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We know that the resistance is the opposition created to the current flowing through the circuit. The resistance is prevention to the major disaster like short-cit 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 mid 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.

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 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.

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 R_{c}, 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) implied that on doubling the length, the resistance of the combined slabs, i.e., Rc becomes the double of the old resistance ‘R’.

Now, considering 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 that 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 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 equation (5) and (5), 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.

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

R ∝ L/A

Now, removing the sign of proportionality, and 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).

FAQ (Frequently Asked Questions)

Question 1: How you can Derive the SI Unit of Resistivity?

Answer: 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 ‘m^{2}'. So, putting these values:

= Ohm m^{2}/m = Ohm.m.

So, the unit of resistivity is Ohm.m.

Question 2: What is the Relationship Between Length and Resistivity?

Answer: 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).

Question 3: What is the Relationship Between Resistance and Charge?

Answer: We can find a direct relationship between the amount of resistance encountered by 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.

Question 4: What Determines the Resistance of a Wire?

Answer: The resistance of the wire can be ascertained by the following four factors:

Cross-sectional area.

Wire’s length.

Resistivity.

The temperature of the wire.