{"id":5593,"date":"2019-10-31T20:30:33","date_gmt":"2019-10-31T15:00:33","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5593"},"modified":"2022-11-04T12:06:29","modified_gmt":"2022-11-04T06:36:29","slug":"dimension-of-permittivity","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/dimension-of-permittivity\/","title":{"rendered":"Dimension of permittivity in Vacuum or free space"},"content":{"rendered":"\n
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Dimensional Formula of Permittivity <\/strong><\/span><\/h2>\n\n\n\n

with its Derivation<\/strong><\/span><\/p>\n\n\n\n

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In this article, we will find the dimension of permittivity in Vacuum or free space.<\/p>\n\n\n\n

\"dimension<\/a><\/figure><\/div>\n\n\n\n

Dimensional formula for absolute electrical permittivity of free space is<\/p>\n\n\n\n

We would now derive this dimensional formula.<\/p>\n\n\n\n

Derivation for expression of Dimension of permittivity<\/h3>\n\n\n\n

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

\\[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)<\/p>\n\n\n\n

From Equation (1) we have<\/p>\n\n\n\n

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

dimensions of electric charge<\/a> \\(q = [AT]\\)  where \\([A]\\) is the dimension of electric current and \\([T]\\) is the dimensions of time<\/p>\n\n\n\n

\\(4\\) and \\(\\pi\\) are constants and does not have any dimension<\/p>\n\n\n\n

Since, \\[Force= mass \\times acceleration = ma\\]<\/p>\n\n\n\n

Dimension of force is \\([MLT^{-2}]\\)<\/p>\n\n\n\n

Dimension of \\(r\\) is the dimensions of length i.e., \\([L]\\)<\/p>\n\n\n\n

Now we would use equation (2) to find the dimensions of permittivity in free space<\/p>\n\n\n\n

\\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*}<\/p>\n\n\n\n

Above formula also gives the dimensional formula of electrical permittivity of any given medium.<\/p>\n\n\n\n

Try the free Quiz given below to check your knowledge of Dimension Analysis:-<\/p>\n\n\n\n

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Quiz on Dimensional Analysis<\/strong><\/span><\/h4>\n
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