Electromagnetism - Dielectric

A dielectric of dielectric constant K is placed in contact with a conductor having surface charge density σ, as shown above. What is the polarization (bound) charge density σ_p on the surface of the dielectric at the interface between the two materials? 

A. σK / (1−K)
B. K / (1 + K)
C. σK
D. σ(1 + K) / K
E. σ(1 − K) / K

(GR8677 #54)

Solution:

Polarization, P = ɛ₀χ_eE = ɛ₀(K − 1)E
Thus, K  1.
If K = 1 → P = 0 and σ_p = 0 → no polarized dielectric

Check answers for K = 1→ σ_p = 0

(A) σ_p = σK/(1 − K) = ∞ → FALSE
(B) σ_p = K/(1 + K) = 1/2 → FALSE
(C) σ_p = σK = σ → FALSE
(D) σ_p = σ(1 + K)/K = 2σ → FALSE
(E) σ_p = σ(1 − K)/K = 0 → TRUE

Answer: E 


Alternative Solution:

In dielectric, E = σ/ɛ = σ/ɛ₀K
Polarization, P = ɛ₀(K − 1)E
σ_p = ɛ₀(K − 1)(σ/ɛ₀K)
= (K − 1)(σ/K)
= σ(1 − K)/K