Light Bulbs in Series and in Parallel. Two light bulbs have constant resistances of 400Ω and 800Ω. The two light bulbs are now connected in parallel across the 120-V line. Find (e) the power dissipated in each bulb.

Verified step by step guidance

1. Step 1: Calculate the total resistance of the two bulbs when connected in parallel. Use the formula for resistors in parallel: \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}\), where \(R_1 = 400\Omega\) and \(R_2 = 800\Omega\).
2. Step 2: Determine the total current flowing through the circuit using Ohm's Law. Apply the formula \(I = \frac{V}{R_{total}}\), where \(V = 120V\) is the voltage of the line.
3. Step 3: Calculate the current through each bulb. Since the voltage across each bulb in a parallel circuit is the same, use Ohm's Law for each bulb separately: \(I_1 = \frac{V}{R_1}\) and \(I_2 = \frac{V}{R_2}\).
4. Step 4: Calculate the power dissipated by each bulb using the power formula \(P = IV\), where \(I\) is the current through the bulb and \(V = 120V\) is the voltage across it.
5. Step 5: Substitute the values of current from Step 3 into the power formula for each bulb to find the power dissipated in each.

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 as V = IR, which is fundamental for analyzing electrical circuits, including those with resistors in series and parallel.

Power in Electrical Circuits

The power (P) dissipated in an electrical component can be calculated using the formula P = IV, where I is the current through the component and V is the voltage across it. In the context of resistors, power can also be expressed as P = I²R or P = V²/R, allowing for the calculation of power based on either current or voltage, depending on the circuit configuration.

Resistors in Parallel

When resistors are connected in parallel, the total or equivalent resistance (R_eq) can be found using the formula 1/R_eq = 1/R1 + 1/R2 + ... + 1/Rn. This configuration allows for multiple paths for current to flow, resulting in a higher total current and lower equivalent resistance compared to individual resistances. The voltage across each resistor in parallel remains the same, which is crucial for calculating the power dissipated in each bulb.