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Question
Resistances R1, 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.
- Both S1 and S2 are closed.
- Both S1 and S2 are open.
- S1 is closed but S2 is open.
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Solution
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)`
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