Light Bulbs in Series and in Parallel. Two light bulbs have constant resistances of 400Ω and 800Ω. If the two light bulbs are connected in series across a 120 V line, find the total power dissipated in both bulbs.

Verified step by step guidance

1. First, understand that when resistors (or light bulbs in this case) are connected in series, the total resistance \( R_{\text{total}} \) is the sum of the individual resistances. So, calculate the total resistance: \( R_{\text{total}} = R_1 + R_2 = 400\,\Omega + 800\,\Omega \).
2. Next, use Ohm's Law to find the total current \( I \) flowing through the circuit. Ohm's Law states \( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. Rearrange this to find \( I = \frac{V}{R_{\text{total}}} \), where \( V = 120\,\text{V} \).
3. Now, calculate the total power dissipated in the circuit using the formula for power: \( P = IV \). Substitute the current \( I \) you found in the previous step and the total voltage \( V = 120\,\text{V} \) into this formula.
4. Alternatively, you can use the formula \( P = I^2R_{\text{total}} \) to find the total power dissipated, where \( I \) is the current calculated earlier and \( R_{\text{total}} \) is the total resistance.
5. Finally, verify your calculations by ensuring that the power calculated using both methods (\( P = IV \) and \( P = I^2R_{\text{total}} \)) are consistent, which confirms the accuracy of your solution.

Key Concepts

Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance. It is expressed as V = IR, where V is voltage, I is current, and R is resistance. This law is fundamental for calculating the current in circuits with known voltage and resistance.

Series Circuit

In a series circuit, components are connected end-to-end, so the same current flows through each component. The total resistance in a series circuit is the sum of individual resistances. This concept is crucial for determining the equivalent resistance and current in the circuit, which are necessary for calculating power dissipation.

Power Dissipation in Resistors

Power dissipation in resistors is calculated using the formula P = I^2R, where P is power, I is current, and R is resistance. Alternatively, it can be calculated using P = V^2/R when voltage and resistance are known. Understanding this concept allows us to determine how much electrical energy is converted into heat in resistors, which is essential for finding the total power dissipated in the circuit.