Dimensional Analysis

Energy and torque have same dimensions

The dimensional formula of resistance is and the dimensional formula of capacitance is . To find the dimension of , we multiply the dimensions of and : Thus, the dimension of is , which is a dimensionless quantity. Therefore, the correct answer is (B).

The given equation for is: In this equation, is a physical quantity, and is the initial value of . The exponential function is dimensionless, as the exponent of an exponential function must be dimensionless. This means that the term inside the exponential must be dimensionless: Therefore, the dimensions of must cancel out the dimensions of , which means: So, has dimensions of . Thus, the correct answer is that has dimensions of .

The dimension represents a quantity that involves mass, length, and time. To identify what physical quantity this dimension corresponds to, we can analyze the options:
1. Pressure Area: Pressure has the dimension , and Area has the dimension , so the product would have the dimension: This is not the same as , so this option is incorrect.
2. Force Pressure: Force has the dimension , and Pressure has the dimension , so the product would have the dimension: This is also not the same as , so this option is incorrect.
3. Power Time: Power has the dimension , and Time has the dimension , so the product would have the dimension: This is not the same as , so this option is incorrect.
4. Energy Density: Energy has the dimension , and Density has the dimension , so the product (Energy Density) has the dimension: This matches the given dimension , so the correct answer is Energy Density.
Thus, the physical quantity corresponding to is Energy Density.