Resistance of wire vs wire with wide middle

[User29013]

Homework Statement

There are two wires:
Wire one looks like --=========--
Wire two looks like ---------------
Do they have the same resistance?

So this isn't actually one of my homework problems, but I'm just debating this with a friend and neither of us can come to an agreement. So here are the arguments.
1: The wires have the same resistance
2: The top wire has less resistance than the bottom wire

Homework Equations

The middle portion of the top wire has less resistance due to larger cross sectional area
The resistance of wires doubles as the length doubles

The Attempt at a Solution

For position 1: Because the wire on the top has thinner ends, the flow of electrons is limited to however many can flow through the thin end, making the thicker middle section of the wire useless, like a pipe with water flowing through it. So therefore, the resistances of the wires are equal.

For position 2: The middle section of the top wire has less resistance than a thin piece of wire of equal length. If you add up the resistance of the short wires on either side plus the middle like resisters in series, it would be less than adding up the whole length of the bottom wire.

 

[davenn]

if this is just standard copper wire and the short lengths or thinner bits are really short, less than a few inches or so the difference in resistance of those thin bits isn't likely to be measurable.

Dave

 

[User29013]

if this is just standard copper wire and the short lengths or thinner bits are really short, less than a few inches or so the difference in resistance of those thin bits isn't likely to be measurable.

Dave

Let's say the wires are magnified or we have the equipment to measure very precisely, would the resistance of the top wire be less? I believe it is less, but my friend keeps using a water pipe as an analogy and that the thin wire (pipe) will limit the flow of electrons. I say that this isn't the same as water because the longer the thin wire is, the more resistance it will have so the water analogy would not be applicable.

 

[davenn]

...

The Attempt at a Solution

For position 1: Because the wire on the top has thinner ends, the flow of electrons is limited to however many can flow through the thin end, making the thicker middle section of the wire useless, like a pipe with water flowing through it. So therefore, the resistances of the wires are equal.

this is correct, it will be really noticeable when you want to increase the current flow through the thin sections
(this is the basis of a fuse design) ... The fuse wire is a thinner section of wire that cannot handle ( carry) the same amount of current as the rest of the wiring. So it will fuse with a given amount of current before the wiring of the rest of the circuit ( well that's the objective, but everyone who has dabbled in electronics for more than a few years will know that a transistor will blow long before the fuse that is protecting it haha)

For position 2: The middle section of the top wire has less resistance than a thin piece of wire of equal length. If you add up the resistance of the short wires on either side plus the middle like resisters in series, it would be less than adding up the whole length of the bottom wire.

yes that is true, the middle section is lower resistance, Yes lower total resistance

the effect of this on the circuit really depends on the overall lengths of the wires and the voltage and current the circuit is operating on.

For a given length of wire at a given thickness and voltage, as the current is increased through that wire, even a relatively small resistance will start to cause a voltage drop.


Practical example ( approximate figures)
My high power radio transmitter for 100W output, the radio draws 18 Amps at 13.8V DC
The difference between a 5 ft. lead and a 10ft. lead between the power supply and the radio can result in a 1V voltage drop difference

cheers
Dave

 

[User29013]

Thanks for the replies! This debate is actually for a real life application. We're just building a caveman circuit; all it is is a coil being shorted out to heat a wick. I'm thinking of using copper bars to connect the terminals of the battery (don't worry, they're rated high enough for this) to an adapter (with low surface area to connect to) that holds the coil. I said the bars lower overall resistance to get the coils to heat up faster/hotter, but he's saying that the bottleneck is in the small surface area of the adapter and the current is limited anyways. So the bars won't do anything and we can just use a wire of the same surface area as the adapter.