Understanding the Dielectric Constant of Vacuum

The question asks about the value of the dielectric constant for a vacuum. The dielectric constant, also known as relative permittivity, is a property of a material that describes how it responds to an applied electric field. It essentially measures the ability of an insulating material to store electrical energy in the form of an electric field.

What is Dielectric Constant?

Mathematically, the dielectric constant (often represented as $\kappa$ or $\epsilon_r$) of a medium is defined as the ratio of the permittivity of that medium ($\epsilon$) to the permittivity of free space (vacuum, $\epsilon_0$):

$$ \epsilon_r = \frac{\epsilon}{\epsilon_0} $$

Here:

• $\epsilon_r$ is the relative permittivity (dielectric constant).
• $\epsilon$ is the permittivity of the material.
• $\epsilon_0$ is the permittivity of free space (vacuum), a fundamental constant approximately equal to $8.854 \times 10^{-12} \, \text{F/m}$.

The dielectric constant indicates how much the electric field is reduced inside the material compared to a vacuum when the same external field is applied. Materials with higher dielectric constants are better at reducing the electric field.

Dielectric Constant of Vacuum

By definition, the vacuum serves as the baseline or reference medium for measuring dielectric properties. The permittivity of a vacuum is denoted as $\epsilon_0$. When we calculate the dielectric constant for a vacuum itself using the formula:

$$ \epsilon_r (\text{vacuum}) = \frac{\epsilon_0}{\epsilon_0} $$

This simplifies to:

$$ \epsilon_r (\text{vacuum}) = 1 $$

Therefore, the dielectric constant of a vacuum is exactly 1.

Analyzing the Options

• Less than 1: This is incorrect. No material has a dielectric constant less than 1, as vacuum is the reference point.
• Exactly 1: This is correct, as explained above. The dielectric constant of vacuum is 1 by definition.
• Slightly greater than 1: This applies to many common dielectric materials like air, paper, glass, and plastic. Their permittivity is slightly greater than that of vacuum, making their dielectric constants slightly greater than 1.
• Zero: This is incorrect. A dielectric constant of zero is physically meaningless in this context.

Conclusion

Based on the definition and the fundamental constants involved, the dielectric constant of a vacuum is precisely 1.