Understanding the Relation Between Resistance and Length

We know that resistance is the opposition created to the current flowing through the circuit. The resistance is the prevention of major disasters like short-cut or high damage to the property.

However, the resistance has a good relationship with the length.

Let’s suppose that the resistance is a speed breaker and the speed of your vehicle is the current. Now, when the speed breaker is in the middle of the road, not on its ends. You will try to take out your speedy vehicle from the side of the road, hit by a vehicle, and meet an accident.

What is Resistance?

An electron traveling through the wires encounters resistance, which is basically a hindrance to the flow of charge. For a moving electron, the journey from one end to the other end is not a direct route, but a zig-zag path, because they collide with the ions in the conducting material. So, the electrons encounter a hindrance to their movement, making it difficult for the current to flow. This causes resistance.

While the electric potential difference between two terminals encourages the movement of charge, the resistance discourages it. The rate at which charge flows from one terminal to another is thus a result of the combined effect of these two factors.

Relationship Between Length and Resistance

In the above example, we discussed how length and resistance are related to each other. Now, let’s talk about it in detail.

Now, you encounter a road that has twice the speed breakers as that were earlier. Now, you will have to be very sure before you reach the edge of the speed breaker because at this time, your very high-speed vehicle will pass through many resistors (speed breakers) and your vehicle will slow down eventually.

So, mathematically, the equation can be expressed as:

R ∝ L ……(1)

You are driving your vehicle on the road and it is compulsory to cross the speed breakers because in front of you there is a big jam on the road. Now, if the length is less and instead of spreading these breakers by a distance, these are joined end-to-end, so what you observed here is, the area is halved but if you drive it fast, your vehicle will jump, again there is a risk.

So, here even if the length is lesser; however, the area is halved, still you have to be slow. It means the resistance is directly proportional even if the area is halved.

So, mathematically, we can write the equation as:

R ∝ 1/A ……(2)

Now, let’s understand the resistance length of wire in terms of Physics.

Relationship Between Resistance and Length of Wire

Let’s suppose that there are two conductors in the form of cuboidal slabs (they are identical in shape and size) joined end-to-end. Each of these has a length as ‘L’ and the area of cross-section as ‘A’.

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When the potential difference ‘V’ is applied across either slab, the current ‘I’ starts flowing. So, by Ohm’s law, we have the relation as:

ROLD = V/I….(3)

Where R is the resistance across conductors, which is the same in each and it is measured in Ohms. As these two conductors are placed side-by-side, so the total length becomes ‘2L’, while the current them becomes ‘I/2’ because if ‘I’ is the total current flowing through both conductors and ‘V’ is the same potential difference across the conductors, so each of these conductors gets ‘I/2’ current.

So, the new resistance of the combination is Rc, and mathematically, we derive our expression in the following manner:

      \[R_{c} = \frac{V}{I/2} = \frac{2V}{I}\]

Looking at equation (3), we find a unique relationship between the old resistance and the resistance of combination, which is as follows:

Rc =  2 ROLD …..(4)

Equation (4) implies that on doubling the length, the resistance of the combined slabs, i.e., Rc becomes the double of the old resistance ‘R’.

Resistance and Length of Wire

Now, consider the same two slabs again. Here, instead of placing them side-by-side, we place them one above the other. We can see this arrangement below:

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We can notice one thing here the length of each conductor remains ‘L’, however, the area of cross-section, i.e., ‘A/2’ instead of ‘A’ because the area of each conductor added to become ‘A’. One thing is common here and that the total current is ‘I’ across both the conductors, so across each conductor, again the current will be ‘I/2’.

Using Ohm’s law again, we get the equation as:

ROLD1 = V/I….(5)

Now, writing the equation for the resistance of the combination as:

        \[R_{p} = \frac{V}{I/2} = \frac{2V}{I}\] ……(6)

From equations (5) and (6), we get a new relationship as:

RP = 2 ROLD1 …..(7)

From equation (7), we can notice that on halving the area, the resistance doubles.

We came to the conclusion that on doubling the length and halving the area of cross-section, the resistance doubles in each case, which means we proved the relationships in equations (1) and (2). Now, we will find a new relationship, so let’s get started.

Relation Between Resistance and Length

Here, we will combine equations (1) and (3):

R ∝ L/A

Now, removing the sign of proportionality, we get the following resistance per unit length formula:

R = ⍴ L/A …..(8)

Or,

⍴ = RA/L

Here, ⍴ is called the proportionality constant or the resistivity or the specific resistance of the material conductor. It is measured in Ohm-m.

So, the resistance per unit length is also called the resistivity of the material (conductor).

⍴ = R/L (Where A is a constant value).

Other Factors that Affect Resistance

Resistance has a relationship with two other factors apart from the length. 

Cross-Sectional Area and Resistance: The cross-sectional area of the wires has a direct effect on the amount of resistance. Wider wires have a larger cross-sectional area, and the wider the wire, the less the resistance to the flow of electric charge. When all other variables are left unchanged, the charge will flow at higher rates through wider wires than through thinner wires. The resistance of a thinner wire is less than the resistance of a thick wire as the thin wire has fewer electrons to carry the current. Thus, the relationship between resistance and the area of the cross-section of a wire is inversely proportional.

Material and Resistance: Not all materials are equal in terms of their ability to conduct. Some materials offer less resistance to the flowing charge than others and that’s why they are better conductors. So, the conducting ability of a material depends on its resistivity. The resistivity depends on the material's electronic structure and temperature. Resistivity increases with rising temperature for most materials. 

Silver is not used in wires even if it is the best conductor because of its high cost. Copper and aluminum have a high conducting ability and are also among the least expensive materials. 

Summary

In short, the resistance in a wire increases as the length of the wire increases. A long wire has higher resistance than a short wire. This is because of the fact that in a long wire, the electrons end up colliding with a lot of ions as they pass through. So, the relationship between resistance and wire length is proportional.