Question

The dimensions of the block are \[1\,cm \times 1\,cm \times 100\,cm\] . If the specific resistance of its material is \[3 \times {10^{ - 7}}\Omega m\] , then the resistance between the opposite rectangular face is
A. $3 \times {10^{ - 9}}\Omega $
B. $3 \times {10^{ - 7}}\Omega $
C. $3 \times {10^{ - 5}}\Omega $
D. $3 \times {10^{ - 3}}\Omega $

Answer 1

Hint: We are given a cuboid with its dimensions and its specific resistance. To find the value of the resistance between the opposite rectangular faces of the given block we need to use the relationship between resistance, resistivity, length and area of cross section of the given block and substitute the known values in it.

Formula used:
Area of rectangle, $A = l \times b$
where $l$ is the length and $b$ is the breadth of the rectangle.
And formula of resistance is,
$R = \dfrac{{\rho l}}{A}$
where $\rho $ is the resistivity and $R$ is the resistance of the given material.

Complete step by step solution:
Given: dimensions of the given block is \[1\,cm \times 1\,cm \times 100\,cm\]
And specific resistance (or resistivity) is, $\rho = 3 \times {10^{ - 7}}\Omega m$
Since we need to find the resistance between the opposite rectangular faces so the area of cross section would be $A = 1 \times 100\,c{m^2}$.

Converting length and area into SI unit, we get
$l = {10^{ - 2}}m$ and $A = {10^{ - 2}}m$ (since $1\,cm = {10^{ - 2}}\,m$ )
Now we know that,
$R = \dfrac{{\rho l}}{A}$ . . .(1)
Substituting all the values in equation (1), we get,
$R = \dfrac{{\left( {3 \times {{10}^{ - 7}}} \right)\left( {{{10}^{ - 2}}} \right)}}{{{{10}^{ - 2}}}} \\ $
This gives,
$R = 3 \times {10^{ - 7}}\Omega $

Hence, option B is the correct answer.

Note: While solving this question one must carefully notice whether the resistance is asked between the square faces or the rectangular faces of the block because the value of length and the area of cross section of the block would be different for the two cases. As a result they might give different values of resistance.