Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery ε = 15.000 V with a 10.00-MΩ resistor in series with an unknown resistor R. As shown in Fig. 26–92, a particular voltmeter reads V₁ = 366 mV when connected across the 10.00 -MΩ resistor and this meter reads V₂ = 7.317 V when connected across R. Determine the value of R. [Hint: Define R_V as the voltmeter’s internal resistance.]

Verified step by step guidance

1. Step 1: Understand the problem setup. The circuit consists of a battery with an electromotive force (E) of 15.000 V, a 10.00-MΩ resistor, and an unknown resistor R in series. A voltmeter with internal resistance R_V is used to measure the voltage across each resistor. The voltmeter readings are V₁ = 366 mV across the 10.00-MΩ resistor and V₂ = 7.317 V across the unknown resistor R.
2. Step 2: Analyze the voltmeter's behavior. When the voltmeter is connected across a resistor, it forms a parallel combination with the resistor. The effective resistance of this parallel combination affects the voltage reading. Use the formula for parallel resistance: $R_{\mathrm{eff}}=\frac{R_V{R}_{\mathrm{resistor}}}{R_V+R_{\mathrm{resistor}}}$, where R_resistor is the resistance being measured and R_V is the voltmeter's internal resistance.
3. Step 3: Relate the voltage readings to the circuit. The voltage across the 10.00-MΩ resistor (V₁) and the unknown resistor (V₂) can be expressed using Ohm's Law: $V=IR$, where I is the current through the circuit and R is the resistance. The total resistance in the circuit determines the current, which can be calculated using the battery voltage and the total resistance.
4. Step 4: Write equations for the circuit. The total resistance in the circuit is the sum of the 10.00-MΩ resistor, the unknown resistor R, and the voltmeter's internal resistance R_V when connected. Use the voltage divider rule to express the voltages V₁ and V₂ in terms of the resistances and the current. For V₁: $V_1=I{R}_{\mathrm{eff}}$. For V₂: $V_2=I{R}_{\mathrm{eff}}$ (for the unknown resistor R).
5. Step 5: Solve for the unknown resistor R. Combine the equations for V₁ and V₂ with the parallel resistance formula to express R in terms of the given voltages, the battery voltage, and the known resistance values. Substitute the values for V₁, V₂, and the 10.00-MΩ resistor into the equations. You will also need to account for the voltmeter's internal resistance R_V, which can be determined from the circuit behavior. Rearrange the equations to isolate R and solve algebraically.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = IR. Understanding this law is crucial for analyzing circuits, as it allows us to relate voltage, current, and resistance, which is essential for solving the given problem.

Voltage Divider Rule

The Voltage Divider Rule is a fundamental principle used in circuit analysis that describes how the voltage is distributed across resistors in series. According to this rule, the voltage across a resistor in a series circuit is a fraction of the total voltage, proportional to the resistance of that resistor. This concept is particularly relevant in the context of the problem, as it helps in determining the voltage across the unknown resistor R based on the known voltage across the 10.00-MΩ resistor.

Internal Resistance of a Voltmeter

The internal resistance of a voltmeter affects its accuracy when measuring voltage in a circuit. A voltmeter with a high internal resistance is preferred as it minimizes the impact on the circuit being measured. In this problem, the internal resistance (R_V) of the voltmeter must be considered when calculating the voltage readings across the resistors, as it can alter the effective resistance in the circuit and thus influence the measured voltages V₁ and V₂.