Dimensional Analysis

We are given: Since both and are terms of the same expression for energy, their dimensions must be the same as that of energy:
Let’s analyze term-by-term: 1.
2.

To find the dimensional formula of (Energy speed), we first determine the dimensional formulas of energy and speed.
- Energy: Energy can be represented by kinetic energy, , where is mass and is velocity. The dimensional formula of mass is . Velocity is distance per time, so its dimensional formula is . Thus, the dimensional formula of energy is:
- Speed: Speed is distance per time, so its dimensional formula is: Now, calculate the dimensional formula of (Energy speed): Combine the dimensions: - Mass:
- Length:
- Time:
Thus, the dimensional formula of (Energy speed) is: This corresponds to , where , , and . Comparing with the options, this matches option (1): .
Thus, the correct answer is (1).

We are given three fundamental quantities: - Speed of light in vacuum: - Acceleration due to gravity: - Mass of the electron: To find the unit of time in this system, we can apply the concept of dimensional analysis. The idea is to express the fundamental quantities in terms of their dimensions and then calculate the time unit. ### Step 1: Express the dimensions of the given quantities We know that the dimensions of the physical quantities are as follows: Where: - is the dimension of mass, - is the dimension of length, - is the dimension of time. ### Step 2: Form a dimensionless quantity for time We aim to express the unit of time in terms of these fundamental quantities. Using the dimensional formula for each quantity, we form a dimensionless expression for time, say . Assume: The dimensions of will be: Simplifying: For this expression to represent the unit of time, the exponents of , , and must match the dimensions of time, i.e., . This gives us the system of equations: ### Step 3: Solve the system of equations From the first equation, we get: From the second equation: Substituting in the third equation: Thus, and . ### Step 4: Substitute the values of , , and Now that we have the values for , , and , we can substitute them back into the expression for : Substitute the given values: Hence, the unit of time in this system is approximately .

The dimensional formula for charge , permittivity , and Planck’s constant .
So,

Step 1: Determine the dimensional formula of Stefan-Boltzmann constant ( ).
Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of the black body's absolute temperature (T). So, Where:
P = Power (Energy/Time) = [ML\textsuperscript{2}T\textsuperscript{-3}] A = Area = [L\textsuperscript{2}]
T = Temperature = [K]
= Stefan-Boltzmann constant
Therefore, Step 2: Determine the dimensional formula of Heat Capacity (S).
Heat capacity (S) is the amount of heat energy required to raise the temperature of a substance by one degree Celsius (or Kelvin).
Where:
Q = Heat Energy = [ML\textsuperscript{2}T\textsuperscript{-2}] T = Change in Temperature = [K] Therefore, Step 3: Calculate the dimensional formula of . Now, simplify the powers of M, L, T, and K:
For M: M\textsuperscript{1-1} = M\textsuperscript{0}
For L: L\textsuperscript{2-0} = L\textsuperscript{2}
For T: T\textsuperscript{-2 - (-3)} = T\textsuperscript{-2 + 3} = T\textsuperscript{1}
For K: K\textsuperscript{-1 - (-4)} = K\textsuperscript{-1 + 4} = K\textsuperscript{3}
So, the dimensional formula of is [M\textsuperscript{0}L\textsuperscript{2}TK\textsuperscript{3}]. Step 4: Select the correct option.
The calculated dimensional formula [M\textsuperscript{0}L\textsuperscript{2}TK\textsuperscript{3}] matches option (2).

Step 1: Write the dimensional formulas of A, B, and C.
Planck's constant (A): [ML\textsuperscript{2}T\textsuperscript{-1}]
Mass (B): [M]
Velocity (C): [LT\textsuperscript{-1}]
Step 2: Compute the dimensional formula of .
So the dimensional formula is: [M\textsuperscript{0}L\textsuperscript{1}T\textsuperscript{0}] Step 3: Select the correct option.
The calculated dimensional formula [M\textsuperscript{0}L\textsuperscript{1}T\textsuperscript{0}] matches option (3).