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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.



a. Both S1 and S2 are closed.
b. Both S1 and S2 are open.
c. S1 is closed but S2 is open.

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Solution

\fraca. The circuit for this case is shown below.

Here, R1 and R2 are in parallel. Their effective resistance is
R′=R1×R2R1+R2R'=\frac{R_1×R_2}{R_1+R_2}R=R1+R2R1×R2
Also, R4 is in parallel with a conductor of 0 resistance. Their equivalent resistance is therefore 0. The given circuit thus becomes as shown below.

Now, R' and R3 are in series. Thus, the equivalent resistance of the circuit is
Req=R1×R2R1+R2+R3R_eq = \frac{R_1×R_2}{R_1+R_2}+R_3Req=R1+R2R1×R2+R3

Req=R1×R2+R1R3+R2R3R1+R2R_eq = \frac{R_1×R_2 + R_1R_3+R_2R_3}{R_1+R_2}Req=R1+R2R1×R2+R1R3+R2R3


Hence, the current flowing in the circuit is
I=V×R1+R2R1×R2+R1R3+R2R3I = V \times\frac{R_1+R_2}{R_1×R_2 + R_1R_3+R_2R_3}I=V×R1×R2+R1R3+R2R3R1+R2
where, V is the potential difference of the battery.
b. The circuit for this case is shown below.

Here, R1, R4 and R3 are in series. Therefore, the equivalent resistance of the circuit is
Req=R1+R2+R3Req = R_1+R_2+R_3Req=R1+R2+R3

Hence, the current flowing in the circuit is
I=VReq=VR1+R2+R3I = \frac{V}{R_eq} =\frac{V}{R_1+R_2+R_3}I=ReqV=R1+R2+R3V
where, V is the potential difference of the battery.
c. The circuit for this case is shown below.

Here, R1 and R2 are in parallel. Their effective resistance is
R′=R1×R2R1+R2R'=\frac{R_1×R_2}{R_1+R_2}R=R1+R2R1×R2

Now, R3, R' and R4 are in series. Thus, the equivalent resistance of the circuit is

Req=R1×R2R1+R2+R3+R4

Req=R1R2+R1R3+R3R2+R1R4+R4R2R1+R2

Hence, the current flowing in the circuit is
I=VReq=V×R1+R2R1R2+R1R3+R3R2+R1R4+R4R2
where, V is the potential difference of the battery.


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