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Chapter 4: Problem 5

What is the sensitivity of the galvanometer (that is, what current gives a full-scale deflection) inside a voltmeter that has a \(1.00-\mathrm{M} \Omega\) resistance on its \(30.0-\mathrm{V}\) scale?

Short Answer

Expert verified
The galvanometer's sensitivity is 30.0 碌A.

Step by step solution

01

Understand the Problem

The problem asks for the sensitivity of a galvanometer inside a voltmeter. Sensitivity here refers to the current that causes a full-scale deflection. We are given the resistance of the voltmeter (1.00 M惟) and its maximum voltage scale (30.0 V).
02

Use Ohm's Law for Voltmeter

The voltmeter's total resistance is 1.00 M惟, and it measures a maximum voltage of 30.0 V. According to Ohm's Law, voltage \( V \), current \( I \), and resistance \( R \) are related by \( V = I \times R \). Here, we need to find the current \( I \) that causes the full deflection, using \( I = \frac{V}{R} \).
03

Calculate the Current

Substitute the given values into the formula \( I = \frac{V}{R} \). So, \( I = \frac{30.0 \text{ V}}{1.00 \times 10^6 \text{ 惟}} \). Evaluating this gives \( I = 30.0 \times 10^{-6} \text{ A} \).
04

Convert Units

The current obtained is in amperes. To express this in microamperes, note that 1 A = 1,000,000 碌A. Therefore, \( 30.0 \times 10^{-6} \text{ A} = 30.0 \text{ 碌A} \). Thus, the current sensitivity is 30.0 碌A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ohm's Law
Understanding Ohm's Law is essential when dealing with electrical circuits. This law creates a relationship between voltage, current, and resistance in a circuit. It is usually expressed as \( V = I \times R \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.

In simpler terms, if you know any two of these values, you can easily find the third one. This law forms the foundation for analyzing many electrical components, including voltmeters and ammeters.
  • Voltage (V): It is the potential difference between two points in a circuit.
  • Current (I): This refers to the flow of electric charge across a circuit, measured in amperes (A).
  • Resistance (R): It is the opposition to the flow of current, measured in ohms (惟).
Ohm's Law helps to determine how these quantities influence each other, making it a crucial tool for solving circuit problems.
Voltmeter
A voltmeter is an instrument used for measuring the electrical potential difference between two points in an electric circuit. Its primary purpose is to display voltage readings.

High resistance in a voltmeter is crucial because it prevents it from drawing significant current from the circuit. This keeps the circuit's characteristics unchanged, ensuring accurate voltage measurements.
  • High Resistance: Typically measured in megaohms (M惟), to minimize circuit disturbance.
  • Connection Method: Always connected in parallel with the component across which you need to measure the voltage.
By using a voltmeter, engineers and technicians can identify potential differences in circuits and troubleshoot more effectively.
Full-Scale Deflection
Full-scale deflection in an instrument like a galvanometer indicates the maximum measurement capacity before the current exceeds its limit. In a voltmeter, understanding full-scale deflection is essential for determining its sensitivity.

This concept tells us how much current will cause the maximum reading on the instrument's scale, thus helping in calibrating and designing measurement devices.
  • Understanding Maximum Range: It helps determine the scale limit for accurate measurements.
  • Sensitivity Measurement: The smaller the current needed for full-scale deflection, the more sensitive the device is.
By knowing the full-scale deflection value, technicians can ensure that measurements stay within the safe operational limits of the instrument.
Current Measurement
Measuring current, which refers to the flow of electric charge in a circuit, is crucial for analyzing and designing electrical systems. To measure current, instruments like ammeters and galvanometers are used.

For a voltmeter, measuring sensitivity in terms of current entails identifying what current level causes the full-scale deflection. In our problem:
  • Sensitivity: Determined by the minimal current that results in a full-scale reading; calculated using Ohm鈥檚 Law.
  • Unit Conversion: Often required, for instance, converting amperes (A) to microamperes (碌A) for finer precision.
Precisely measuring current allows us to ensure that circuits are functioning correctly and also to prevent overloading or potential damage.

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