The quantities and are defined where -capacitance, - Resistance, - length, - Electric field, - magnetic field and - free space permittivity and permeability respectively. Then:
and have the same dimension
Step 1: Given data:
The given physical quantities-
where -capacitance, - Resistance, - length, - Electric field, - magnetic field and - free space permittivity and permeability respectively.
Step 2: Formula used:
We know that the speed of wave is given by-
where is speed of light
Step 3: Used dimensions
Length's dimension
Dimension of speed using formula ,
Dimension of capacitance is calculated using the formula, Where, is charge, is voltage
The dimension of charge
The dimension of voltage is calculated using the formula,
The dimension of electric field is calculated using the formula,
The dimensional formula of the electric field will be-
The dimensional formula for voltage will be-
Thus the dimensional formula for capacitance is calculated as-
The dimension of resistance is calculated using the ohm's law
Resistance's dimension
The magnetic field can be calculated using the formula, where is velocity
Thus, the dimensional formula for magnetic field is as follows-
Step 4: Find the dimension of
As we know that,
Using equation (6)-
Now substituting, the dimension of speed in equation (7)
Thus, the dimension of will be-
Step 4: Compute the dimension of
It is understood that . So,
Substitute the known dimensions of electric field and magnetic field from equations (2) and (5) in the relation,
Step 5: Compute the dimension of
We know that .
To find the dimension of , substitute the known dimension from equation (3) and (4) in the relation,
Thus, have the same dimensions.
Hence, option B is the correct answer.