Problem 80 - Physics

Question

The coil of wire in a galvanometer has a resistance of \(R_{C}=60.0 \Omega\). The galvanometer exhibits a full-scale deflection when the current through it is \(0.400 \mathrm{~mA}\). A resistor is connected in series with this combination so as to produce a voltmeter. The voltmeter is to have a full- scale deflection when it measures a potential difference of \(10.0 \mathrm{~V}\). What is the resistance of this resistor?

Step-by-Step Solution

Answer

The resistance required is 24,940 \(\Omega\).

Step 1: Understanding the Problem

We need to determine the resistance required in series with a galvanometer coil to function as a voltmeter with a full-scale deflection at 10 V. We know the coil resistance and full-scale current of the galvanometer.

Step 2: Relationship between Voltage, Current, and Resistance

Using Ohm's Law, we know that the full-scale voltage \( V \) is related to the current \( I \) and the total resistance \( R_{total} \) by: \[ V = I \times R_{total} \]

Step 3: Expressing the Total Resistance

The total resistance \( R_{total} \) is the sum of the coil resistance \( R_C \) and the resistance \( R \) we need to find:\[ R_{total} = R_C + R \]

Step 4: Applying Ohm's Law to the Given Values

Substituting known values into Ohm's Law gives:\[ 10.0 = 0.0004 imes (60 + R) \]

Step 5: Solving for the Unknown Resistance

Rearrange and solve the equation:\[ 10.0 = 0.0004 imes R + 0.0004 imes 60 \]\[ 10.0 = 0.0004R + 0.024 \]Subtract 0.024 from both sides:\[ 9.976 = 0.0004R \]Solve for \( R \) by dividing both sides by 0.0004:\[ R = \frac{9.976}{0.0004} = 24940 \Omega \]

Step 6: Conclusion

The required resistance to make the galvanometer act as a voltmeter with a full-scale deflection at 10 V is \( 24,940 \Omega \).

Key Concepts

Galvanometer

A galvanometer is a sensitive instrument designed to detect and measure small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque proportional to the current. This causes a needle or pointer to move across a scale, indicating the current's magnitude. A galvanometer is quite similar to a basic moving coil meter which makes it suitable for detecting minute currents.

Essential features of a galvanometer include:

• High sensitivity: Capable of detecting current as small as microamperes.
• Coil resistance: The coil offers some inherent resistance, denoted as \( R_{C} \), usually measured in ohms.
• Full-scale deflection: This indicates the maximum current a galvanometer can measure without getting damaged or misreading, measured in milliamps (mA).

Series Resistance

In electrical measurements, series resistance is essential when converting a galvanometer to a voltmeter. This added resistance prolongs the range of measurable voltages without altering the galvanometer's sensitivity to the current. When a resistance is added in series to a galvanometer, the overall resistance of the circuit increases, enhancing the device's potential handling capacity.

The significance of series resistance lies in:

• Protecting the galvanometer: By limiting how much current can flow through the galvanometer, preventing potential damage.
• Adjusting sensitivity: Allows the measurement of higher potential differences by increasing total resistance.
• Enabling voltmeter range extension based on required scales, such as from 0 V to 10 V in this exercise.

Voltmeter Conversion

Converting a galvanometer into a voltmeter is a common procedure due to the galvanometer's sensitivity. This conversion involves adding a precise value of series resistance so that the combined setup can measure higher voltages. The prime tool responsible for this is Ohm's Law, which relates voltage (\( V \)), current (\( I \)), and resistance (\( R \)) with the formula: \( V = I \times R \).

For voltmeter conversion:

• Determine the necessary full-scale voltage the new voltmeter needs to measure.
• Use the coil's known resistance and full-scale deflection current to calculate the total desired resistance.
• The series resistance \( R \) is found by subtracting the galvanometer's coil resistance \( R_{C} \) from the total resistance, derived by rearranging and solving the equation \( V = I \times (R_{C} + R) \).

This process allows us to convert delicate galvanometers into robust voltmeters with specific voltage ranges suitable for various applications.