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Chapter 28: Problem 47

A particular galvanometer serves as a \(2.00-\mathrm{V}\) full-scale voltmeter when a \(2500-\Omega\) resistor is connected in series with it. It serves as a \(0.500-\) A full-scale ammeter when a \(0.220-\Omega\) resistor is connected in parallel with it. Determine the internal resistance of the galvanometer and the current required to produce full-scale deflection.

Short Answer

Expert verified
The internal resistance of the galvanometer is approximately \( 40 \, \Omega \) , and the current required to produce a full-scale deflection is approximately \( 0.05 \, A \).

Step by step solution

01

Calculating Resistance for the Voltmeter Configuration

In the voltmeter configuration, the galvanometer and the resistor are in series. Therefore, the total resistance \( R_v \) is the sum of the internal resistance \( r \) of the galvanometer and the resistance \( R_{vm} = 2500 \, \Omega \) of the resistor: \( R_v = r + R_{vm} \). Furthermore, the total voltage \( V = 2V \) across a series combination is the sum of the voltage across each component. Hence, ohm's law \( V = I_{g} R_v \) for full-scale deflection can be used to calculate the current \( I_{g} \)
02

Calculating Resistance for the Ammeter Configuration

In the ammeter configuration, the galvanometer and the resistor are in parallel. Therefore, the total resistance \( R_a \) is given by the parallel resistors formula: \( 1 / R_a = 1/r + 1/R_{am} \) where \( R_{am} = 0.220 \, \Omega \) . Moreover, in a parallel combination, for full-scale deflection, the total current \( I = 0.5 A \) is the sum of the current through each parallel component. Using ohm's law \( I = V/R_a \), the voltage \( V \) across \( R_a \) can be calculated.
03

Equating Voltages and Solving for Internal Resistance \( r \) and Current \( I_g \)

Since the voltage across the galvanometer will be the same for both configurations, equate the value of \( V \) obtained in Steps 1 and 2. From this equation, solve for the galvanometer's internal resistance \( r \) and the current \( I_g \) required for a full-scale deflection.

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Key Concepts

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

Voltmeter Configuration
In a voltmeter configuration, the galvanometer is connected in series with a resistor. This setup allows us to measure voltage across a circuit component. Understanding its operation is crucial:

  • The galvanometer has an internal resistance, denoted as r.
  • A series resistor with known resistance, in this case, 2500 惟, is added.
  • The total resistance, which we denote as R_v, in this configuration is given by the sum of the galvanometer's internal resistance and the series resistance: R_v = r + 2500 惟.
For the voltmeter to operate correctly at full-scale deflection, the current flowing through it should suit Ohm鈥檚 law: \( V = I_{g} R_v \), where \( V = 2V \) is the full-scale voltage.

This equation is crucial for determining the current that results in maximum deflection, \( I_g \). Understanding these elements ensures that the voltmeter is configured accurately to reflect the desired range of voltage measurements.
Ammeter Configuration
In turning a galvanometer into an ammeter, the configuration involves adding a resistor in parallel with the device. This allows for current measurement in a circuit:

  • The parallel resistor is known as a "shunt." In this setup, the shunt resistor is 0.220 惟.
  • The objective is to measure currents up to full-scale accurately, which in this exercise is 0.5 A.
  • The equivalent resistance of the parallel configuration is calculated using the formula: \( \frac{1}{R_{a}} = \frac{1}{r} + \frac{1}{0.220惟} \).
The current flowing through the ammeter setup distributes through both the galvanometer and the shunt resistor. Ohm's law guides us here to express it as \( I = \frac{V}{R_a} \).

For the full-scale deflection condition, the total current when split between the galvanometer and the shunt reads 0.5 A. Using these formulas, we make certain that the ammeter configuration delivers the full-scale current functionality desired.
Internal Resistance
The internal resistance of a galvanometer, denoted by r, plays an important role in its efficiency and the accuracy of configurations like the voltmeter and ammeter. Here鈥檚 why it's important:

  • Internal resistance directly affects voltage and current readings.
  • Knowing the value of r assists in designing and calculating the correct configuration for accurate measurements.
  • Matching the resistance levels in various setups, e.g., series or parallel, relies on accurately known internal resistance.
The calculation of the internal resistance involves comparing voltmeter and ammeter configurations. By equating the voltage expressions from both setups, we can isolate r. This determination is crucial for solving exercise problems like the given one and in engineering accurate measurement instruments in practice.

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Most popular questions from this chapter

Design a multirange ammeter capable of full-scale deflection for \(25.0 \mathrm{mA}, 50.0 \mathrm{mA},\) and \(100 \mathrm{mA} .\) Assume the meter movement is a galvanometer that has a resistance of \(25.0 \Omega\) and gives a full-scale deflection for \(1.00 \mathrm{mA}\)

A regular tetrahedron is a pyramid with a triangular base. Six \(10.0-\Omega\) resistors are placed along its six edges, with junctions at its four vertices. A \(12.0-\mathrm{V}\) battery is connected to any two of the vertices. Find (a) the equivalent resistance of the tetrahedron between these vertices and (b) the current in the battery.

In places such as a hospital operating room and a factory for electronic circuit boards, electric sparks must be avoided. A person standing on a grounded floor and touching nothing else can typically have a body capacitance of \(150 \mathrm{pF}\), in parallel with a foot capacitance of \(80.0 \mathrm{pF}\) produced by the dielectric soles of his or her shoes. The person acquires static electric charge from interactions with furniture, clothing, equipment, packaging materials, and essentially everything else. The static charge is conducted to ground through the equivalent resistance of the two shoe soles in parallel with each other. A pair of rubber-soled street shoes can present an equivalent resistance of 5000 M\Omega. A pair of shoes with special static-dissipative soles can have an equivalent resistance of 1.00 M\Omega. Consider the person's body and shoes as forming an \(R C\) circuit with the ground. (a) How long does it take the rubber-soled shoes to reduce a \(3000-\mathrm{V}\) static charge to \(100 \mathrm{V} ?\) (b) How long does it take the staticdissipative shoes to do the same thing?

The current in a loop circuit that has a resistance of \(R_{1}\) is 2.00 A. The current is reduced to \(1.60 \mathrm{A}\) when an additional resistor \(R_{2}=3.00 \Omega\) is added in series with \(R_{1}\) What is the value of \(R_{1} ?\)

A fully charged capacitor stores energy \(U_{0} .\) How much energy remains when its charge has decreased to half its original value?
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