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 zero resistance of FG in 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 PQ 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"_1I)/("R"_1 + "R"_2))`

= `("R"_1I)/("R"_1 + "R"_2)`