Question

A steel trolley-car rail has a cross-sectional area of $56.0\,cm^2$. What is the resistance of \[10\,km\] of rail? The resistivity of the steel is \[3.00 \times {10^{ - 7}}\Omega m\]

Answer 1

Hint: The length, cross – sectional area and the resistivity of steel is already given to us. Using the formula for relation between resistance and resistivity, we could easily find out the resistance of the said rail.

Formulae used:
$R = \rho \dfrac{l}{A}$
where $R$ is the resistance of the object, $\rho $ is the resistivity of the object, $l$ is the total length of the object and $A$ is the cross sectional area of the object.
Useful Conversions :
For converting $c{m^2}$ to ${m^2}$ , we use $1c{m^2} = 1 \times {10^{ - 4}}{m^2}$
For converting $km$ to $m$ , we use $1km = 1 \times {10^3}m$

Complete step by step answer:
When an electric current flows through a bulb or any conductor, the conductor offers some obstruction to the current and this obstruction is known as electrical resistance and is denoted by R. Every material has an electrical resistance and this is the reason why conductors give out heat when current passes through it. According to the given question,
Cross – sectional area of the trolley – car rail $A = 56\,c{m^2}$
After conversion to SI unit, the cross – sectional area becomes
$A = 56 \times {10^{ - 4}}{m^2}$

The total length of the rail for which the resistance is to be calculated is $l = 10\,km$.After conversion to SI unit, the total length becomes $l = 10 \times {10^3} = {10^4}$
Resistivity of steel rail $\rho = 3.00 \times {10^{ - 7}}\Omega m$
Using the formula $R = \rho \dfrac{l}{A}$ to equate resistance with resistivity, we get
$R = \rho \dfrac{l}{A} \\
\Rightarrow R = 3.00 \times {10^{ - 7}} \times \dfrac{{{{10}^4}}}{{56 \times {{10}^{ - 4}}}}\Omega \\
\Rightarrow R = \dfrac{{30}}{{56}}\Omega \\
\therefore R= 0.536\Omega \\ $
Therefore, resistance of the $10km$ long steel rail is found to be $0.536\Omega $.

Note:Many students commit the blunder of not converting the respective units used in the question to SI units. This practice might lead to correct answers when all of the units are equivalent to each other. But, it mostly leads to wrong results. So, students should make it a point to convert all values to SI units before solving a question given to them.