Parallel Circuit: Current with 4 vs 3 Resistors

[Durfys]

Hey I came across this question in my textbook and the answer provided just does't make sense.

A circuit consists of a battery and 4 identical resistors in parallel. The current in each resistor is 12 A. If one of the resistors burns out, what will be the current in each of the remaining 3 resistors?

Book's explanation:
The battery remains the same and will provide the same voltage both with the 4 resistors and with the 3 resistors. Since the resistors are in parallel, the voltage drop on each must be the same as the voltage provided by the battery, and will be unchanged when one resistor burns out. Since both V and R are unchanged for each resistor by the burned out resistor, I is also unchanged for each resistor, and the current in each resistor is still 12 A. Note that what does change is the total current on the circuit, from 48 A with 4 resistors (4 × 12 A) to 36 A with 3 resistors (3 × 12 A). Since the resistors are in parallel, fewer resistors will mean a higher equivalent resistance, so the total current is reduced, even though the current on each resistor is unchanged.

I don't understand how the total current could change without any change to a single resistor's current. Can someone please clarify. Thanks very much in advance.

 

[azizlwl]

Let total resistance of parallel circuit is R_t.

With 4 identical resistors of resistance RΩ,
1/R_t =1/R₁+ 1/R₂+ 1/R₃+ 1/R₄
R_t=R/4

With 3 identical resistors of resistance RΩ,
R_t=R/3

 

[BruceW]

I don't understand how the total current could change without any change to a single resistor's current. Can someone please clarify. Thanks very much in advance.

Hi, welcome to physics forums Durfys! Normally on this forum, people need to show their attempt at figuring out the problem first, and write that down, so that we know how to help. But this is your first post, so maybe you didn't know that. No worries.

I'm not certain what you're stuck on. Is it this sentence from the book: Since the resistors are in parallel, fewer resistors will mean a higher equivalent resistance, so the total current is reduced, even though the current on each resistor is unchanged. Um, It is true that the current on each resistor is unchanged, but one of the resistors has been removed, due to it being burned out! So the circuit does change. Maybe the sentence in the book made it seem like the circuit did not change at all. But the circuit does change.

 

[Durfys]

Hey, sorry about that. Thank you very much for the replies, after spending more time thinking about the question I now understand what I missed.

 

[Nickie]

I can understand your confusion with the explanation provided in the textbook. Let me try to clarify it for you.

Firstly, in a parallel circuit, the total current is divided among the different branches of the circuit. This means that the current in each resistor is not dependent on the other resistors in the circuit. Therefore, even if one resistor burns out, the current in the remaining resistors will not change.

Secondly, when one resistor burns out, the total resistance in the circuit increases. This is because there are now fewer paths for the current to flow through. This increase in resistance leads to a decrease in the total current in the circuit.

To better understand this, let's look at an analogy. Imagine you have a water pump connected to four different pipes, each with the same diameter and length. The pump is able to push a certain amount of water through each pipe. But if one of the pipes suddenly gets clogged, the pump will still be able to push the same amount of water through the remaining three pipes. However, since there is now one less pipe for the water to flow through, the overall flow of water will decrease.

Similarly, in a parallel circuit, when one resistor burns out, the current in the remaining resistors will remain the same, but the total current in the circuit will decrease due to the increase in resistance.

I hope this helps to clarify the concept for you. Remember, as a scientist, it is important to think critically and question explanations that do not make sense to you. Keep up the good work!