Question

In the circuit shown, readings shown of the voltmeter connected across $400\;\Omega $ is $30$V. If it is connected across $300\;\Omega $ resistance then reading will be:

A. 45 V
B. 32.5 V
C. 22.5 V
D. 18 V

Answer 1

Hint: The internal resistance of the voltmeter will be in parallel with the resistance across which it is connected. Find the equivalent resistance in the first case using the information provided. Use this to find the internal resistance and then the equivalent resistance in the second case and the voltage drop consequently.

Complete step by step solution:
Given,
$\begin{align}
  & {{R}_{1}}=400\;\Omega \\
 & {{R}_{2}}=300\;\Omega \\
\end{align}$
${{V}_{i}}_{n}=60$V
Let $r$ be the internal resistance of the voltmeter, ${{V}_{1}}$ be the potential measured across ${{R}_{1}}$ and ${{V}_{2}}$ is the potential measured across ${{R}_{2}}$.
If the voltmeter is connected across ${{R}_{1}}$, then
\[\begin{align}
  & \dfrac{1}{{{R}_{e}}_{q}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{r} \\
 & \Rightarrow {{V}_{1}}=\dfrac{{{V}_{i}}_{n}{{R}_{e}}_{q}}{{{R}_{e}}_{q}+{{R}_{2}}} \\
 & \Rightarrow 30=\dfrac{60{{R}_{e}}_{q}}{{{R}_{e}}_{q}+300} \\
 & \Rightarrow 30{{R}_{e}}_{q}+30\times 300=60{{R}_{e}}_{q} \\
 & \Rightarrow {{R}_{e}}_{q}=300\;\Omega \\
 & \Rightarrow r={{(\dfrac{1}{{{R}_{e}}_{q}}-\dfrac{1}{{{R}_{1}}})}^{-}}^{1}=1200\;\Omega \\
 & \\
\end{align}\]
If the voltmeter is connected across ${{R}_{2}}$, then
\[\begin{align}
  & \Rightarrow \dfrac{1}{{{R}_{e}}_{q}}=\dfrac{1}{{{R}_{2}}}+\dfrac{1}{r}=\dfrac{1}{300}+\dfrac{1}{1200}=\dfrac{5}{1200} \\
 & \Rightarrow {{R}_{e}}_{q}=240\;\Omega \\
 & \Rightarrow {{V}_{2}}=\dfrac{{{V}_{i}}_{n}{{R}_{e}}_{q}}{{{R}_{e}}_{q}+{{R}_{1}}} \\
\end{align}\]
\[{{V}_{2}}=\dfrac{60\times 240}{240+400}=22.5\]V

The correct answer is option C.

Additional information:
The circuit in this problem is known as voltage divider circuit when the output is measured in between the two resistors. It can be used as a variable voltage source if one of the resistances is variable. They are used for adjusting signal levels, biasing of active elements in amplifiers and for measuring voltages. A potentiometer is a variable resistance device and is utilised as voltage dividers. They are commonly used in volume control of audio devices.

Note: Voltmeters are connected in parallel to measure potential difference as they have a high internal resistance. If it is connected in parallel with the load resistance, then the equivalent resistance will be nearly equal to the load resistance. Also the voltage drop will be the same across the voltmeter and the load. In an ideal voltmeter, the internal resistance is infinite such that no current is drawn and as a result there is no power loss in the instrument.