So, will the bulb light up in this parallel circuit with a zero ohm resistor?

[Kyoma]

For example, you have an electrical circuit, which consists of a battery of 6V, a zero ohm resistor and a light bulb of 2 ohms that are connected in parallel. Assume that the battery has no internal resistance and that the wire has no resistance, will the bulb light up?

At first, my answer is 'no' since all of the current would go through the zero ohm resistor, instead of going through the bulb. However, according to my textbook, each branch of the parallel circuit experiences the same voltage as the battery, which is 6V, so thus, the light bulb would experience 6V, and it would light up.

So, which is which?

 

[tiny-tim]

Hi Kyoma!

Since there's no such thing as a zero resistance, let's assume it is 10^-6 Ω.

Then the power going through the bulb is V²/R = 18 W, but the power going through the wire is 36 MW !

The wire will burn out rather fast!

So yes you're correct, most of the current is going through the wire, but the same current is going through the bulb, no matter what is in parallel with it.

(Why doesn't that usually happen to wires? Because the voltage across a small section of wire is usually extremely small … unless you're silly enough to put something in parallel with it! )​

 

[Evil Bunny]

In theory, the bulb would light. In practice, it won't.

The battery probably doesn't have the capacity to handle a dead short and light the bulb. As soon as you short it out with the wire your light will go out. The theory works if you have enough power available.

 

[BackEMF]

The theory cannot answer this case. What you have asked to solve is a paradox! Let me explain in more detail:

Your ideal voltage source demands that there is 6 V across it.
Your ideal short circuit demands that there is 0 V across it.

This situation is not solvable as both of these demands cannot be met simultaneously.

However in practise the battery will have an internal resistance, your short circuit will have a much much smaller resistance, hence most of the voltage will be dropped in the batteries internal resistance and no bulb will light.

 

[tiny-tim]

However in practise the battery will have an internal resistance, your short circuit will have a much much smaller resistance, hence most of the voltage will be dropped in the batteries internal resistance and no bulb will light.

But the total resistance of the parallel pair (the bulb and the short circuit) will be only very slightly less than that of the bulb … why should that make any difference?

 

[nasu]

The problem is not in the zero resistance of the wire but in the zero internal resistance of the source.

IF the source had zero internal resistance, then the bulb will experience the same current no matter what you put in parallel to it. The current through the wire will be infinite but the voltage across it will be 6 V (zero resistance times infinite current can give any value) same as on the bulb and same as the voltage across the terminals of the battery.
The current through the source will be also infinite, but this is again OK. If the internal resistance is zero, 6 V are enough to produce infinite current.

Now in practice, the source has a finite resistance. This will limit the current even for a short circuit wire. If the wire has zero resistance, the short-circuit current will be E/r where r is the internal resistance. The voltage drop across the internal resistance will be r*I = E and the voltage across the terminals of the battery will be E-Ir= 0. So both wire and bulb will have zero voltage. The bulb will have zero current at zero voltage which is OK.

 

[BackEMF]

But the total resistance of the parallel pair (the bulb and the short circuit) will be only very slightly less than that of the bulb … why should that make any difference?

I'm not exactly sure what you mean here. The resistance of the parallel combination of the bulb and the short-circuit should be considerably less than the resistance of the bulb on its own. The exact value will be slightly less than the resistance of the short-circuit - and we can make short circuits that are extremely low resistance, particularly for DC circuits.

Making an ideal voltage source with a low-internal resistance on the other hand, would probably be quite a bit harder.

 

[Studiot]

Actually, in theory, the equivalent resistance of two resistors connected in parallel is less than either of the original two.

Thus the equivalent resistance of parallel combination of the zero ressitor and the light bulb is less than zero!

This type of reasoning shows the problems you can encounter if you insist on dividing by zero.

 

[BackEMF]

Actually, in theory, the equivalent resistance of two resistors connected in parallel is less than either of the original two.

Thus the equivalent resistance of parallel combination of the zero ressitor and the light bulb is less than zero!

This type of reasoning shows the problems you can encounter if you insist on dividing by zero.

Yes strange things have happened there! The parallel combination of two resistors is only less than either individually if no resistance is zero (or negative).

In my post above I was talking about the practical case where my "short-circuit" was an approximation to a short-circuit, and so it doesn't have zero resistance. I should have made that clear.