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.

(i)   Both S₁and S₂are closed.

(ii)  Both S₁ and S₂ are open.

(iii) S₁ is closed but S₂ is open.

Answer 1

(i) The circuit for this case is shown below.

R₄ is shunted by S₂ . Hence the Resistance of this combination will be practically zero

R₁ and R₂ in parallel

∴ \(\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2}\)

∴ \(R_p = \frac{R_1 R_2}{R_1 + R_2}\)

R₃ and R_p in series

∴ R_s = R₃ + R_p

∴ I₃ = \(\frac{V}{R_s}\) = \(\frac{V}{R_3 + R_p}\)

V₁= V-I₃R₃

= V- \(\frac{R_3 V}{R_3 + R_p}\)

= V \((1-\frac{R_3}{R_3 + R_p})\)

= V\((\frac{R_p}{R_3 + R_p})\)

∴ I₁= V₁/R₁ = \(\frac{V}{R_1}(\frac{R_p}{R_3 + R_p})\)

Similarly

I₂ = \(\frac{V}{R_2}(\frac{R_p}{R_3 + R_p})\)

 

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(ii)  Both S₁ and S₂ are open.

Answer

The circuit for this case is shown below.

Here, R₁, R₄ and R₃ are in series. Therefore, the equivalent resistance of the circuit is

R_eq=R₁+R₃+R₄

Hence, the current flowing in the circuit is

I= \(\frac{V}{R_(eq)}\) =  \(\frac{V}{R_1+R_3+R_4}\)

where, V is the potential difference of the battery.

 

 

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(iii) S₁ is closed but S₂ is open.

 

The circuit for this case is shown below.

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

Now, R₃, R' and R₄ are in series.  Thus, the equivalent resistance of the circuit is
Req= \(\frac{R_1×R_2}{R_1+R_2} + R_3 + R_4 \)

= \(\frac{R_1 R_2 +R_3 R_1 + R_3 R_2 + R_4 R_1 +R_4 R_2}{R_1+R_2}\)

Hence, the current flowing in the circuit is
I=V/R_eq= \(\frac{v}{\frac{R_1 R_2 +R_3 R_1 + R_3 R_2 + R_4 R_1 +R_4 R_2}{R_1+R_2}}\)

= \(\frac{V(R_1+R_2)}{R_1 R_2 +R_3 R_1 + R_3 R_2 + R_4 R_1 +R_4 R_2}\)

where, V is the potential difference of the battery.