Question

If l₁ and l₂ are the two length of same material and same area of cross-section such that R₁ and R₂ are their respective resistances then derive relation between R₁, R₂, l₁, l₂.

Answer 1

The resistance of a conductor is given by:

\[R=\rho\frac{l}{A}\]

where ρ is the resistivity, (l) is its length and (A) is its area of cross-section.

For the first conductor:

\[R_1=\rho\frac{l_1}{A}\]

For the second conductor:

\[R_2=\rho\frac{l_2}{A}\]

Since both conductors are of the same material and have the same area of cross-section, ρ and (A) are the same.

Dividing the two equations:

\[\frac{R_1}{R_2}=\frac{l_1}{l_2}\]

Hence, the required relation is \[{\frac{R_1}{R_2}=\frac{l_1}{l_2}}\] or equivalently, \[{R_1l_2=R_2l_1}\]

Thus, resistance is directly proportional to length when the material, cross-sectional area and temperature remain the same.