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.

Answer 1

\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​+R2​R1​×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_3Re​q=R1​+R2​R1​×R2​​+R3​

Req=R1×R2+R1R3+R2R3R1+R2R_eq = \frac{R_1×R_2 + R_1R_3+R_2R_3}{R_1+R_2}Re​q=R1​+R2​R1​×R2​+R1​R3​+R2​R3​​

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​+R1​R3​+R2​R3​R1​+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=Re​qV​=R1​+R2​+R3​V​
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​+R2​R1​×R2​​

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

$R_{e}q=\frac{R_1\times R_2}{R_1+R_2}+R3+R4$

$R_{e}q=\frac{R_1{R}_2+R_1{R}_3+R_3{R}_2+R_1{R}_4+R_4{R}_2}{R_1+R_2}$

Hence, the current flowing in the circuit is
$I=\frac{V}{R_{e}q}=V\times \frac{R_1+R_2}{R_1{R}_2+R_1{R}_3+R_3{R}_2+R_1{R}_4+R_4{R}_2}$
where, V is the potential difference of the battery.