Calculate the magnetic flux density including units brarr

Question

Question 1. A circular current loop of radius a carries a current I Amperes. The magnetic field at point P located a distance z above the center of the disk can be found from the Law of Biot-Savart:

B^→ = μ₀/4Π∫Idl^→ x αˆR/R²

where dl^→ = adΦαˆ_Φ.

(a) Write the unit vector fin from dl^→ to P.

(b) Calculate Idl^→ x αˆ_R. What term goes to zero due to symmetry?

(c) Showing all work, calculate the magnetic flux density (including units) B^→ at P, and the flux density when z >> a. (Note that z ≠ ∞ so you cannot say B = 0.) Use back of page if necessary.

Question 2.The current in the circuit below is I. What is the value of the integral ∫H^→.dl^→ calculated around the closed loops A, B, C and D?

Question 3. The diagram below shows the trajectory of a positive charge +q in a crossed electric and magnetic field. Determine the direction of the electric and magnetic fields. Note that the charge is initially at rest at the origin.

Question 4. A wire of radius a =1 mm, length I = 100 meters, and conductivity σ = 5 x 10⁷ Siemans/m is connected to a battery of 10 volts. The wire is oriented in the a^{^}z direction.

(a) Including units, calculate the electric field E^→ in the wire, and the resistance R of the wire.

(b) Including units, calculate the total current I and the current density J^→

(c) If the number density of charge carriers is N = 5 x10²⁸ per m³ and the charge is 1.602x 10^-19 Coulombs, calculate the drift velocity of the charge carriers.

(d) Calculate the magnetic field intensity H^→(p) inside the wire (p < a) and outside the wire (p > a). (You can leave answer in terms of p and a.) Plot the field intensity field H vs. p. Use back of page if necessary

Question 5. A ferromagnetic toroid with relative permeability μ_r = 250 is shown below. The toroid has a .5 cm gap. The radius of the toroid is 12 cm, the cross sectional-area is 8 cm², and the core is wrapped with N windings carrying 100 mA.

Draw the equivalent circuit and calculate the total magnetic reluctance R.

Calculate the number of turns required to support a flux density of 1.5 Tesla in the air gap (neglecting fringing). Use back of page if necessary.

Question 6. The magnetic vector potential in free space (μ = μ_o) A^→ is

A^→ = μ₀ (xya^{^}_x, -y²a^{^}_y -xza^{^}_z)

(a) Determine the magnetic flux density B^→ (including units).

(b) Determine the volume current density J^→ (including units).