Dimension of permittivity in Vacuum or free space

Dimensional Formula of Permittivity

with its Derivation

In this article, we will find the dimension of permittivity in Vacuum or free space.

Dimensional formula for absolute electrical permittivity of free space is

We would now derive this dimensional formula.

Derivation for expression of Dimension of permittivity

From Coulomb’s Law, electrical force acting between two charges \(q_1\) and \(q_2\) kept at a distance \(r\) is given by

\[F = \frac{1}{4\pi \varepsilon_{0}} \frac{{{q_1}{q_2}}}{{{r^2}}} \tag{1}\] Here, \(\varepsilon_{0}\) is called absolute electric permittivity of the free space (or vacuum)

From Equation (1) we have

\[\varepsilon_{0} = \frac{1}{4\pi F } \frac{{{q_1}{q_2}}}{{{r^2}}} \tag{2}\] Now since, \[\text{electric charge} = current\times time\]

dimensions of electric charge \(q = [AT]\)  where \([A]\) is the dimension of electric current and \([T]\) is the dimensions of time

\(4\) and \(\pi\) are constants and does not have any dimension

Since, \[Force= mass \times acceleration = ma\]

Dimension of force is \([MLT^{-2}]\)

Dimension of \(r\) is the dimensions of length i.e., \([L]\)

Now we would use equation (2) to find the dimensions of permittivity in free space

\begin{align*}
\text{Dimensions of } \varepsilon_{0} &= \frac{\left[ AT \right] \left[ AT \right]}{\left[ MLT^{-2} \right] \left[ L^2 \right]} \\
&=\left[ M^{-1}L^{-3}T^4A^2 \right]
\end{align*}

Above formula also gives the dimensional formula of electrical permittivity of any given medium.

Try the free Quiz given below to check your knowledge of Dimension Analysis:-

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Quiz on Dimensional Analysis

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1. The dimensions of universal gravitational constant are

\([M^{-2}L^{2}T^{-1}]\)

\([M^{-1}L^{3}T^{-2}]\)

\([M^{-2}L^{3}T^{-2}]\)

\([ML^{2}T^{-1}]\)

Question 1 of 5

2. Which of the following pair does not have similar dimensions

tension and surface tension

angle and strain

stress and pressure

Planck's Constant and angular momentum

Question 2 of 5

3. Which of the following physical quantity as the dimension of \([ML^2T^{-3}]\)

impulse

Power

Pressure

Work

Question 3 of 5

4. Choose the correct statement(s)

A dimensionally correct equation may be incorrect

A dimensionally incorrect equation may be correct

A dimensionally incorrect equation may be incorrect

A dimensionally correct equation may be correct

Question 4 of 5

5. Which of the following is a dimensionless quantity

Quantity of heat

Strain

Stress

Specific Heat

Question 5 of 5

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Related Articles

1. https://physicscatalyst.com/elec/coulombs-law.php :- visit this page to know more about Coulomb’s law and dielectric constant or relative permittivity.
2. New Simplified Physics by SL Arora : I highly recommend this book for class 11 Physics students. It is easy to understand with lots of solved problems.
3. Dimensional Analysis:- a very good website for physics concepts
4. dimension of Voltage
5. dimension of Resistance
6. dimensional formula of pressure

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