Question 187: Current Electricity

Question

The reading of the ammeter $(A)$ in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is $\_\_\_\_$ A.

Accepted Answer

In a DC circuit, a capacitor reaches a steady state after being connected for a long time. In this state the capacitor is fully charged and it acts as an infinite resistance (an open circuit). The means no current flows through the branch containing the $10 \mu \mathrm{~F}$ capacitor and the $8 \Omega$ resistor connected in series with it.
The two $8 \Omega$ resistors $\mathrm{R}_4$ and $\mathrm{R}_5$ are in parallel.
$$ \mathrm{R}_{4,5}=\frac{8 	imes 8}{8+8}=\frac{64}{16}=4 \Omega $$
This $\mathrm{R}_{4,5}=4 \Omega$ equivalent is in series with the $\mathrm{R}_3=4 \Omega$ resistor.
$$ \mathrm{R}_{3,4,5}=4 \Omega+4 \Omega=8 \Omega $$
This $\mathrm{R}_{3,4,5}=8 \Omega$ combination is now in parallel with the $\mathrm{R}_2=8 \Omega$ vertical resistor in the middle.
$$ \mathrm{R}_{2,3,4,5}=\frac{8 	imes 8}{8+8}=4 \Omega $$
Finally, this $\mathrm{R}_{2,3,4,5}=4 \Omega$ is in series with the $\mathrm{R}_1=1 \Omega$.
$$ \mathrm{R}_{	ext {total }}=1 \Omega+4 \Omega=5 \Omega $$
Using Ohm's Law for the whole circuit:
$$ \mathrm{i}=\frac{\mathrm{V}}{\mathrm{R}_{	ext {total }}}=\frac{10}{5} \mathrm{~A}=2 \mathrm{~A}=\mathrm{i}_1 $$
The reading of the ammeter is equal to the value of $\mathrm{i}_3$.
Since the resistances $\mathrm{R}_2$ and $\mathrm{R}_{3,4,5}$ are equal to $8 \Omega$, the current splits equally, $\mathrm{i}_2=\mathrm{i}_3$
Using Kirchhoff's current law at junction, $\mathrm{i}_1=\mathrm{i}_2+\mathrm{i}_3=2 \mathrm{i}_2=2 \mathrm{i}_3 \Rightarrow \mathrm{i}_2=\mathrm{i}_3=\frac{\mathrm{i}_1}{2}$
So, the reading of the ammeter is $\mathrm{i}_3=\frac{\mathrm{i}_1}{2}=\frac{\mathrm{i}}{2}=1 \mathrm{~A}$
Therefore, the ammeter reading is 1A in steady state. Hence, the correct option is (B).

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Current Electricity

The reading of the ammeter (A) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is \_\_\_\_ A.

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