Question

A copper wire of resistance $R$ is pulled so that its length is doubled (temperature is constant). Find the resistance of the wire in terms of its original resistance $R$.

Answer 1

Resistance offered by a material is dependent on its resistivity which is unique for each material. In-order to find the change in resistance the formula for the resistance in terms of its resistivity, area of cross section and length of wire is applied. When there is a change in length there is a change in area as well and this is found out and substituted to determine the new resistance.

Formula used:
The resistance of a material is given by the equation:
$R = \rho \dfrac{l}{A}$
Where, $l$ is the length, $A$ is the area of cross section and $\rho$ is the resistivity.

Complete step by step answer:
We are asked to find the new resistance value when its length of the copper wire is changed.To find this out we first look at the concept of resistance. Based upon the nature of a conductor a quantity known as the resistance came into existence. Each material has a value of resistance associated with it based on its ability to conduct or in other words its ability to pass current through its material.

Hence the resistance of a material is defined as the property by virtue of which opposes or hinders the flow of current or the flow of charges through it. This was determined from the ohm’s law from the concept that voltage is directly proportional to current. The value of resistance varies from material to material. Here, in this problem a copper wire is taken into consideration which means that this wire will have a low value of resistance since copper is said to be a good conductor.

Even though the material is a conductor it will offer some amount of resistance. This resistance value is dependent on a quantity known as resistivity. The resistance is independent of the voltage and current of a circuit but depends mainly on the nature of the conductor and its physical quantities such as its length and area of cross section.Since it was dependent on the length and area, that is, directly proportional to length and inversely proportional to the area of cross section a new constant of proportionality was obtained which was known as the resistivity. The equation relating the two quantities is given by:
$R = \rho \dfrac{l}{A}$ ------($1$)
Here $\rho$ denotes the resistivity.

The resistivity or the specific resistance of a material is defined as the resistance of a conductor of that material having a unit length and area of cross section or in other words is known as the resistance offered by a unit cube of the material. It is basically a quantity that specifies the amount of resistance offered by every unit of the material. Knowing this, let us extract the data given in the question. It is given that the length is doubled. Hence the following equation is made:
${l_1} = 2l$ ------($2$)
Where, ${l_1}$ denotes the change in length, that is, the new length while $l$ is the original length of the copper wire.

However, when there is a change in length there will be a change in area of the wire as well. We now need to determine this change in area. In order to do this we must construct an equation relating the area of the length of the wire. The area relates the length by the following equation:
$A = \dfrac{V}{l}$ -----($3$)
Where, $V$ is the voltage and $l$ is the length.

Based upon the above equation let us construct the equation for the new value of area:
${A_1} = \dfrac{V}{{{l_1}}}$
Now, we substitute equation ($2$) in the above equation to get the change in area:
${A_1} = \dfrac{V}{{2l}}$
$\Rightarrow {A_1} = \dfrac{1}{2}\left[ {\dfrac{V}{l}} \right]$
Putting the value from equation ($3$) we get:
${A_1} = \dfrac{A}{2}$ ------($4$)
Hence we can clearly see that when the length is doubled the area is halved. The resistivity is independent of the length and the area of cross section values but however it is dependent on the temperature of the material. Since, the temperature is said to be constant in the question, there will be no change in resistivity and its original value is retained.

The new changed value of resistance is given by the equation:
From equation ($1$):
${R_1} = \rho \dfrac{{{l_1}}}{{{A_1}}}$ -----($5$)
Where, ${R_1}$ denotes the new change in resistance that we are required to find.
We are required to find the new resistance in terms of its original resistance.
By substituting equations ($2$) and ($4$) in equation ($5$) we get:
${R_1} = \rho \dfrac{{2l}}{{\left( {\dfrac{A}{2}} \right)}}$
By simplifying we get:
${R_1} = \rho \dfrac{{2 \times 2 \times l}}{A}$
$\Rightarrow {R_1} = \rho \dfrac{{4l}}{A}$
$\Rightarrow {R_1} = 4\left[ {\dfrac{{\rho l}}{A}} \right]$
By substituting the equation ($1$) in the above equation we get:
$\therefore {R_1} = 4R$
Hence the new resistance changes by four times its original resistance.

Note: The resistivity of a material is the quantity which was derived from the concept of the factors which the resistance was dependent on. However, the electrical resistivity of materials varies over a wide range and has a specific value for each set of substances. The resistivity of conductors are very low while the resistivity of insulators are relatively high whereas the semiconductors have resistivity values ranging between the conductors and insulators.