Question

A light bulb is connected to a source of emf. There is a \(6.20 \mathrm{~V}\) drop across the light bulb, and a current of 4.1 A flowing through the light bulb. a) What is the resistance of the light bulb? b) A second light bulb, identical to the first, is connected in series with the first bulb. The potential drop across the bulbs is now \(6.29 \mathrm{~V},\) and the current through the bulbs is \(2.9 \mathrm{~A}\). Calculate the resistance of each light bulb. c) Why are your answers to parts (a) and (b) not the same?

Short answer

Answer: The difference in resistance values could be due to factors such as internal resistance of the voltage source, different power settings of the light bulbs when connected in series, or accuracy of the measured voltage and current values. Connecting the light bulbs in series affects how they share the voltage and current, ultimately influencing their individual resistance values.

Step-by-step solution

a) Find the resistance of the light bulb

Use Ohm's Law formula, V = IR, and solve for Resistance (R): R = V/I R = (6.20 V)/(4.1 A) R = 1.51 Ω The resistance of the light bulb is approximately 1.51 Ω.

b) Calculate the resistance of the two light bulbs connected in series

When two light bulbs are connected in series, the total resistance is the sum of their individual resistances (R_total = R1 + R2). Since the light bulbs are identical, R1 = R2 = R. The total resistance is R_total = 2R. Using Ohm's Law for the series circuit: V_total = I_total * R_total Now, substitute the given values for voltage and current, and the expression for total resistance from above: 6.29 V = 2.9 A * (2*R) Solve for R: R = (6.29 V)/(2.9 A * 2) R = 1.08 Ω The resistance of each light bulb in the series connection is approximately 1.08 Ω.

c) Explain the difference between the resistances in part a) and b)

The difference in resistance values could be due to a few reasons like the internal resistance of the voltage source, different power settings of the light bulbs when connected in series, or accuracy of the measured voltage and current values. One key aspect is that connecting the light bulbs in series can affect the way they share the voltage and current, which ultimately affects their individual resistance values.

Key concepts

Electrical Resistance

Electrical resistance is a measure of how much an object opposes the flow of electric current. Think of it like water flowing through a pipe: if the pipe is narrow or has obstructions, it is harder for the water to flow through, much like how resistance makes it difficult for current to flow through a material.

Using a formula called Ohm's Law, represented as \( V = IR \) — where \( V \) is voltage, \( I \) is current, and \( R \) is resistance — one can calculate the resistance of an object. In the light bulb exercise, resistance was determined by the formula \( R = \frac{V}{I} \), showing that resistance is directly proportional to voltage and inversely proportional to current. In a practical sense, if we increase the voltage or decrease the current, we observe a higher resistance.

In typical scenarios, factors that impact resistance include the material's nature (its resistivity), its temperature, length, and cross-sectional area. However, in the given exercise, the implied resistance changes when extra elements are introduced, indicating external factors affecting resistance, such as the circuit configuration.

Series Circuit

A series circuit is one in which components are connected in a single path. This setup ensures that the same electric current flows through all components. Picture this as a single track where runners (electrons) must follow one after the other; there's no alternative route.

In the context of our textbook problem, when two identical light bulbs are connected in series, they share the current and the total resistance is the sum of the two resistances (\( R_{\text{total}} = R1 + R2 \)). With identical bulbs, we then say \( R_{\text{total}} = 2R \). Understanding the impact of a series placement is essential as it determines not only the total resistance but also how voltage is divided among the components. It's like two friends splitting a pizza; they each receive an equal share of the pie, similar to the bulbs each getting an equal share of the total voltage.

Voltage Drop

The concept of a voltage drop refers to the reduction in voltage across a component in an electrical circuit. Imagine you have a certain amount of energy, and as you complete tasks throughout the day, that energy is used up. Similarly, as electric current moves through a circuit component, it loses some voltage due to the resistance of that component.

In the exercise, a voltage drop is observed across the light bulb. When one bulb is connected, it accounts for the whole voltage drop across itself. However, when a second bulb is added in series, the total voltage drop remains the same, but it must be shared between the two bulbs. This alteration in distribution can lead to an apparent change in resistance measured for individual bulbs, as addressed in parts (a) and (b) of the exercise. Hence, the exercise improvement advice is to understand that though the resistance remains constant for a given piece of material under the same conditions, the measured value may differ due to the circuit configuration and the way voltage is distributed.