The circuit at right shows four identical bulbs connected to an ideal battery. a. Rank the bulbs according to brightness, in order from brightest to dimmest. If two bulbs have the same brightness, indicate that explicitly. Explain how you determined the ranking of the bulbs
b. Suppose that a switch has been added to the circuit as shown. The switch is initially closed. When the switch is opened, will the current through bulb A increase, decrease, or remain the same? Explain how you determined your answer.
The Correct Answer and Explanation is :
a. Ranking the Bulbs by Brightness
In a circuit with four identical bulbs connected to an ideal battery, the brightness of each bulb depends on the power dissipated, which is determined by the current flowing through each bulb. The power dissipated by a bulb is given by ( P = I^2 R ), where ( I ) is the current through the bulb and ( R ) is its resistance. Since the bulbs are identical, their resistances are equal. Therefore, the brightness ranking is determined by the current through each bulb.
Ranking from Brightest to Dimmest:
- Bulb A: If Bulb A is directly connected to the battery, it will have the maximum current flowing through it, making it the brightest.
- Bulb D: If Bulb D is also directly connected to the battery, it will have the same current as Bulb A, making it equally bright.
- Bulb B and Bulb C: If Bulb B and Bulb C are connected in series with each other, the total current through them will be less than that through Bulbs A and D. In a series circuit, the same current flows through all components, but the total resistance is the sum of individual resistances. Since the bulbs are identical, the total resistance of the series combination is ( 2R ), leading to a lower current and thus less brightness. Therefore, Bulb B and Bulb C will have the same brightness, which is dimmer than that of Bulbs A and D.
Explanation:
In a parallel circuit, each component has the same voltage across it. If Bulbs A and D are connected directly to the battery, they each experience the full battery voltage, resulting in higher current and brightness. Bulbs B and C, being in series, share the battery voltage, leading to a lower voltage across each bulb and consequently a lower current and brightness.
b. Effect of Opening the Switch on Bulb A
When a switch is added to the circuit and initially closed, it provides an alternative path for current. If the switch is opened, the current through Bulb A will increase.
Explanation:
When the switch is closed, it offers a parallel path for current, which divides the total current between the switch and Bulb A. Opening the switch removes this parallel path, causing all the current to flow through Bulb A. According to Kirchhoff’s Current Law, the total current entering a junction equals the total current leaving the junction. With the switch open, the current that would have flowed through the switch now flows through Bulb A, increasing the current through it. Since the power dissipated by a bulb is proportional to the square of the current (( P = I^2 R )), the increased current results in greater power dissipation and thus increased brightness of Bulb A.
This analysis assumes that the battery can supply the increased current without significant voltage drop, and that the resistance of the switch is negligible when closed.
For a visual explanation of how a switch affects the current through bulbs in a circuit, you might find the following video helpful:
videoFour Light Bulbs Circuit with Switch – Effect of Closing Switchturn0search5