Question

What is the effect of the length on the resistance of a conductor? How does the resistance of a wire change when its length is

b.halved?

Answer 1

Step 1- Relationship between resistance and length of the conductor

The resistance $R$ is directly proportional to the length $l$ of the wire.

$R=\frac{\rho l}{A}$

Where $\rho$ is the resistivity and $A$ is the cross-section area of the wire,

Hence, the resistance of a conductor increases with an increase in length, and the resistance of a conductor decreases with a decrease in length.

Step 2- Finding the formula for the resistance after the length of the conductor is changed

Before the change in the length of the conductor, the above expression is reduced to

$R_1=\frac{\rho l_1}{A}...\left(1\right)$

Where $R_1$ is the resistance of the conductor before the change and $l_1$ is the initial length of the conductor.

After the change in the length of the conductor, the above expression is reduced to

$R_2=\frac{\rho l_2}{A}...\left(2\right)$

Where $R_2$ is the resistance of the conductor after the change and $l_2$ is the final length of the conductor.

Divide equation $\left(2\right)$ by equation $\left(1\right)$ to obtain the expression for $R_2$.

$$\begin{gathered} \frac{R_2}{R_1}=\frac{{\displaystyle \frac{\rho l_2}{A}}}{{\displaystyle \frac{\rho l_1}{A}}} \\ {R}_2=R_1\left(\frac{l_2}{l_1}\right)...\left(3\right) \end{gathered}$$

Step 3- (b) Finding the value of resistance after the length of wire is halved

Here, the length of a conductor is halved so, $l_2=\frac{1}{2}{l}_1$

Substitute $\frac{1}{2}{l}_1$ for $l_2$ into eqiuation $\left(3\right)$.

$$\begin{gathered} R_2=R_1\left(\frac{{\displaystyle \frac{1}{2}}{l}_1}{l_1}\right) \\ {R}_2=\frac{1}{2}{R}_1 \end{gathered}$$

Hence, the resistance also gets halved when the length of a wire is halved.