Question

You want to make an ohmmeter to measure the resistance of unknown resistors. You have a battery with voltage \(\mathrm{V}_{\mathrm{emf}}=9.00 \mathrm{~V}\), a variable resistor, \(R,\) and an ammeter that measures current on a linear scale from 0 to \(10.0 \mathrm{~mA}\) a) What resistance should the variable resistor have so that the ammeter gives its full-scale (maximum) reading when the ohmmeter is shorted? b) Using the resistance from part (a), what is the unknown resistance if the ammeter reads \(\frac{1}{4}\) of its full scale?

Short answer

Answer: The variable resistor should have a resistance of 900 Ω for the ammeter to show its maximum reading. The unknown resistor's resistance is 2700 Ω when the ammeter reads one-quarter of its full scale.

Step-by-step solution

Ohm's Law

First, let's recall Ohm's Law: \(V = IR\) where V is voltage, I is current, and R is resistance.

Find the variable resistor's resistance for a maximum ammeter reading

For part (a), we want to find the resistance of the variable resistor when the ammeter reads its maximum value, which is 10.0 mA. Since the circuit is shorted, there is no other resistance present, and we can use Ohm's Law. First, convert the maximum current to amperes: \(I = 10.0 \times 10^{-3} A = 0.010 A.\) Now, apply Ohm's Law to find the resistance: \(R = \frac{V_{emf}}{I} = \frac{9.00 V}{0.010 A} = 900 \Omega\) The variable resistor should have a resistance of 900 Ω such that the ammeter shows its maximum reading when the circuit is shorted.

Find the unknown resistance when the ammeter reads one-quarter of its full scale

For part (b), when the ammeter reads one-quarter of its full scale, the current passing through the circuit can be calculated as follows: \(I = \frac{1}{4}(10.0 \times 10^{-3} A) = 0.00250 A.\) Let's now use Ohm's Law again, this time to find the voltage across the unknown resistor: \(V_{unknown} = V_{emf} - V_{variable} = V_{emf} - IR_{variable} = 9.00 V - (0.00250 A)(900 \Omega) = 6.75 V\) Now, we can find the unknown resistor's resistance, \(R_{unknown}\), by applying Ohm's Law once more: \(R_{unknown} = \frac{V_{unknown}}{I} = \frac{6.75 V}{0.00250 A} = 2700 \Omega\) So the unknown resistor has a resistance of 2700 Ω when the ammeter reads one-quarter of its full scale.

Key concepts

Resistance Measurement

Measuring resistance is a fundamental task when dealing with electric circuits. This process helps determine how much a resistor opposes the flow of electric current. To perform a resistance measurement, we commonly use devices like an ohmmeter. The core idea is based on Ohm's Law, given by the formula: \[ V = IR \]where \(V\) represents voltage, \(I\) is the current, and \(R\) is the resistance. In practical applications, understanding this relationship allows us to find unknown resistances by measuring current when voltage is constant. The specific arrangement in a circuit and the method of connection are crucial for accurate measurements. Below are things to consider:

• Ensure the circuit is not powered with external voltages when testing.
• The accuracy of the resistance measurement often hinges on the precision of voltage and current readings.
• Short circuits or open circuits can significantly skew results.

Ammeter

An ammeter is an instrument used to measure electric current in a circuit. It provides readings typically in amperes or milliamperes. The ammeter is an essential tool when constructing an ohmmeter since measuring the current is vital for determining resistance via Ohm's Law.
An ideal ammeter should have negligible resistance to avoid altering the current it aims to measure. When using an ammeter:

• Always connect it in series with the circuit component whose current you wish to gauge.
• Be mindful of the ammeter's maximum range to prevent damage by excessive current.
• For accurate readings, ensure the ammeter is properly calibrated.

Electric Circuits

Electric circuits are pathways through which electric current flows. They consist of various components like resistors, batteries, and conductive wires that connect these elements. Understanding circuits is crucial for creating devices like ohmmeters which are designed to measure resistance by manipulating the current flow.
Basic circuit concepts include:

• Closed and open circuits, where current can only flow completely in closed paths.
• Series and parallel configurations affecting how components interact with each other.
• Total circuit resistance can be found using rules for combining series or parallel resistors.

Ohm's Law is applied throughout circuit analysis, allowing us to predict how changes in voltage or resistance affect the current.

Variable Resistor

A variable resistor, commonly called a potentiometer or rheostat, allows you to adjust resistance within a certain range. This is particularly useful for tuning circuits without altering their design.
Variable resistors play a critical role in creating a functional ohmmeter. They enable calibrating the device for different measurement ranges, for instance ensuring the ammeter reaches full scale to see its maximum reading.
The setting of the variable resistor directly impacts the resistance measurement in the following ways:

• It ensures the circuit can reach desired current levels by adjusting the overall resistance.
• Helps in achieving precise control over the current flowing in the circuit.
• Vital during calibration, especially when optimizing the scale of measurement tools like ammeters.

Through careful adjustment, variable resistors offer flexibility and precision in electronic measurements.