A battery with emf \( \varepsilon \) and internal resistance of \(30~\Omega\) is being recharged by connecting it to an outlet with a potential difference of \(120~\mathrm{V}\), as shown above. While it is being recharged, \(3~\mathrm{A}\) flows through the battery. Determine the emf of the battery.
Correct Answer: \( \boxed{\mathrm{D}} \)
Since the battery is being charged, the charging current flows into its positive terminal.
Apply Kirchhoff’s Loop Rule around the circuit:
\( 120-\varepsilon-Ir=0 \)
Substitute the given values:
\( 120-\varepsilon-(3)(30)=0 \)
\( 120-\varepsilon-90=0 \)
\( \varepsilon=30~\mathrm{V} \)
Therefore, the emf of the battery is
\( \boxed{30~\mathrm{V}} \)
Hence, the correct answer is (D).
In the circuit shown above, what is the current through the \(3~\Omega\) resistor?
Correct Answer: \( \boxed{\mathrm{E}} \)
Assume the current \(I\) flows clockwise around the loop.
Applying Kirchhoff’s Loop Rule, the algebraic sum of the potential changes around the loop is zero:
\( +7-I(3)+3-I(2)=0 \)
Simplifying,
\( 10-5I=0 \)
\( I=\dfrac{10}{5}=2.0~\mathrm{A} \)
Since the circuit contains only one loop, the same current flows through every component, including the \(3~\Omega\) resistor.
Therefore, the current through the \(3~\Omega\) resistor is
\( \boxed{2.0~\mathrm{A}} \)
Hence, the correct answer is (E).
In the circuit above, the emfs and the resistances have the values shown. The current in the circuit is \(2~\mathrm{A}\).
What is the potential difference between points \(X\) and \(Y\)?
Correct Answer: \( \boxed{\mathrm{C}} \)
From the previous part, the unknown resistance is
\( R=1~\Omega \)
Apply Kirchhoff’s Loop Rule from point \(X\) to point \(Y\) through the right-hand side of the circuit.
The potential changes are:
Across the \(6~\mathrm{V}\) battery: \( -6~\mathrm{V} \)
Across the \(0.2~\Omega\) resistor: \( -(2)(0.2)=-0.4~\mathrm{V} \)
Across the \(1~\Omega\) resistor: \( -(2)(1)=-2.0~\mathrm{V} \)
Thus,
\( V_Y-V_X=-6-0.4-2=-8.4~\mathrm{V} \)
Therefore, the magnitude of the potential difference between \(X\) and \(Y\) is
\( \boxed{|V_X-V_Y|=8.4~\mathrm{V}} \)
Hence, the correct answer is (C).
