the combination is equal to the sum of resistances of individual resistors.

 

Properties of the series combination

• The same amount of current flows through all resistors
• The potential is divided and a potential difference is different depending upon the resistance of the resistor
• The value of equivalent resistance is equal to the sum of individual resistance.
• The value of equivalent resistance is greater than that of even the greatest resistance.

 

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

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 across each resistor is same (i.e. V).

Let I be the total current supplied by the battery. The current I is divided into I1, I2, and I3 and flows through R1, R2, and R3 respectively. Let Rp be the equivalent resistance of the circuit. Then,

I = I₁ + I₂ + I₃ ……(_1)
From Ohm’s law, we have,
I₁ =V/R₁
I₂ =V/R₂
I₃= V/R₃

Substituting these in the above equation we get,

I =V/R₁+V/R₂+VR₃
I/V=1/R₁+1/R₂+1/R₃ …….. (2)

Let R_P be the equivalent resistance in this combination, then,

V=IR_P
I/V= 1/R_P……..... (2)

From equations (1) and (2), we have,

1/R_P= 1/R₁+ 1/R₂+ 1/R₃

Thus, when resistors are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of reciprocal of the resistance of individual resistors.

 

Properties of parallel combination:
• The potential difference is the same across each resistor.
• The circuit is divided and the value of current is different depending upon the resistance of the resistor.
• The reciprocal of equivalent resistance is equal to the sum of the reciprocal of individual resistance.
• The value of equivalent resistance is less than that of even the smallest resistance.

 

Voltage Divider Circuit
 An electric circuit that contains a series combination of resistors is known as a voltage divider circuit. Let R₁ and R₂ resistors, connected in series. I remain constant in each resistor but the potential difference V provided by the cell is divided into each resistor. Let V₁ and V₂ be the potential difference across resistors with resistance R₁ and R₂ respectively.

Here the equivalent resistance is given by: R=R₁+R₂
Total current = I

Thus, total voltage (V)=I(R₁+R₂)

                               I=V/(R₁+R₂)………(1)

Now, the Potential difference cross R₁ is given by :

                                              V₁=IR₁   …… (2)

Substituting (1) in equation (2), we get,

V₁={V/(R₁+R₂)}*R₁

V₁={R₁/(R₁+R₂)}*V

Similarly,

V₂={R₂/(R₁+R₂)}*V

If the internal resistance is neglected, V is the sum of V₁ and V₂.

 

Current Divider Circuit
An electrical circuit that contains a parallel combination of resistors is known as a current divider circuit. Let R1 and R2 be the resistances of two resistors connected in parallel. In the circuit connection, the potential difference V in each resistor remains constant, but the current I is divided into each part of a parallel circuit. Let I1 and I2 be the current passing through resistors with resistances R1 and R2 respectively.

Here the equivalent resistance (R) is calculated as;

1/R  =  1/R₁  +  1/R₂

1/R   =(R₁+R₂)/(R₁R₂)

R=(R₁R₂)/ (R₁+R₂)

Now, the current passing through R₁,

I₁ = V / R₁

I₁ = IR / R₁

I₁= (I/R₁) * {(R₁R₂)/ (R₁+R₂)}

I₁=  {R₂/ (R₁+R₂)}*I

Similarly,

I₂=  {R₁/ (R₁+R₂)}*I

 

 

The total current (I) is the sum of I₁ and I_2.

 

·     Potential difference and Energy
The potential difference between two points X and Y in an electric field. E is given by the work done in moving a unit positive charge between these points in an external circuit.

      Potential difference = work done / charge carried
      V   = W   / q₀
Dimensional formula for Volt (V)= [ML² T^-3A^-1 ]

·     Expression for heat developed in a resistor.
 Let us consider an electric device, carrying a current I from end X to Y

for time t. Let Q be the charge carried from end X to Y.
    I = Q / t
    Q = I *t

If W be the amount of energy liberated (heat, light, sound, mechanical

work etc) in time t, P.d (V) between the ends X and Y is given by,

          V= W / Q
         W = Q*V
         W = I*t*V
         W = I²Rt

Here, W may be the work done to transfer the charge from X to Y. In

passive resistor, all electrical energy is converted into heat energy H.

H = I²Rt =V²t / t

·    Electrical power
 The electrical power of an electric device is defined as energy electrical liberated for second in it.

       Let, V be the potential difference across two ends A and B of a conductor and I be the current flowing through it for time t. Then electrical energy liberated is given by,

                 W = I²Rt =IVt
            Electrical power supplied is,
                      P = W / t = IVt / t = IV
 In a passive resistor (that converts all electrical energy into heat) of resistance R all the power supplied appears as heat.
        P = IV = I²R =V² / R

SI unit of power is watt (W). Kilowatt (KW) is larger unit of power.

One watt is power consumed in a device through which one ampere current flows through when a potential difference of one volt is maintained across it.

 

 

Electromotive force
To maintain continuous electric current in a closed circuit, it is necessary to maintain a potential difference between any two points of the conductor in the circuit. The devices, such as batteries for electric generators are able to do so. The property of the devices which maintain potential difference between two points of a conductor connected across it, is called electromotive force (emf). The devices are called sources of emf and are able to do work on the charge carriers, so a current flows through the circuit.

The emf of a source is defined as the work done in moving a unit

positive charge from low potential to high potential end throughout the circuit.

Let, W be the work done in moving a charge Q then, emf E of the

source is

     E = W / Q

SI unit of is volt (V). The potential difference between two poles of a cell in an open circuit gives emf of a cell.

 

·     Internal resistance of a cell
The electrolyte between electrodes of the cell offers certain resistance. when a current flows through it. This resistance is called the internal resistance of the cell. It depends on the following factors.

i. It depends on the nature of the electrolyte
ii. It is directly proportional to the concentration of electrolytes
iii. It is directly proportional to the distance between electrodes
iv. It is inversely proportional to the area of electrodes inside the electrolyte
v. It depends on the temperature of the electrolyte.

 

·    Terminal potential difference
The potential difference between two poles of a cell in a closed circuit is called a terminal potential difference of the cell. It is equal to the total potential difference across all the external resistance of the circuit. Relation between Internal resistance and Electromotive force of a cell. The internal resistance is connected in series with the resistance of the external circuit. In the above diagram, a cell of emf E and internal resistance (r) are connected in series with a key while a voltmeter is connected in parallel with the cell. So, the total resistance of the circuit is given by;

   R_TOTAL = (R + r)

Normally, the value of internal resistance is very low. If the current in the circuit

is I, then we get,

   Total current =  Total /  Total resistance
        I  =   E  /   (  R +  r )

       E = I ( R + r )
        E = IR + Ir
      E   =  V   + Ir
      E – V = Ir………..(1)

Here, V and Ir are respectively the potential difference between the two ends of the resistance of the external circuit and potential drop for r inside the cell. V is called the terminal potential difference or voltage.

That means V is the potential difference between the terminals of the cell during the flow of current in the circuit.

From equation (1), it is clear that V < E.

The reason for the smaller value of terminal voltage than the electromotive force is due to the fact that some energy is needed for the current to flow through the internal resistance of the cell. As a result, a potential drop takes place inside the cell. So it is seen that the potential difference between the two terminals of a cell decreases by an amount of ‘Ir’, which means ‘Ir’ amount of voltage does not contribute in the circuit, rather becomes lost. For this reason, ‘Ir’ is called ‘lost volt’.

     r= (E – V) / I =( E – V ) / (V/R)

    r = {( E – V ) / V } * R

This relation is known as circuit formula and gives the relation between internal

resistance (r), emf (E), voltage (V) & Resistance (R)