Question

Assertion-
The dimensional formula of surface energy is ${M^1}{L^2}{T^{ - 2}}$.
Reason-
Surface energy has the same dimensions as that of potential energy.
$
  A{\text{ Both Assertion and Reason are correct and reason is the correct explanation of Assertion}}{\text{.}} \\
  {\text{B Both Assertion and Reason are correct but Reason is not the correct explanation of Assertion}}{\text{.}} \\
  {\text{C Assertion is correct but Reason is incorrect}}{\text{.}} \\
  {\text{D Both Assertion and Reason are incorrect}}{\text{.}} \\
 $

Answer 1

Hint:Here we will proceed by using the dimensional formula of surface energy to check whether the given assertion is right or wrong. Then we will calculate the dimensional formula of potential to check whether the given reason is right or wrong. Hence, we get our desired result.
Formula used-
1. $PE = {M^1}{L^2}{T^{ - 2}}$
2. dimensional formula of area$ = {M^0}{L^2}{T^0}$
3. dimensional formula of energy$ = \left[ {{M^1}{L^2}{T^{ - 2}}} \right]$
4. $Surface{\text{ area = energy}} \times {\left[ {Area} \right]^{ - 1}}$

Complete step by step answer:
Firstly, we will find the dimensional formula of surface energy to check whether the given assertion is right or wrong.
As we know that the dimensional formula of energy is given by-
$\left[ {{M^1}{L^2}{T^{ - 2}}} \right]$
Where M is mass
L is length
T is time
And the dimensional formula of area$ = {M^0}{L^2}{T^0}$
On substituting the dimensional formula of area and dimensional formula of energy,
We get-
$Surface{\text{ area = energy}} \times {\left[ {Area} \right]^{ - 1}}$
Or $E = \left[ {{M^1}{L^2}{T^{ - 2}}} \right] \times {\left[ {{M^0}{L^2}{T^0}} \right]^{ - 1}} = \left[ {{M^1}{L^0}{T^{ - 2}}} \right]$
Therefore, the surface energy is dimensionally represented as ${M^1}{L^0}{T^{ - 2}}$.
So, this means the given Assertion is wrong.
Now we will calculate the dimensional formula of potential to check whether the given reason is right or wrong.
Potential energy is the product of mass, acceleration due to gravity and altitude.
The dimensional formula of altitude and mass $ = {M^0}{L^1}{T^0}and{\text{ }}{{\text{M}}^1}{L^0}{T^0}$.
For acceleration due to gravity,
The dimensional formula is ${M^0}{L^1}{T^{ - 2}}$.
Now substituting the values of dimensional formula of acceleration due to gravity, altitude and mass in the formula of potential energy-
We get-
$PE = {{\text{M}}^1}{L^0}{T^0} \times {M^0}{L^1}{T^{ - 2}} \times {M^0}{L^1}{T^0}$
Or $PE = {M^1}{L^2}{T^{ - 2}}$
Potential energy is dimensionally represented as ${M^1}{L^2}{T^{ - 2}}$.
So, we can see that the surface energy does not have the same dimensions as that of potential energy.
Therefore, we conclude that both assertion and reason are incorrect.
Hence, option D is correct.

Note- While solving this question, we must know the dimensional formula of acceleration due to gravity, altitude and mass. Also, we must know the concept of potential energy i.e. Potential energy is the product of mass, acceleration due to gravity and altitude to get the required result.