a) State Kirchhoff's rules and explain on what basis they are justified.

(b) Two cells of emfs E₁ and E₂ and internal resistances r₁ and r₂ are connected in parallel.

Derive the expression for the

(i) Emf and

(ii) internal resistance of a single equivalent cell which can replace this combination.

OR

 

(a) "The outward electric flux due to charge +Q is independent of the shape and size of the surface which encloses is." Give two reasons to justify this statement.

(b) Two identical circular loops '1' and '2' of radius R each have linear charge densities -and + C/m respectively. The loops are placed coaxially with their centre distance apart. Find the:

magnitude and direction of the net electric field at the centre of loop '1'.

a)

i) Junction Rule: The algebraic sum of currents meeting at a point in an electrical circuit is always zero.
                                      

This law is in accordance with law of conservation of charge.

ii) Loop Rule: In a closed loop, the algebraic sum of emfs is equal to the algebraic sum of the products of the resistances and the current flowing through them.
                                

This law is based on the conservation of energy.

b) Consider the circuit, 
 

Here, E₁ and E₂ are the emf of two cells,

r₁ and r₂ are the internal resistance of cell,

I₁ and I₂ current due to two cells.

Terminal potential difference across the first cell is given by, 

For the second cell, terminal potential difference will be equal to that across the forst cell.

So, 
 

Let E be the effective emf and r the resultant internal resistance.

Consider, I as the current flowing through the cell.

Therefore, 

Now, comparing the equation with V = E –Ir, we have
 
                                               OR

a) The outward electric flux due to the charge enclosed inside a surface is the number of electric field lines coming out of the surface. Outward flux is independent of the shape and size of the surface because:

i) Number of electric field lines coming out from a closed surface is dependent on charge which does not change with the shape and size of the conductor.

ii) Number of electric lines is independent of the position of the charge inside the closed surface.
b) Magnitude of electric field at any point on the axis of a uniformly charged loop is given by, 
 
 

Electric field at the centre of loop 1 due to charge present on loop 1 = 0

Electric field at the centre of loop1 due to charge present on loop 2 is given by, 

 
, is the required electric field.