00 Chapter 2 Active-Recall Diagnostic Quiz

00 Chapter 2 Active-Recall Diagnostic Quiz (Electrostatics)

Overview: Test your conceptual understanding and mathematical recall of electrostatic postulates, Gauss’s law applications, electric dipole derivations, dielectric polarization, boundary conditions, and Poisson/Laplace equations.

Question 1: Electrostatic Fundamental Postulates

State the differential and integral forms of the two fundamental postulates of electrostatics in free space.

Solution:

1. Gauss’s Postulate: ∇ · D = ρv ⟺ ∮S D · ds = Qenclosed
2. Conservative Postulate: ∇ × E = 0 ⟺ ∮C E · dl = 0

Question 2: Field Inside a Spherical Charge Cloud

What is the electric field intensity E at radius r inside a uniformly charged spherical cloud of radius a and volume charge density ρv (r < a)?

Solution:

Ein = âr (ρv r) / (3ϵ0)

The field grows linearly with radius r from 0 at the center to maximum (ρv a) / (3ϵ0) at r=a.

Question 3: Electric Dipole Potential Formula

Write the expression for the electric potential V at a distant point P(r, θ) due to an electric dipole with dipole moment p = qd.

Solution:

V = (p cosθ) / (4πϵ0 r²) = (p · âr) / (4πϵ0 r²)

Potential decays quadratically (1/r²) with distance.

Question 4: Polarization Charge Densities

Define surface bound charge density ρps and volume bound charge density ρp in terms of the polarization vector P.

Solution:

• Surface Bound Charge Density: ρps = P · ân
• Volume Bound Charge Density: ρp = -∇ · P

Question 5: Total Flux Density in Dielectrics

Write the constitutive relation relating total electric flux density D, electric field E, and polarization P.

Solution:

D = ϵ0 E + P

Question 6: Tangential Boundary Condition

State the boundary condition for the tangential component of electric field intensity E across a dielectric interface.

Solution:

E1t = E2t (or D1t / ϵ1 = D2t / ϵ2)

The tangential component of E is continuous across the boundary.

Question 7: Normal Boundary Condition

State the boundary condition for the normal component of electric flux density D across a dielectric interface with surface charge density ρs.

Solution:

D1n - D2n = ρs (or ϵ1 E1n - ϵ2 E2n = ρs)

If no free surface charge exists (ρs = 0), D1n = D2n (normal D is continuous).

Question 8: Dielectric Refraction Law

State the law of refraction for electric field lines passing across a charge-free boundary between medium 1 (ϵ1) and medium 2 (ϵ2).

Solution:

tanα1 / tanα2 = ϵ1 / ϵ2

Where α1, α2 are the angles the field lines make with the normal to the interface.

Question 9: Poisson’s and Laplace’s Equations

Write Poisson’s equation and Laplace’s equation for electric potential V.

Solution:

• Poisson’s Equation: ∇² V = -ρv / ϵ
• Laplace’s Equation: ∇² V = 0 (for charge-free region ρv = 0)

Question 10: Electrostatic Energy Density

Write the formula for the electrostatic energy density we stored in an electric field.

Solution:

we = 1/2 ϵ E² = 1/2 D · E [J/m³]