Units And Measurement And Dimensions

To determine which quantity constructed from the given fundamental constants has the dimensions of an electric charge, let's start by identifying the dimensions of each constant and then evaluate each expression.We have the following fundamental constants:Permittivity constant :Derived from Coulomb's law: .Dimensions: .Planck's constant :Derived from the energy formula: .Dimensions: .Velocity of light :As represents a speed, the dimensions are: .Newton's constant :Derived from the gravitational force formula: .Dimensions: .Boltzmann's constant :Derived from the relation .Dimensions: .Now, let's check each option to see if it results in the dimensions of electric charge, which is :Option A: .Option B: .Option C: .Option D: .Upon evaluating each option, onlyOption D: has the dimensions of an electric charge .

To determine the expression for in SI units, use the given relation: From this, the dimensional formula for is: Substituting the dimensional formulas, we have: Next, consider the dimensions of the permittivity of free space , which can be derived from Coulomb's law: The dimensional formula for is: Taking the square root of , we get: By comparing the dimensions of with , it is apparent that matches the dimensions of except for a factor of . Thus, the expression for in terms of and is:

The measured value contains three significant figures:The zeros before the digit 2 are not significant as they only help to place the decimal point.The digits 2 and 3 are significant.The trailing zero after the digit 3 is also significant because it is located after the decimal point and follows a non-zero digit.