A 1mA D' Arsonval movement has resistance of 100 Ω. It is to be converted to a 10 V voltmeter. The value of multiplier resistance is 

Explanation:

Problem Statement:

A 1mA D'Arsonval movement has a resistance of 100 Ω, and it needs to be converted into a 10 V voltmeter. The problem requires us to determine the value of the multiplier resistance for this conversion.

To solve this, we use the concept of a series resistor (multiplier resistor) to extend the voltage range of the D'Arsonval galvanometer. The following explanation provides the detailed solution:

Solution:

When designing a voltmeter using a galvanometer, a multiplier resistor is connected in series with the galvanometer. The multiplier resistor ensures that the galvanometer does not exceed its current rating while measuring higher voltages. The total resistance of the voltmeter circuit limits the current through the galvanometer to its full-scale deflection current.

Given Data:

• Full-scale deflection current of the D'Arsonval movement, I_fs = 1 mA = 1 × 10^-3 A
• Internal resistance of the galvanometer, R_g = 100 Ω
• Voltage to be measured, V = 10 V

Formula:

The total voltage across the voltmeter is shared between the galvanometer and the multiplier resistance. Mathematically, this relationship can be expressed as:

V = I_fs × (R_g + R_m)

Where:

• R_m is the multiplier resistance to be calculated.

Rearranging the equation to solve for R_m:

R_m = (V / I_fs) - R_g

Substitute the values:

R_m = (10 / 1 × 10^-3) - 100

R_m = 10,000 - 100

R_m = 9,900 Ω

Conclusion:

The value of the multiplier resistance required to convert the 1mA D'Arsonval movement into a 10V voltmeter is 9,900 Ω. Thus, the correct option is:

Option 3: 9,900 Ω

Important Information

To further understand the analysis, let’s evaluate why the other options are incorrect:

Option 1: 999 Ω

This value is much lower than the calculated multiplier resistance. A multiplier resistor of 999 Ω would result in a much higher current flowing through the galvanometer, exceeding its full-scale deflection current, potentially damaging the galvanometer.

Option 2: 9,999 Ω

Although close to the correct value, this option is slightly higher than the required resistance of 9,900 Ω. Using a resistance of 9,999 Ω would limit the current to less than 1 mA, leading to an underestimation of the measured voltage.

Option 4: 990 Ω

This value is far too low, similar to Option 1. A resistance of 990 Ω would result in excessive current through the galvanometer, well beyond its full-scale deflection current, potentially causing damage.

Key Points:

• The purpose of the multiplier resistance is to ensure that the galvanometer operates within its safe current range while accurately measuring higher voltages.
• The formula R_m = (V / I_fs) - R_g is crucial for designing voltmeters using galvanometers.
• It is essential to choose the correct multiplier resistance to avoid inaccurate readings or damage to the galvanometer.