Neglect the internal resistance of a battery unless the problem refers to it. Eight bulbs are connected in parallel to a 120-V source by two long leads of total resistance 1.4 Ω. If 210 mA flows through each bulb, what is the resistance of each, and what fraction of the total power is wasted in the leads?

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Determine the total current flowing through the circuit. Since there are 8 bulbs connected in parallel and each bulb draws 210 mA, the total current is given by:

I total = 8 \times 210 mA = 1.68 A

Calculate the resistance of each bulb. Since the voltage across each bulb is the same (120 V) and the current through each bulb is 210 mA, use Ohm's Law:

R = \frac{V}{I} = \frac{120}{}

. Simplify to find the resistance of each bulb.

Determine the total resistance of the circuit. Since the bulbs are connected in parallel, the equivalent resistance of the bulbs can be calculated using the formula:

\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{8}}

. Substitute the resistance of each bulb to find the equivalent resistance.

Calculate the power wasted in the leads. The total current flowing through the leads is 1.68 A, and the resistance of the leads is 1.4 Ω. The power wasted in the leads can be calculated using the formula:

P = I^{2} R

. Substitute the values of current and resistance to find the power wasted.

Determine the fraction of the total power wasted in the leads. First, calculate the total power supplied by the source using the formula:

P = V I

. Then, divide the power wasted in the leads by the total power supplied to find the fraction:

\frac{P}{P_{total}}

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 = I × R. Understanding this law is crucial for calculating the resistance of each bulb and the total current in the circuit.

Power in Electrical Circuits

The power (P) consumed by an electrical device is calculated using the formula P = V × I, where V is the voltage across the device and I is the current flowing through it. In this problem, calculating the power consumed by each bulb and the total power in the circuit is essential to determine how much power is wasted in the leads due to their resistance.

Parallel Circuits

In a parallel circuit, multiple components are connected across the same voltage source, allowing each component to operate independently. The total current in the circuit is the sum of the currents through each component. This concept is important for understanding how the bulbs share the voltage and how to calculate the total current flowing through the circuit, which affects the power distribution and losses in the leads.