What Are the Dimensions of Permittivity?

The concept of dimensions of permittivity is fundamental in electromagnetism and plays a key role in understanding electric fields in different media. Permittivity quantifies a material’s ability to permit the formation of an electric field within it and appears in several core equations of electrostatics.

Definition and Physical Meaning of Permittivity

Permittivity, typically represented by the symbol $\varepsilon$, measures how much electric field is reduced inside a medium compared to a vacuum. The absolute permittivity of free space is denoted as $\varepsilon_0$ and provides the baseline reference for all materials. The permittivity of a medium determines the force between charges and the capacitance when that medium is used as a dielectric in a capacitor.

Dimensional Formula of Permittivity

The dimensional formula of permittivity is derived using Coulomb’s law, which gives the force between two point charges:

$F = \dfrac{1}{4\pi \varepsilon_0} \dfrac{q_1 q_2}{r^2}$

Solving for $\varepsilon_0$:

$\varepsilon_0 = \dfrac{1}{4\pi} \dfrac{q_1 q_2}{F r^2}$

Ignore constants while determining dimensions. The relevant physical quantities and their dimensional formulas are:

• Force ($F$): $[M\,L\,T^{-2}]$
• Charge ($q$): $[I\,T]$
• Distance ($r$): $[L]$

Substitute these into the equation:

$\varepsilon_0 = \dfrac{q^2}{F\, r^2}$

$\Rightarrow [\varepsilon_0] = \dfrac{[I\,T]^2}{[M\,L\,T^{-2}] \, [L^2]}$

$\Rightarrow [\varepsilon_0] = \dfrac{I^2 T^2}{M L^3 T^{-2}}$

$\Rightarrow [\varepsilon_0] = M^{-1} L^{-3} T^4 I^2$

SI Units and Dimensional Representation

The SI unit of permittivity is farad per meter ($\text{F}\,\text{m}^{-1}$). Alternatively, it can also be expressed as coulomb squared per newton per meter squared. The dimensional formula, as established, is $M^{-1} L^{-3} T^4 I^2$.

Dimension of Permittivity for Materials

The dimensional formula of permittivity remains the same for any medium, not only for free space. For a dielectric material or any other medium, permittivity is denoted as $\varepsilon$ and is generally expressed in the same dimensional terms as $\varepsilon_0$.

Relative Permittivity and Its Dimensional Nature

Relative permittivity, also called the dielectric constant and represented as $\varepsilon_r$, is a ratio of the permittivity of a medium to that of free space: $\varepsilon_r = \dfrac{\varepsilon}{\varepsilon_0}$. As it is a ratio of two quantities with identical dimensions, relative permittivity is dimensionless.

Significance in Electromagnetism

Permittivity directly affects the magnitude of electrostatic forces and capacitance in materials. In designing capacitors or analysing electric fields, knowing the dimensions of permittivity ensures dimensional consistency in equations and helps in checking the correctness of derived relationships.

Applications of Permittivity

Permittivity values are crucial in electronics and material science. They are used to determine capacitance, analyze electromagnetic wave propagation, and characterize materials as insulators or dielectrics. For further reading on electrostatics and related concepts, refer to Introduction to Electrostatics.

Summary Table: Dimensional Formulas Related to Permittivity

Key Points for JEE Preparation

• Permittivity dimensions are $M^{-1}L^{-3}T^4I^2$
• SI unit is farad per meter (F\,m$^{-1}$)
• Relative permittivity has no dimension
• Dimensions help verify and analyze electrostatics equations

A clear understanding of the dimensions of permittivity is essential for verifying expressions involving electric fields and capacitance. For advanced study on capacitor concepts, refer to What is Capacitance?.