Problem 54 - Physics for Scientists and Engineers with Modern Physics

Question

(I) What is the resistance of a voltmeter on the \(250-\mathrm{V}\) scale if the meter sensitivity is \(35,000 \Omega / \mathrm{V} ?\)

Step-by-Step Solution

Answer

The resistance is 8,750,000 Ω or 8.75 MΩ.

Step 1: Understand the Sensitivity Concept

The sensitivity of a voltmeter is given in ohms per volt, and it indicates the load resistance for each volt of measurement capability. In this case, the voltmeter has a sensitivity of \( 35,000 \, \Omega / \text{V} \). This means that for every volt, the meter has a resistance of \( 35,000 \, \Omega \).

Step 2: Calculate Total Resistance

To find the total resistance of the voltmeter when set to a particular voltage range, multiply its sensitivity by the voltage range. Here, the voltage range is \( 250 \, \text{V} \). Thus, the total resistance \( R \) is given by:\[R = \text{sensitivity} \times \text{voltage scale} = 35,000 \times 250\]

Step 3: Compute the Resistance

Perform the multiplication calculation to find the voltmeter resistance:\[R = 35,000 \times 250 = 8,750,000 \, \Omega\]Thus, the resistance of the voltmeter on the \(250-\text{V}\) scale is \( 8,750,000 \, \Omega \), or \( 8.75 \, \text{M}\Omega \).

Key Concepts

sensitivity

Sensitivity is a key characteristic of a voltmeter that defines how responsive it is to different voltages. When we talk about a voltmeter's sensitivity, we refer to how much resistance the device has for each volt it measures. This is usually expressed in ohms per volt, which simply means how many ohms of resistance there are per one volt.
For instance, if a voltmeter has a sensitivity of 35,000 ohms per volt, this means that for each volt, the voltmeter effectively adds a resistance of 35,000 ohms.
This principle helps determine the voltmeter's load on a circuit, which is essential because a high load can alter the circuit’s behavior when measurements are taken, leading to inaccurate readings. Thus, understanding sensitivity ensures that you use a voltmeter with an appropriate resistance for your needs.

ohms per volt

The term 'ohms per volt' is a metric used to express the sensitivity of voltmeters. It is an easy and straightforward way to understand how much resistance a voltmeter design includes for every volt in its measured range.
If the sensitivity of a voltmeter is expressed as 35,000 ohms per volt, then in practice, for any given volt the meter reads, there is 35,000 ohms of resistance introduced by the meter.
This can be important when dealing with circuits where introducing high resistance could influence the functionality of the circuit itself. Therefore, using a voltmeter with an appropriate ohms per volt value is necessary for obtaining accurate measurements, particularly in delicate electronic circuits where even slight changes in current paths could lead to significant signal distortions.

resistance calculation

To calculate the total resistance of a voltmeter using its sensitivity and a specific voltage scale is straightforward. This involves multiplying the voltmeter's sensitivity, expressed in ohms per volt, by the maximum voltage for that scale.
Let's consider a scenario where you have a voltmeter with a sensitivity of 35,000 ohms per volt set to measure up to 250 volts. The total resistance, denoted as \( R \), can be found using:
\[R = ext{sensitivity} \times ext{voltage scale}\]\[R = 35,000 \times 250 = 8,750,000 \, \Omega\]
So, when your voltmeter is set to a 250 V scale, it will have a total resistance of 8,750,000 ohms or 8.75 Megaohms (\( \text{M}\Omega \)). Doing this calculation correctly ensures the voltmeter is appropriately configured for taking measurements at that voltage range without unintentionally affecting the circuit.