Problem 63 - Physics for Scientists and Engineers with Modern Physics

Question

(III) Two \(9.4-\mathrm{k} \Omega\) resistors are placed in series and connected to a battery. A voltmeter of sensitivity 1000\(\Omega / \mathrm{V}\) is on the \(3.0-\mathrm{V}\) scale and reads 2.3 \(\mathrm{V}\) when placed across either resistor. What is the emf of the battery? (Ignore its internal resistance.)

Step-by-Step Solution

Answer

The emf of the battery is 4.6 V.

Step 1: Understanding the Components

We have two 9.4 kΩ resistors in series connected to a battery. When placed across each resistor, the voltmeter reads 2.3 V on its 3.0 V scale setting.

Step 2: Determine Voltmeter Resistance

The voltmeter has a sensitivity of 1000 Ω/V. At a 3.0 V scale, the voltmeter's internal resistance is calculated as \( 1000 \, \Omega/\text{V} \times 3.0 \text{ V} = 3000 \, \Omega \), or 3.0 kΩ.

Step 3: Equivalent Resistance Calculation

Placing the voltmeter across a resistor forms a parallel circuit: one 9.4 kΩ resistor in parallel with the 3.0 kΩ voltmeter. We calculate the equivalent resistance using the formula: \( R_{eq} = \frac{9.4 \, \text{k}\Omega \times 3.0 \, \text{k}\Omega}{9.4 \, \text{k}\Omega + 3.0 \, \text{k}\Omega} \approx 2.238 \, \text{k}\Omega \).

Step 4: Voltage Across One Parallel Segment

The voltmeter reads 2.3 V across this equivalent resistance. The total voltage across both resistor-voltmeter combinations can be expressed using the voltage divider formula based on two equal equivalent resistances.

Step 5: Total Resistance and Voltage Calculation

The total circuit resistance \( R_{total} \) is \( 2 \times 2.238 \, \text{k}\Omega = 4.476 \, \text{k}\Omega \). Knowing the voltage (V) across this is the sum of the voltages across each equivalent resistor combination, use \( 2 \times 2.3 \, V = 4.6 \, V \), which is the emf of the battery.

Step 6: Verifying the Emf Value

Thus, with an equivalent resistance and given voltages, we confirm that the emf of the battery is precisely \( 4.6 \, V \).

Key Concepts

Resistor

Resistors are fundamental components in electrical circuits that restrict the flow of electrical current. They are specified by their resistance value measured in ohms (Ω). The resistance signifies how much the resistor resists the electrical current—greater resistance means less current flow.
When resistors are placed in a circuit, they consume power. This occurs because resistors convert electrical energy into heat, which is why they often have power ratings that determine how much energy they can safely dissipate without damage.
Additionally, the value of a resistor is crucial for controlling voltages and currents throughout a circuit. For example, if you have a resistor with a value of 9.4 kΩ, it means it offers a resistance of 9,400 ohms, which profoundly influences how the current flows through that part of the circuit.

Voltmeter

Voltmeters are devices used to measure the electric potential difference between two points in an electric circuit. This potential difference is commonly referred to as voltage. A voltmeter is typically connected in parallel with the circuit element across which the voltage is being measured.
Voltmeters have a certain sensitivity, usually expressed in terms of ohms per volt (Ω/V). This sensitivity indicates how much resistance the voltmeter adds to the circuit when measuring voltages. A higher sensitivity or internal resistance is desirable because it minimizes the impact on the circuit it measures.
In our exercise, the voltmeter with a sensitivity of 1000 Ω/V was used across a 3.0 V scale, giving an internal resistance of 3,000 Ω. This helps ensure the voltmeter doesn’t significantly alter the readings by loading the circuit.

Electromotive Force (emf)

Electromotive force, often abbreviated as emf, refers to the voltage created by a battery or other power source when no current is flowing in the circuit. It's what drives current through the circuit. Even though it's called a 'force,' it is measured in volts, not newtons.
In practical terms, emf represents the energy supplied to each coulomb of charge. For instance, our solution discovered that the battery has an emf of 4.6 V, which tells us this energy level propels electric charge through the circuit.
Understanding emf is crucial for analyzing circuits, especially when predicting how voltage and current will behave across various components in different configurations of resistors.

Series Circuit

A series circuit has components connected end-to-end, so the same current flows through each component. In such a setup, the total resistance of the circuit is the sum of the individual resistances. This is because resistors add up in series directly, limiting the current based on their total value.
In our example, two resistors are in series, making the total resistance equal to the sum of their individual resistances if no measuring devices were altering the circuit.
The way series circuits handle voltage is using a principle called the voltage divider rule. This rule states that the voltage is divided across resistors in proportion to their resistances, which means larger resistors take up more voltage in a series configuration.

Parallel Circuit

Parallel circuits differ from series circuits in that they have multiple paths for current to travel. Each component is connected to the same two points of the circuit, meaning the voltage across each component is the same.
The total resistance in a parallel circuit is less than the smallest resistor in the combination due to the nature of parallel connections. The formula for calculating total resistance is complex but involves inversely summing the resistances.
In the exercise, when the voltmeter is shunted across a resistor, it forms a parallel circuit, reducing the effective resistance measured by the voltmeter. By re-calculating the resistance taking into account this parallel layout, accurate voltage readings across the components can be determined.