Problem 92 - A Complete Resource Book in Physics for JEE Main

Question

\(A, B\), and \(C\) are voltmeters of resistances \(R, 1.5 R\), and \(3 R\), respectively. When some potential difference is applied between \(X\) and \(Y\), the voltmeter readings are \(V_{A}, V_{B}\), and \(V_{C}\), respectively. (A) \(V_{A}=V_{B}=V_{C}\) (B) \(V_{A} \neq V_{B}=V_{C}\) (C) \(V_{A}=V_{B} \neq V_{C}\) (D) \(V_{B} \neq V_{A}=V_{C}\)

Step-by-Step Solution

Answer

The correct option is (A) \( V_{A} = V_{B} = V_{C} \)

Step 1: Analyze the circuit

First, we need to understand how the circuit is arranged. The three voltmeters \(A\), \(B\), and \(C\), are connected in parallel across points \(X\) and \(Y\). In a parallel circuit, the voltage is the same across all components. It's irrelevant the resistance of each voltmeter.

Step 2: Apply Ohm's Law

Applying Ohm's Law, voltage across each voltmeter will remain equal regardless of their resistances. So the readings on voltmeters \(A\), \(B\), and \(C\) will be same, irrespective of their resistances. Hence \(V_{A} = V_{B} = V_{C} \).

Step 3: Identify the Answer

From the options given, we can see that option (A) \( V_{A} = V_{B} = V_{C} \) is true. This is the correct answer.

Key Concepts

Voltmeter

A voltmeter is an essential tool when working with electrical circuits. Its main function is to measure the electrical potential difference, or voltage, between two points in an electric circuit. By doing so, it helps in assessing how much work is needed to move a unit charge from one point to another.

• Voltmeter Connection: A voltmeter must always be connected in parallel with the circuit component you wish to measure. This is because it needs to measure the potential difference across the component, not through it.
• Internal Resistance: Voltmeter should have high internal resistance to ensure it draws very little current from the circuit. This way, it doesn't affect the overall circuit and provides an accurate measurement of voltage.
• Types: There are analog and digital voltmeters. Digital voltmeters are commonly used as they provide quick and precise readings.

When using a voltmeter, keep in mind that the placement and type of voltmeter can influence its readings. The goal is always to accurately determine the potential differences in the circuit.

Parallel Circuit

A parallel circuit is a key concept in understanding how voltmeters work and how they are connected. In a parallel circuit, all components share the same two nodes, forming multiple paths for the current to flow.

• Equal Voltage: The most significant feature of a parallel circuit is that the voltage across each component is the same. This happens because all paths receive an equal share of the total voltage supplied to the circuit.
• Current Distribution: While the voltage is constant throughout a parallel circuit, the current is divided among the different paths based on the individual resistances.
• Applications: Household wiring is a common example where parallel circuits are used, ensuring all electrical appliances receive the same voltage.

Understanding parallel circuits is crucial, especially when analyzing problems involving voltmeters and ensuring accurate measurements of voltage across components connected in parallel.

Resistance in Parallel

When dealing with circuits, understanding how resistance behaves in parallel is essential. In a parallel circuit, each component has its own path to the voltage source, which affects the total resistance.

• Formula: The reciprocal of the total resistance (\( R_{total} \)) is the sum of the reciprocals of each individual resistance. Mathematically, this is represented as: \[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \, ... \, \]
• Effect: Adding more paths (or resistances in parallel) decreases the total resistance of the circuit. This is because the total current flowing through the circuit increases with more available paths.
• Importance: High resistance components like voltmeters need to be connected in parallel to minimize their impact on the circuit and to maintain the desired voltage measurement.

Grasping how resistance in parallel works will allow you to predict the total current flow and ensure accurate circuit analyses, especially when measuring voltages with voltmeters.