Question

A hollow cylinder \[(\rho = 2.2 \times {10^{ - 8}}\Omega - m)\] of length $3\,m$ has inner and outer diameter as $2\,mm\,and\,4\,mm$ respectively. The resistance of the cylinder is:
A. $0.35 \times {10^{ - 3}}\Omega $
B. $3 \times {10^{ - 3}}\Omega $
C. $7 \times {10^{ - 3}}\Omega $
D. none of these

Answer 1

Hint:In order to this question, to calculate the resistance of the cylinder, first we will rewrite the given facts and then we will find the outer and inner radius (as outer and inner diameter is given). Then we will apply the formula of resistance in terms of resistivity, area and length.

Complete step by step answer:
Given that, Outer diameter $ = 4mm$
Let the outer radius be $R$ .
$R = \dfrac{4}{2} = 2mm$
Again, inner diameter (given) $ = 2mm$
Let the inner radius be $r$ .
$r = \dfrac{2}{2} = 1mm$
Length of hollow cylinder, $l = 3m$
\[\rho = 2.2 \times {10^{ - 8}}\Omega - m\] (given)

We have to find the resistance of the given cylinder:
So, as we know the formula to find the resistance of the cylinder-
$\text{resistance} = \text{resistivity}(\dfrac{\text{length}}{\text{area}})$
In the above formula, to find the resistance, we have to first find the area of the cylinder, and we already have the value of resistivity(constant) and the length.
$\text{Area of hollow cylinder} = \pi ({R^2} - {r^2})$
$\Rightarrow \text{Area of hollow cylinder} = \pi ({2^2} - {1^2}) \\
\Rightarrow \text{Area of hollow cylinder} = 3.14 \times 3 \times {10^{ - 6}} \\
\Rightarrow \text{Area of hollow cylinder} = 9.42 \times {10^{ - 6}} \\ $
And, we have value of resistivity, \[\rho = 2.2 \times {10^{ - 8}}\Omega - m\]
Now, we can find the resistance of the given cylinder:-
$\text{resistance} = \text{resistivity}(\dfrac{\text{length}}{\text{area}}) \\
\Rightarrow R = 2.2 \times {10^{ - 8}}[\dfrac{3}{{9.42 \times {{10}^{ - 6}}}}] \\
\Rightarrow R = 7 \times {10^{ - 3}}\Omega $
Therefore, the required resistance of the given hollow cylinder is $7 \times {10^{ - 3}}\Omega $ .

Hence, the correct option is C.

Note: We can also compare the resistance of cylinders based on the length of the cylinders. The longer the cylinder is, the more collisions between charges and its atoms will occur. Hence there will be more resistance for the large cylinders.