A cell of unknown EMF E and internal resistance r, two unknown resistances R1 and R2 (R2>R1) and a perfect ammeter are given. The current in the circuit is measured in five different situations : 

1. without any external resistance in the circuit,
2. with resistance R1 only,
3. with resistance R2 only,
4. with both R1 and R2 used in series combination, and 
5. with both R1 and R2 used in parallel combination.

The current obtained in the five cases are 0.42 A, 0.6 A, 1.05 A, 1.4 A and 4.2 A, but not necessarily in that order.

In terms of r, what is the value of resistance R2 ?

6r

The current will be maximum when the resistance is in the lowest value configuration.

$ \frac{E}{r} = 4.2$ .........(1)

$\frac{E}{r+R_1} = 1.05 A$ .......(2)

$\frac{E}{r+R_2} = 0.6 A$ ...........(3)

$\frac{E}{r+R_1+R_2} = 0.42A$ ........(4)

$\frac{E}{r + \frac{R_1 R_2}{R_1+R_2}} = 1.4A$ ...........(5)

From Equation (1) ,$ E = 4.2r$

$ \text{From equation (2) } , \Rightarrow \frac{4.2r}{r+R_1} = 1.05$

$\Rightarrow r+R_1 = 4r , R_1 = 3r$

 $\text{ From equation 1 and 3 } \frac{4.2r}{r+R_2} = 0.6 $

$\Rightarrow 4.2 r = 0.6 r + 0.6 R_2 , R_2 = \frac{3.6}{0.6} = 6\Omega$