Problem 12 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Eight identical bulbs are connected in series across a \(110-\mathrm{V}\) line. \((a)\) What is the voltage across each bulb? \((b)\) If the current is \(0.42 \mathrm{A},\) what is the resistance of each bulb, and what is the power dissipated in each?

Step-by-Step Solution

Answer

(a) 13.75 V per bulb. (b) 32.74 ohms resistance and 5.775 W power per bulb.

Step 1: Recognize Series Connection

Since the bulbs are connected in series, the total voltage across all the bulbs is the voltage of the line, which is 110 V. In a series circuit, the voltage is distributed equally if the components are identical. Therefore, each bulb will have the same voltage drop across it.

Step 2: Calculate Voltage per Bulb

Since there are 8 identical bulbs in series, and the total voltage is 110 V, the voltage across each bulb can be calculated by dividing the total voltage by the number of bulbs. So, the voltage across each bulb is \( \frac{110}{8} = 13.75 \, \text{V} \).

Step 3: Use Ohm's Law to Find Resistance of Each Bulb

The formula for Ohm's Law is \( V = IR \). We know the current \( I = 0.42 \, \text{A} \) and the voltage across each bulb \( V = 13.75 \, \text{V} \). Therefore, \( R = \frac{V}{I} = \frac{13.75}{0.42} \approx 32.74 \, \text{ohms} \).

Step 4: Calculate Power Dissipated in Each Bulb

The power dissipated in an electrical component can be calculated using the formula \( P = VI \). For each bulb, \( V = 13.75 \, \text{V} \) and \( I = 0.42 \, \text{A} \), thus \( P = 13.75 \times 0.42 \approx 5.775 \, \text{W} \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental concept in electrical engineering and physics, which describes the relationship between voltage, current, and resistance in an electrical circuit. The formula for Ohm's Law is expressed as \( V = IR \), where \( V \) stands for voltage, \( I \) represents current, and \( R \) denotes resistance. This equation states that the voltage across a component is the product of the current flowing through it and its resistance.
Understanding Ohm's Law is crucial for analyzing circuits, especially when you want to determine one of these three properties if the other two are known. For example, if you know the voltage across a bulb and the current flowing through it, you can quickly find the resistance by rearranging the formula: \( R = \frac{V}{I} \). Similarly, if the resistance and current are known, voltage can be calculated as \( V = IR \).
Learning how to manipulate Ohm's Law is key to diagnosing circuits and predicting how they will behave under different conditions. Whether designing or troubleshooting circuits, this law serves as a foundational tool.

Electrical Power

Electrical power is the rate at which electrical energy is transferred or used by a component in a circuit. The formula to calculate power is \( P = VI \), where \( P \) is power measured in watts (W), \( V \) is voltage in volts, and \( I \) is current in amperes (A). This relationship stipulates that power in a circuit is the product of the voltage across a component and the current flowing through it.
In practical terms, if you're looking at a light bulb like those in the exercise, knowing how much power it dissipates can help understand how bright it will shine or how much electrical energy it consumes over time. For the bulbs in the exercise, given a constant voltage and current, their power usage was computed using this formula to find each bulb dissipates approximately 5.775 W. This info can also inform design decisions or energy calculations.
These insights into electrical power not only pertain to circuit analysis but also extend to the efficient use of power in homes and industries. It's critical when discussing battery life, heat generation, and overall electrical efficiency.

Voltage Division

Voltage division is a rule used in series electrical circuits to determine the voltage drop across a particular component. In a series circuit, the total voltage supplied is equal to the sum of the individual voltage drops across all components. This implies that voltage is divided among the components in proportion to their resistance. For identical components, such as the bulbs in the exercise, each will have an equal share of the total circuit voltage.
In our case of eight identical bulbs in series across a 110 V line, the voltage division principle allows us to calculate that each bulb has a voltage of \( \frac{110}{8} = 13.75 \, \text{V} \) across it. This calculation assumes no other components or losses in the circuit, making it ideal for clear demonstrations like in the textbook problem.
Understanding how voltage is divided in series circuits can be advantaged in designing circuits or troubleshooting. Regardless of how complex a circuit becomes, the voltage division rule always hinges on the basic ratio principle, facilitating the calculation and distribution of voltages among components in series.