Unit 11: Electric Circuits
Weightage : 15–18%
When a battery is connected to an external resistance \(R\), as shown above on the left, there is a current \(I\) in the circuit. When the external resistance is changed to \(3R\), the current changes to \( \dfrac{I}{2} \), as shown above on the right. What is the internal resistance \(r\) of the battery?
Correct Answer: \( \boxed{\mathrm{C}} \)
Let the emf of the battery be \( \varepsilon \).
When the external resistance is \(R\),
\( I=\dfrac{\varepsilon}{R+r} \)
When the external resistance is changed to \(3R\), the current becomes \( \dfrac{I}{2} \):
\( \dfrac{I}{2}=\dfrac{\varepsilon}{3R+r} \)
Substitute \( I=\dfrac{\varepsilon}{R+r} \):
\( \dfrac{\varepsilon}{2(R+r)}=\dfrac{\varepsilon}{3R+r} \)
Cancel \( \varepsilon \) and cross-multiply:
\( 3R+r=2(R+r) \)
\( 3R+r=2R+2r \)
\( R=r \)
Therefore, the internal resistance of the battery is \( \boxed{R} \).
Answer: (C)
The figure above represents a section of a circuit containing three resistors, \(X\), \(Y\), and \(Z\), of different sizes but made of the same material. Which of the following correctly ranks the current in the resistors?
Correct Answer: \( \boxed{\mathrm{D}} \)
The resistance of a conductor is given by
\( R=\dfrac{\rho L}{A} \)
where \(L\) is the length, \(A\) is the cross-sectional area, and \(\rho\) is the resistivity. Since all three resistors are made of the same material, they have the same resistivity.
For a given potential difference, the current is
\( I=\dfrac{V}{R}\propto\dfrac{A}{L} \)
Resistors \(X\) and \(Y\) have the same cross-sectional area, but \(Y\) is shorter, so
\( I_{Y}>I_{X} \)
Resistors \(X\) and \(Z\) have approximately the same length, but \(X\) has a larger cross-sectional area, so
\( I_{X}>I_{Z} \)
Combining these results,
\( I_{Y}>I_{X}>I_{Z} \)
Therefore, the correct ranking is (D).
An engineer has four wires made of the same material and wants to determine the material’s resistivity. The engineer measures the length \(L\) and cross-sectional area \(A\) of each wire. The engineer then applies a potential difference \(V\) across each wire and measures the resulting current \(I\).
To estimate the resistivity of the material using only the slope of a graph of the data, which of the following should be graphed as a function of \( \dfrac{L}{A} \)?
Correct Answer: \( \boxed{\mathrm{C}} \)
The resistance of a uniform wire is
\( R=\rho\dfrac{L}{A} \),
where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length, and \(A\) is the cross-sectional area.
Using Ohm’s law,
\( R=\dfrac{V}{I}. \)
Substituting into the resistance equation gives
\( \dfrac{V}{I}=\rho\dfrac{L}{A}. \)
Therefore, if \( \dfrac{V}{I} \) is plotted on the vertical axis as a function of \( \dfrac{L}{A} \), the graph is a straight line with
Slope \(=\rho\).
Thus, the resistivity of the material can be determined directly from the slope of the graph.
Therefore, the correct answer is (C).
