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प्रश्न
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.
उत्तर
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)`
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