a. For a given field V, show that;
i. \(\nabla \times (\nabla V) = 0\)
ii. \(\nabla \cdot (\nabla \times V) = 0\)
b. Calculate the charge within the regions
i. \(\rho_v = \frac{1}{x^3 y^3 z^3}, 0.1 \le x, y, z \le 0.2\)
c. The electrostatic field intensity E is the negative gradient of a scalar electric
potential, that is \(E = -\nabla V\). Determine E at point (1,1,0) if \(V = V_0 e^{-x} \sin\frac{\pi y}{4}\)
d. Boundary conditions of Time-varying electric fields between two media.
e. Derive the equation for electric field intensity due a uniform infinite line
charge density \(\rho_l\).
f. Classify the following fields.
1. \(\nabla \times F = 0\) and \(\nabla \cdot F = 0\)
2. \(\nabla \times F \ne 0\) and \(\nabla \cdot F = 0\)
3. \(\nabla \times F \ne 0\) and \(\nabla \cdot F \ne 0\)
g. Calculate the capacitance of a square parallel plate capacitor having two
dielectrics \(\epsilon_{r1} = 2.5\) and \(\epsilon_{r2} = 3.5\) each comprising one half of the area between
the plates. The area of a plate is \(4.0 \times 10^4 cm^2\) and the two plates separated by
a distance of 20mm.
h. If a 24V volts electric potential is applied across the terminals of this
capacitor described above, find the electric potential energy stored in the
capacitor.
