Courses
Courses for Kids
Free study material
Free LIVE classes
More

Relation Between Resistance and Length

Resistance and Length Relation

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’.


(Image will be uploaded soon)


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:


(Image will be uploaded soon)


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.

FAQs on Relation Between Resistance and Length

1. How can you Derive the SI Unit of Resistivity?

We know that the formula for the resistivity is:

⍴ = RA/L

The unit of the resistance is Ohm, the length is meter, and the area of cross-section is ‘m2'. So, putting these values:

= Ohm m2/m = Ohm.m.

So, the unit of resistivity is Ohm.m.

2. What is the Relationship Between Length and Resistivity?

We define length as the physical dimensional measurement of extension between any two given points. The relation between length and resistivity is given by the resistivity formula, i.e, ⍴ = R/L. Resistance varies directly with the length of the wire. It means that any variation in the length of the material will change the value of resistance (or resistivity).

3. What is the Relationship Between Resistance and Charge?

We can find a direct relationship between the amount of resistance encountered by the charge and the length of wire it must pass through.


Since the resistance occurs because of collisions between charge carriers (electrons) and the so there is likely to be more collisions in a longer wire. More collisions mean more resistance to the charge flow.

4. What Determines the Resistance of a Wire?

The resistance of a wire can be determined on  the following four factors:

  • The resistance is inversely proportional to the cross-sectional area. When the cross-sectional area of a wire is larger, the number of electrons experiencing the electric slope is also greater. As the length of the wire and the potential gradient do not change, the drift velocity of electrons also does not change. So, while the electrons don’t move faster, the bigger cross-sectional area causes more of them to move, increasing the total charge movement and the current flow. In short, the resistance decreases. 

  • Wire’s Length: The longer the wire the fewer volts each centimeter of it will get. This means that the electric slope which makes the electrons move gets less steep as the length of the wire increases. Also, the average drift velocity of electrons decreases. 

  • The Resistivity of Material: Resistance depends on the material of the wire as if the atom is held tightly then it is harder to make a current flow. 

  • The Temperature of the Wire: Resistance increases as the temperature of the wire rises. A hotter wire has larger resistance because of increased vibration of the atomic lattice. This will be difficult for the electrons to move without interaction with an atom, in turn increasing resistance. 

5. What is current?

An electric current is a  charged particle moving through an electrical conductor or space. The charge that is flowing in a wire is carried by the electrons that orbit the atoms of the wire. When the potential difference is applied the electrons drift in one direction. The higher potential difference will cause the faster the electrons move and there will be more charge in the given time. Thus, creating a bigger current. Current is measured as the net rate at which electric charge flows through a surface. 

Comment