Part -1

P1)
Let A^→ = x^ Sin(z)

Find ∇ × A^→ and ∇.A^→

P2)
Φ = Sin(z) Sin(3x)

Find ∇Φ

P3)

If E^→ = y^ Cos(x y), does B^→ vary with time?

P4)
Can the field B^→ = x^ Sin(y) + y^ Sin(x)+ z^ Cos(x y), exist in the region defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1?

P5)

What is the charge density ρv in a region where

D = x^ Cos(x) Sin(y) + y^ Sin(x) Cos(y)

P6)

What is the plasma frequency ω_p where the electron density is 10⁶ electrons/m³ frequency? Find the permittivity ∈ at 1 KHz and 1 MHz.

P7)

If E = y^ E₀e^-jkz, find the corresponding B^→ (assume k is a constant).

P8)

If E^→ = y^ E₀e^-jkz , find the the following:

a) E^→(t)

b) H^→(t)

c) S^→(t)

d) →>

e) U_E

f) U_H

g) < UE>

h) < UH>

Part -2:

P1)

Assume seawater has conductivity σ = 4 Siemens/m, permittivity ∈ = 81 ∈0, and permeability μ = μ₀. Find the complex permittivity ∈^ at 5 Hz, 5 KHz, and 5 MHz.

P2)

An electromagnetic wave with frequency 1 KHz is propagating vertically down in seawater. Assume seawater has conductivity σ = 4 Siemens/m, permittivity ∈ = 81 ∈0, and permeability μ = μ0. Assume the electric field intensity just beneath the seawater surface is 1.0 V/m. What is the intensity of electric field at depth of 100 m?

P3)
In Problem P2, what is the power density just beneath the surface of seawater?

P4)
A top-secret room needs to be shielded from radio detection. The room requires a layer of copper five skin-depths thick. If the shielding frequency is 1 KHz to 100 MHz, then what thickness of copper (in mm) is required?

P5)
What is the total force due to solar radiation pressure on an object on surface of the earth? Compare this force to gravitation force on the object. Assume cross-section of object is roughly 0.5 m2 and it has a weight of 50 kg.