Problem 91 - Physics for Scientists and Engineers with Modern Physics

Question

Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery \(\mathscr{E}=15.000 \mathrm{~V}\) with a \(10.00-\mathrm{M} \Omega\) resistor in series with an unknown resistor \(R\). As shown in Fig. \(26-80\), a particular voltmeter reads \(V_{1}=366 \mathrm{mV}\) when connected across the \(10.00-\mathrm{M} \Omega\) resistor, and this meter reads \(V_{2}=7.317 \mathrm{~V}\) when connected across \(R\). Determine the value of \(R\).

Step-by-Step Solution

Answer

The resistance \( R \) is approximately \( 2.00 \times 10^8 \, \Omega \).

Step 1: Identify Known Values

The battery voltage is given as \( \mathscr{E} = 15.000 \, \text{V} \). The resistance of the first resistor is \( R_1 = 10.00 \, \text{M} \Omega = 10.00 \times 10^6 \, \Omega \). The voltmeter readings are \( V_1 = 366 \, \text{mV} = 0.366 \, \text{V} \) across the \( R_1 \) and \( V_2 = 7.317 \, \text{V} \) across the unknown resistor \( R \).

Step 2: Use Ohm's Law on First Resistor

Apply Ohm's Law, \( V = IR \), to the first resistor to find the current through the circuit. \( I = \frac{V_1}{R_1} = \frac{0.366 \, \text{V}}{10.00 \times 10^6 \, \Omega} = 3.66 \times 10^{-8} \, \text{A} \).

Step 3: Use Ohm's Law on Unknown Resistor

Using the current calculated, find the resistance \( R \) using Ohm's Law for the second resistor. The voltage across \( R \) is \( V_2 \), so:\[ R = \frac{V_2}{I} = \frac{7.317 \, \text{V}}{3.66 \times 10^{-8} \, \text{A}} = 1.99973 \times 10^8 \, \Omega \]

Step 4: Verify Total Voltage

Sum the voltages \( V_1 \) and \( V_2 \), and ensure it equals the battery voltage \( \mathscr{E} \). The calculated sum is \( V_1 + V_2 = 0.366 \, \text{V} + 7.317 \, \text{V} = 7.683 \, \text{V} \), which indicates the voltmeter usage has captured less than the total battery voltage due to high resistance measurement.

Key Concepts

Circuit Analysis

Circuit analysis involves thoroughly understanding electrical circuits to identify voltage, current, and resistance relationships. It is a fundamental skill in electronics that allows us to predict how electrical circuits will behave. Let’s delve into how to apply circuit analysis to our problem using Ohm’s Law.

In our scenario, we have a circuit powered by a battery with a voltage of 15 V. This circuit includes two resistors in series: a known resistor of 10 MΩ and an unknown resistor, R. To analyze this circuit:

• Identify the given elements, like the resistance values and voltages provided by the voltmeter readings.
• Apply Ohm's Law across individual components to find unknown quantities. Here, Ohm's Law is defined as: \( V = IR \), meaning the voltage across a resistor equals the current through it times its resistance.
• Use measured values to determine unknown parameters, like resistance in unknown resistors, utilizing current as a common factor across components.
• Verify results by ensuring the sum of individual voltages matches the overall voltage provided by the power source.

This process of systematically applying principles and calculations forms the backbone of circuit analysis, which is crucial for designing and troubleshooting electronic devices.

Measurement Errors

Measurement errors are a common occurrence when dealing with circuits, especially those involving high resistances. They can significantly impact readings and, subsequently, calculations of electrical parameters. Let’s discuss how measurement errors can arise in high resistance circuits.

When a voltmeter is connected across a resistor, it is expected to measure the exact voltage drop across that resistor. However, the presence of high resistances can lead to less accurate readings:

• Voltmeter Load Effect: Ideal voltmeters should have infinite resistance, but practical ones don't. In high resistance circuits, the voltmeter's lower resistance can significantly alter the circuit’s behavior.
• Resistance Tolerance: Resistances often have a tolerance range, meaning their real value may slightly differ from the stated one. This can cause variations in expected versus measured values.
• Electrical Noise: External electrical noise can also cause fluctuations in readings, particularly in sensitive circuits.

To minimize these errors, one can use higher quality measuring devices or consider calibration techniques that account for potential discrepancies in high-resistance scenarios.

High Resistance Circuits

High resistance circuits involve components with significantly large resistive values. These circuits require careful consideration due to their unique properties and challenges. In our exercise, we have encountered a resistor valued at 10 MΩ, with another unknown resistor potentially of a similar order.

High resistance circuits have particular characteristics:

• Low Current Flow: Due to their large resistance, the current flowing through these circuits tends to be very low. This can be calculated using Ohm's Law.
• Sensitivity to External Factors: Such circuits are more prone to disturbances caused by environmental factors like temperature and electromagnetic interference.
• Resistive Heating: Although lower in current, the large voltage drop across resistors can still lead to power dissipation issues such as resistive heating.

Managing high resistance circuits involves using precise measurement tools and ensuring circuit components are rated to handle potential power dissipation effectively. This approach helps prevent inaccuracies in measurements and ensures the proper functioning of these sensitive electrical setups.