Why does an area increase cause a resistance decrease and a length increase cause a resistance increase?

Question

Please read this:

Source.

How does the resistance of a wire depend on its length?

... The analogy used is that of a crowded place.

If you want to move through a crowd, if there is more space, then it’s easier (hence inversely related to A), and if the crowd is ‘longer’ it takes more time to make it across.

The first part makes sense, more space, means easier to move through, but I always had this problem with the second part of the analogy. It just didn’t make sense.

Let me be more specific.

Current is proportional to the speed of the charged particles. So in the crowd analogy, the current is proportional to the speed at which you can make it through. What does your speed have anything to do with the length of the crowd? ... If you think of it this way, the resistance shouldn’t depend on the length at all. How could the material start resisting more at one place, just by adding another wire at some other place? ...

The key is to understand that charges are being pushed through the wire due to an electric field. When the wire is short, the electric field is more concentrated and hence stronger. If the wire is doubled in length, the electric field gets more diluted and reduces to half its strength.

This is why increasing the length of wire, reduces the speed of the charges and hence translates to increasing the resistance of the wire. ... Let’s get back to that crowd and imagine you and your friends are trying to make it through the crowd. The important thing is there is someone externally providing energy to you and your friends to move through the crowd. Now only those of you need energy who are within the crowd length. Your friends outside the crowd length don’t need any energy as there is no opposition.

Now if the crowd length doubles, then twice the number of people are trying to make it through the crowd. But the external total energy remains the same. And so when distributed, each of you gets half as much as you got before, and hence you end up moving slower. ... And this why resistance increases with length because for a given voltage, lesser energy is transferred to each electron.

I found this answer to be most satisfying for answering of why current decrease when length increase, but if we extend this analogy or explanation towards area, than in area increase there should be atom increase, i.e electron increase, that should lead to high resistance according to above link answer.

Many say that when the area increases, the space to move increases. That gives a low resistance. Don't you think area increase means electron increase so how could there be more space? Really no logic? They are seeing it as amount of electrons remain the same just the area increases. That is not possible. The more metal, the more free electrons?

Could anyone tell me where I am wrong or give a satisfying explanation for both area, length concept all together?

Answer 1

This might be a case where the water analogy is useful.

Imagine a pipe. Voltage is analogous to pressure, so let's say that is fixed. The volume of water that flows is analogous to the current, so the ratio of pressure over volume is analogous to resistance.

The larger the cross-sectional area of the pipe, the more water flows (like a resistance decrease).

The longer the pipe, for a given cross-sectional area, the less water flows (like a resistance increase).

Answer 2

Think of the resistance as something like the number of collisions between the charge carriers and the atoms of the compound/material of the resistor: The longer the material, the more collisions, hence the higher resistance you get. The resistivity (\$\rho\$) which has a unit of \$\mathrm{\Omega\cdot m}\$ may give you a clue about this.

If you increase the cross-sectional area of the resistor, you just decrease the possible number of collisions: The lesser the collisions, the lower resistance you get.

Answer 3

Answering because the comments got long.

If the wire is doubled in length, the electric field gets more diluted and reduces to half its strength ...

Could u explain more what actually "dilution" of electric field u meant?

What the originator of the analogy probably means by dilution is the reduction in electric field when length increases since E = V / L. This of course assumes that V is fixed.

Assuming V is fixed, when the area increases, the electric field doesn't get "diluted". Electric field would be potential difference / length; area doesn't come into picture. When length increases the electric field for the same potential difference would be smaller ("diluted"). So, even if you increase the area, the electric field remains the same^*.

Doesn't electric field depend on electron amount increase as happen in BOTH Length as well as AREA?

The circuit that we must imagine (I assume) while reading this analogy, is a cell or battery connected to a resistive material. The battery fixes the potential difference between the resistor terminals (e.g. 1.5 V AA cell).

The length of the resistor then fixes the electric field inside the resistor (potential difference imposed by the battery divided by length of the resistor).

The area doesn't affect either the V or the E^*.

The number of free electrons also doesn't affect V or E^*.

Doesn't electric field depend on electron amount ...

No^*.

1. If you are talking about free electrons in the resistive material that contribute to it being a (bad) conductor, then remember that their charge and electric field are neutralized by the positive charge of nucleus of the atoms from which they come from.
2. If you are talking about excess electrons (e.g. that we learn in electrostatics), remember that such isolated / excess electrons are not present in most circuits^*.
3. So the cell / battery is the only source of electric field / potential difference^*.

^* There are likely a lot of assumptions left unsaid.

Answer 4

Resistance depends on an object’s size, shape, and material. Electric current flows when electrons move through a conductor, such as a metal wire. The moving electrons can collide with the ions in the metal. This makes it more difficult for the current to flow, and causes resistance.

The resistance of a long wire is greater than the resistance of a short wire because electrons collide with more ions as they pass through. The relationship between resistance and wire length is directly proportional.

The resistance of a thin wire is greater than the resistance of a thick wire because a thin wire has fewer electrons to carry the current. The relationship between resistance and the area of the cross section of a wire is inversely proportional.