Questions: Dielectric Constant and Relative Permittivity

Question 1

A parallel-plate capacitor holds charge Q and has capacitance C₀ in vacuum. A dielectric with κ = 4 is inserted while charge Q is held fixed. What happens to the capacitance and the electric field inside?

• A. Capacitance increases to 4C₀; field decreases to E₀/4
• B. Capacitance decreases to C₀/4; field increases to 4E₀
• C. Capacitance increases to 4C₀; field remains E₀
• D. Both capacitance and field increase by factor κ

Answer: (A)

Inserting a dielectric multiplies capacitance by κ: C = κC₀. At fixed charge Q, voltage V = Q/C decreases by κ, and since E = V/d, the electric field also decreases to E₀/4. This is the key physical effect: the dielectric's polarization creates an internal field opposing the external field, partially canceling it. The dielectric reduces the field — it doesn't block it or leave it unchanged. Options C and D incorrectly assume the field is unaffected or increases.

Question 2

The displacement field D is introduced in Maxwell's equations primarily because:

• A. It is easier to measure directly than the electric field E
• B. It allows Gauss's law to be written in terms of free charges only, absorbing bound charges from polarization into the material constant ε
• C. It replaces ε₀ entirely in all electromagnetic equations
• D. It describes the force on a test charge inside a dielectric

Answer: (B)

Inside a dielectric, polarization creates bound charges that contribute to the total field. Tracking both free and bound charges in Gauss's law is cumbersome. The displacement field D = ε₀κE = εE absorbs the bound-charge contribution into the permittivity ε, so ∇·D = ρ_free — only externally placed charges appear on the right. This preserves the clean mathematical form of Gauss's law. D is a bookkeeping device, not a directly observable force field.

Question 3

Inserting a dielectric into a capacitor increases its capacitance because the dielectric's polarization partially cancels the internal electric field, allowing more charge to be stored at the same voltage.

• T. True
• F. False

Answer: True

Polarization creates an internal field opposing the applied field, reducing the total field by factor κ. Since V = E·d, the voltage across the capacitor drops at fixed charge. To restore the original voltage (in a battery-connected scenario), more charge must flow onto the plates — hence capacitance C = Q/V increases to κC₀. The dielectric reduces the field; that reduction is precisely the mechanism that increases capacitance.

Question 4

The dielectric constant κ represents the absolute permittivity of a material — a quantity with units of farads per meter.

• T. True
• F. False

Answer: False

κ is the relative permittivity — a dimensionless ratio comparing the material's permittivity to that of free space: κ = ε/ε₀. It has no units. The absolute permittivity is ε = κε₀, which has units of F/m. This distinction matters: κ = 80 for water does not mean the permittivity is 80 farads per meter — it means water's permittivity is 80 times that of vacuum.

Question 5

Why does water have a dielectric constant of approximately 80 while nonpolar materials like most plastics have κ ≈ 2–4? Explain in terms of molecular structure.

Model answer: Water molecules are permanent dipoles — the electronegative oxygen atom pulls electron density away from the hydrogen atoms, creating a persistent charge separation. When an external electric field is applied, these permanent dipoles rotate to align with the field, producing very strong polarization. Nonpolar molecules have no permanent dipole; they develop only induced dipoles when the external field slightly distorts their electron distributions. Induced dipoles are much weaker than aligned permanent dipoles, resulting in far smaller polarization and a much lower κ.

Strong polarization means stronger opposition to the external field, which means a larger reduction in E_inside, corresponding to a higher κ. Materials with large κ are especially useful in capacitors (ceramics like barium titanate, κ > 1000, are widely used in electronics) and in biological contexts, where water's high κ screens electrostatic interactions between charged molecules.