Calculate the potential at point p located a distance z

Question

Question 1. The transformer below is a toroid with's diameter r equal to 25 centimeters, a cross-sectional area A = 4 cm², and a relative permeability μ_r = 650. There is an airgap l_g = 0.75 mm. If the primary consists of l_g = 0.75 turns calculate the inductance of the primary.

If the primary carries a current of I(t) = 3.5 sin(75Πt) turns calculate the voltage induced in a secondary coil with N2 = 3500 turns.

Question 2. A circular current loop of radius a carries a current I Amperes. The magnetic field at point P located a distance z from the center of the disk can be found from the Law of Biot-Savart:

(a) Showing all work, calculate the magnetic flux density (inclu'ding units) .6 at P. Use back of page if necessary.
(b) What is the flux density if z -+ 07

Question 3.
(a) In class, the Helmholtz Theorem was presented in two different ways. State either one of these two definitions.

(b) A Vector field is given by

A^→(x, y, z) = 8x²y²a^{^}_x + 2x²yz²a^{^}_z

Calculate the divergence of A^→, and evaluate at point (1, 0, 2).

(c) Calculate the gradient of the scalar function Φ(x, y, z) = 2xy³ - 2x²y -x²yz, and evaluate at (1, 0, 2).

(d) Assuming Φ is in volts and including units, calculate the spatial rate of change of Φ(x, y, z) = 2xy³ - 2x²y - xy²z at the point (1, 0, 2) in the direction

A^→ = 2a^{^}_x - 3a^{^}_y - a^{^}_z

Question 4.

(a) Calculate the potential at point P located a distance z above the center of a charged disk of radius a and surface charge density ρ_s Coulombs per square meter. Please show all work. An integral that you may need is

∫ρdρ/√(ρ² + z²) = √(ρ² + z²)

The elemental area in cylindrical coordinates is dA = ρdρdΦ m².

(b) How much work is required to move charge Q from ∞ to P? Include units.

(c) Calculate the electric field at P. Include units.

Question 5. The integral form of Poynting's Theorem is written as

∫E^→ x H^→.dA^→ + ∫∫∫J^→.E^→dv + 1/2 ∂/∂t∫∫∫∈|E|² + μ|H|²dv = 0

Explain what each term represents. What is the physical meaning of the Poynting vector S^→? What are the units of S^→?

Question 6. A parallel plate capacitor has a plate area of 1500 cm² and plate separation of 2.5 mm (0.0025 meter). Assume a dielectric with relative permittivity ∈_r = 8.

(a) Calculate the capacitance of this capacitor.

(b) A battery is connected to the capacitor, plaring a total charge of Q = 7 x 10^-9 Coulombs on the plates. Determine the electric flux density D and from the electric flux density, calculate the electric field E in the dielectric between the plates, and the voltage V of the battery. Include units.

(c) If a time-varying voltage V = 25 sin(400Πt) is applied to the capacitor, calculate the Dis¬placement Current Density Amperes/m²) and the total Displacement Current I_disp (Amperes) between the plates.

Question 7.

(a) Write the four Maxwell's equations in differential form, and give a name for each. In each equation, define the different fields with units.

(b) From which equation (or equations) can each of the following be derived:
- Kirchhoff's Voltage Law
- Kirchhoff's Current Law

(c) Starting with Ampere's Law, derive the equation of continuity.

(d) Define Voltage. If the voltage difference between two points in a circuit is 25 volts, what does this mean?