Question

Resistances R_1, R₂, R₃ and R₄ are connected as shown in the figure. S₁ and S₂ are two keys. Discuss the current flowing in the circuit in the following cases.

1. Both S₁ and S₂ are closed. 
2. Both S₁ and S₂ are open.
3. S₁ is closed but S₂ is open. 

Answer 1

a.

Due to the zero resistance of FG in a parallel series combination of R₄, the resultant resistance of this combination will also be almost zero, and the entire electric current will flow through the path.

Resultant resistance of parallel series combination of R₁, R₂

∴ `I_3 = V/(R_3 + R_S)`

`V_1 = V - I_3 R_3`

= `V - (R_3 V)/(R_3 + R_p)`

= `V (1 - R_3/(R_3 + R_p))`

= `V (R_p/(R_3 + R_p))`

∴ `I_1 = V_1/R_1 = V/R_1(R_p/(R_3 + R_p))`

similarly, 

`I_2 = V/R_2(R_p/(R_3 + R_p))`

b.

Equivalent resistance of series combination of R₁, R₃, R₄

`R_S = R_1 + R_3 + R_4`

Current in the circuit,

`I = V/(R_1 + R_3 + R_4)`

c.

`R_P = (R_1 R_2)/(R_1 + R_2)`

`R_S = R_3 + R_4 + (R_1 R_2)/(R_1 + R_2)`

`I = V/R_S = I_3 = I_4`

Also, `I = I_1 + I_2` and `R_1 I_1 = R_2 I_2`

∴ `I = I_1 + (I_1 R_1)/R_2`

∴ `I_1(1 + R_1/R_2)`

= `(I_1(R_1 + R_2))/R_2`

`I_1 = (R_2 I)/(R_1 + R_2)`

and `I_2 = (I_1 R_1)/R_2`

= `R_1/R_2((R_1 I)/(R_1 + R_2))`

= `(R_1 I)/(R_1 + R_2)`