Dimension of permittivity in Vacuum or free space

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Dimensional Formula of Permittivity

with its Derivation

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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:-

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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