The diagram above shows a circuit that contains a battery with a potential difference of \(V_B\) and negligible internal resistance, five identical resistors, three ammeters \(A_1\), \(A_2\), \(A_3\), and a voltmeter.
Which of the following correctly ranks the readings of the ammeters?
Correct Answer: \( \boxed{\mathrm{D}} \)
The battery divides the circuit into two parallel branches.
The left branch contains two resistors in series, so its equivalent resistance is
\(R_{\mathrm{left}}=2R\).
The right branch contains one resistor in series with two identical resistors in parallel. The parallel combination has equivalent resistance
\(R_{\mathrm{parallel}}=\dfrac{R\cdot R}{R+R}=\dfrac{R}{2}\).
Therefore, the total resistance of the right branch is
\(R_{\mathrm{right}}=R+\dfrac{R}{2}=\dfrac{3R}{2}\).
Since \(R_{\mathrm{right}}<R_{\mathrm{left}}\), the current through the right branch is greater than the current through the left branch. Thus,
\(A_2>A_1\).
At the junction on the right, the current measured by \(A_2\) splits equally between the two identical parallel resistors. Ammeter \(A_3\) measures the current in only one of these branches, so
\(A_3=\dfrac{A_2}{2}\).
Combining these results gives
\(A_2>A_1>A_3\).
Therefore, the correct answer is (D).
If the ammeter in the circuit above reads zero, what is the resistance \(R\)?
Correct Answer: \( \boxed{\mathrm{E}} \)
Since the ammeter reads zero, there is no current through it. Therefore, the two junctions connected by the ammeter are at the same potential. This is the condition for a balanced bridge.
The potential drop across the \(1~\Omega\) resistor must equal the potential drop across the \(2~\Omega\) resistor:
\( I_1(1)=I_2(2) \)
Thus,
\( I_1=2I_2 \)
The upper branch has resistance \(1+3=4~\Omega\), and the lower branch has resistance \(2+R\).
Since branch current is inversely proportional to branch resistance,
\( 4=\dfrac{1}{2}(2+R) \)
\( 8=2+R \)
\( R=6~\Omega \)
Therefore, the required resistance is
\( \boxed{6~\Omega} \)
Hence, the correct answer is (E).
In the circuit shown above, the resistances have the values given and there is no current in the ammeter \(A\). What is the value of \(R_x\)?
Correct Answer: \( \boxed{\mathrm{E}} \)
Since there is no current through the ammeter, the bridge is balanced. Therefore, the potentials at the top and bottom junctions are equal, and the Wheatstone bridge condition applies:
\( \dfrac{R_x}{75}=\dfrac{100}{25} \)
Solving for \(R_x\),
\( R_x=75\left(\dfrac{100}{25}\right)=75(4)=300~\Omega \)
Thus, the unknown resistor must have a resistance of
\( \boxed{R_x=300~\Omega} \)
Hence, the correct answer is (E).
