Understanding Units in Physics: Farad per Metre

The question asks for the physical quantity whose unit is Farad per metre. Understanding the units of fundamental electrical quantities is crucial in physics.

What is Permittivity?

Permittivity is a material property that describes how an electric field affects, and is affected by, a dielectric medium. It relates the electric displacement field (D) to the electric field (E).

The relationship is given by:

\(\vec{D} = \epsilon \vec{E}\)

where:

• \(\vec{D}\) is the electric displacement field
• \(\vec{E}\) is the electric field
• \(\epsilon\) is the permittivity of the medium

Permittivity (\(\epsilon\)) can be thought of as a measure of how much resistance is encountered when forming an electric field in a medium. In vacuum, this property is represented by the permittivity of free space, \(\epsilon_0\).

Deriving the Unit of Permittivity

The unit of permittivity can be derived from its definition or from formulas involving capacitance. A common formula for the capacitance (C) of a parallel-plate capacitor is:

\(C = \frac{\epsilon A}{d}\)

where:

• \(C\) is the capacitance
• \(\epsilon\) is the permittivity of the dielectric material between the plates
• \(A\) is the area of one plate
• \(d\) is the distance between the plates

Rearranging this formula to solve for permittivity (\(\epsilon\)):

\(\epsilon = \frac{C d}{A}\)

Now, let's look at the units of each term:

• Capacitance \((C)\) is measured in Farads (F).
• Distance \((d)\) is measured in metres (m).
• Area \((A)\) is measured in square metres \((\text{m}^2)\).

Substituting these units into the formula for \(\epsilon\):

\([\epsilon] = \frac{[C][d]}{[A]} = \frac{\text{F} \cdot \text{m}}{\text{m}^2} = \frac{\text{F}}{\text{m}}\)

Thus, the unit of permittivity is Farad per metre (F/m).

Analyzing Other Options

Let's consider the units of the other given options to confirm they are not Farad per metre.

Comparing the derived unit for permittivity (\(\text{F/m}\)) with the units of the other options, it is clear that Farad per metre is the unit specifically associated with permittivity.

Conclusion

Based on the unit derivation from the formula for capacitance and the fundamental definition relating electric displacement and electric field, the unit Farad per metre corresponds to permittivity. The units for watt per steradian, permeability, and electric conductance are distinctly different.

Revision Table: Key Electrical Units

Additional Information on Permittivity and Units

Permittivity is a key concept in electrostatics, especially when dealing with materials placed in electric fields. It determines how much electrical energy can be stored in a material for a given electric field, which is why it appears in the formula for capacitance. The higher the permittivity, the more charge can be stored for a given voltage and geometry. The unit Farad per metre directly reflects this ability to store charge (related to Farad) over a distance (metre) within a material.

It's important not to confuse permittivity (\(\epsilon\)) with permeability (\(\mu\)). While both are material properties, permittivity relates to how a material responds to an electric field, whereas permeability relates to how a material responds to a magnetic field. Their units, F/m and H/m respectively, reflect their distinct roles in electromagnetism.

Understanding the units of physical quantities is fundamental to solving problems in physics and engineering. It helps in checking the consistency of equations and understanding the physical meaning of calculated values. Always pay attention to the units given in a problem statement or derived in a calculation.