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Chapter 1: Problem 150

Find the dimensional formula for energy per unit surface area per unit time (a) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-2}\) (b) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{1}\)

Short Answer

Expert verified
The short answer to the question is: The correct dimensional formula for energy per unit surface area per unit time is (c) \( \mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3} \).

Step by step solution

01

Identify the dimensions of energy, surface area, and time.

The dimensions of energy can be represented as the dimensions of work, which is the same as force multiplied by distance. The dimensions of force are Mass (M), Length (L), and Time (T), represented as \( [M][L][T]^{-2} \), and the dimensions of distance are mainly in Length (L), represented as \( [L] \). The dimensions of surface area can be derived from the square of length and are represented as \( [L]^2 \). The dimensions of time are represented as \( [T] \).
02

Divide the dimensions of energy by the dimensions of surface area and time.

The desired dimensional formula is the dimensions of energy divided by the dimensions of surface area, divided by the dimensions of time. Therefore, the dimensional formula can be expressed as: \( \frac{[M][L][T]^{-2}}{[L]^2 [T]} \)
03

Simplify the dimensional formula

Now, we will simplify the dimensional formula by combining like-terms: \( \frac{[M][L][T]^{-2}}{[L]^2 [T]} = [M]^{1} [L]^{-1} [T]^{-3} \)
04

Compare the result with the given choices and select the correct answer.

Our derived dimensional formula is \( [M]^{1} [L]^{-1} [T]^{-3} \). Comparing this with the given choices, we find that it matches with choice (c). Therefore, the correct answer is (c) \( \mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3} \).

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