Dimensional Analysis

We are given that the dimensions of length are expressed as , where , , and are the universal gravitational constant, speed of light, and Planck's constant, respectively. We need to find the values of , , and such that this expression represents length. Step 1: Analyzing the Dimensions of , , and
We first write the dimensions of each of the physical constants , , and . - The universal gravitational constant has the dimensions of: - The speed of light has the dimensions of: - Planck's constant has the dimensions of: Step 2: Dimensions of
Now, we need to find the dimensions of . Using the dimensions of each constant, we have: This simplifies to: Now, group the terms involving , , and : For this expression to have the dimensions of length, the powers of , , and must match the dimensions of length, which are . Therefore, we must solve the following system of equations: 1. (the power of must be 0), 2. (the power of must be 1), 3. (the power of must be 0). Step 3: Solving the System of Equations From equation (1), , we get: Substitute this into equation (3): Now substitute and into equation (2): Thus, . Using , we find: Thus, the values of , , and are: Step 4: Conclusion Based on the calculations, the correct answer is:

We need to determine the dimension of and compare it with the dimensions of the given options. The formula for is:
Step 1: Analyzing the units and dimensions 1. (Permittivity of free space): The dimension of is: 2. (Length): The dimension of length is: 3. (Potential difference): The dimension of potential difference (Voltage) is: 4. (Time interval): The dimension of time is:
Step 2: Determining the dimension of Substitute the dimensions of each quantity into the formula for : Simplifying the expression: Cancel out common units: Thus, the dimension of is:
Step 3: Conclusion The correct answer is: