Question

There are two concentric conducting spheres of radii $a$ and $b$ is filled with a medium of resistivity $\rho $. The resistance between the two spheres will be:
$\left( a \right).{{ }}\dfrac{\rho }{{4\pi }}\left( {\dfrac{1}{a} - \dfrac{1}{b}} \right)$
$\left( b \right).{{ }}\dfrac{\rho }{{2\pi }}\left( {\dfrac{1}{a} - \dfrac{1}{b}} \right)$
$\left( c \right).{{ }}\dfrac{\rho }{{2\pi }}\left( {\dfrac{1}{a} + \dfrac{1}{b}} \right)$
$\left( d \right).{{ }}\dfrac{\rho }{{4\pi }}\left( {\dfrac{1}{a} + \dfrac{1}{b}} \right)$

Answer 1

Hint So in this question, it is given that there are two spheres and they have a different radius. Also, the space between them is given and we have to find the gap between the two spheres. By using the resistance we will be able to get the solution.
Formula:
Resistance,
$R = \dfrac{{\delta x}}{A}$
Where,
$R$ , will be the resistance and $A$ will be the area.

Complete Step By Step Solution So in this question, it is given that there are two spheres and they have a different radius. Also, the space between them is given and we have to find the gap between the two spheres.
So by the formula, we can say that
$R = \dfrac{{\delta x}}{A}$
And it can also be written as,
Since the area of the sphere is $4\pi {r^2}$
$ \Rightarrow dR = \dfrac{{\delta .dr}}{{4\pi {r^2}}}$
So the total resistance will be equal to
$ \Rightarrow R = \int\limits_a^b {dR} = \dfrac{\delta }{{4\pi }}\int\limits_a^b {\dfrac{{dx}}{{{x^2}}}} $
On substituting the values, the equation will be as
$ \Rightarrow R = \dfrac{\delta }{{4\pi }}\int\limits_a^b {{x^{ - 2}}dx} $
Now on solving the integration, we get
$ \Rightarrow \dfrac{\delta }{{4\pi }}\left[ {\dfrac{{{x^{ - 1}}}}{{ - 1}}} \right]_a^b$
After further solving more we will get the new equation as
$ \Rightarrow \dfrac{\delta }{{4\pi }}\left[ {\dfrac{1}{b} - \left[ {\dfrac{{ - 1}}{a}} \right]} \right]$
And simplifying the above equation, we get
$ \Rightarrow R = \dfrac{\rho }{{4\pi }}\left( {\dfrac{1}{a} - \dfrac{1}{b}} \right)$
Therefore, $\dfrac{\rho }{{4\pi }}\left( {\dfrac{1}{a} - \dfrac{1}{b}} \right)$ will be resistance

between the two given spheres. Hence the option $A$ will be the correct solution for this question.

Note Resistance is the opposing force you exert over anything. Sometimes it’s obvious and necessary and out in the open but most often it’s like a clandestine, subterranean, almost involuntary form of combat.
This resistance comes from my ego, from the sense that through sheer force of will I can bend things so that they go my way. Therefore, resistance is constant for materials obeying Ohm’s law, this type of material is called Ohmic materials. For active elements voltage and current are not proportional to each other, therefore resistance is not constant. In such cases, resistance is calculated from the graph of V vs I, therefore \[r = dv/dt\].