Q1 (5 marks) Consider a solid conducting wire that extends along the z-axis of a cylindrical co- ordinate system. The origin of this co-ordinate system is located at the centre of the circular wire cross-section, at a point that is far away from both ends (i.e., the wire is long). The radius of the cross-section is b =  (1/^π ) . 10 − 1  m, and the conductor carries current I  =  ^π A in the z direction.  It may be assumed that the current density J- is uniform across the conductor cross- section, and that the magnetic field is cylindrically symmetric for lzl < 1.

In Gradescope, upload no more than 2 A4 pages with clearly labelled responses to the following: (a) Show that J-(rρ (φ) + 0z ) = 100^π z A/m2 for 0 < r < b and φ e [0, 2π)?       [1 mark] (b) Use V x  = J- and Stokes’ Theorem, to show that for all φ e [0, 2π), the magnitude of the magnetic field strength (rρ (φ) + 0z ) A/m varies as shown in Figure 1.          [4 marks]

Q2 (7 marks) A length é m rigid bar of small cross-section and very high conductivity lies in the z = 0 plane. One end is fixed at the origin, and the rest of the bar is free to rotate about the z-axis in this plane with no friction. Consider a steady state with the bar rotating at (slow) constant angular velocity ω-  = ω0 z  rad/s, for given ω0   > 0, in the presence of a spatially uniform impressed

magnetic flux density  = B0 z  Wb/m2 , for given B0  > 0. In this steady state, there is a fixed spatial distribution of conduction charge along the bar, and zero current density throughout.         In Gradescope, upload no more than 2 A4 pages with clearly labelled responses to the following:

(a) Let φ(t) = ω0t + φ0 be the angular position of the bar at time t > 0. Show that the induced electric field I (t, r) = B0 ω0r ρ (φ(t)) N/C at the point r ρ (φ(t)) + 0z  inside the bar,

0 < r < é .                                                                                                           [2 marks]

(b) What is the electric potential difference between the two ends of the bar, at éρ (φ(t)) + 0z and at the origin, respectively? Which end has the higher potential?   [2 marks]

(c) To maintain the specified steady state, what net external torque is required?    [1 mark]

(d) With the net external torque unchanged from part (c), and considering the very high conduc- tivity of the bar, qualitatively describe what would happen if the impressed magnetic field abruptly changed sign to become  = _B0 z . Comment on the time it takes to recover a steady state, and its characteristics after the change relative to before the change.   [2 marks] 

Q3 (5 marks) Figure 2 (on page 4) shows a two part ferromagnetic core, with an insulated copper winding that makes N turns around the inner part, which is supported axially in a way not shown. The inner dimensions are such that the height g m of the narrow air gaps is much less than the width w m, which is much less than the depth d m. Take the permeability of air to be µ0  and the core permeability to be µ s o. The winding carries current I A. The leakage of flux outside of the core is negligible.

In Gradescope, upload no more than 1 A4 page with clearly labelled responses to the following:

(a) Explain why the magnitude of the magnetic field strength in the air gaps is approximately

l gap l s NI\(2g) A/m.                                                                                      [2 marks]

(b) What is the magnetic flux across the air gap through the top of the inner part?   [1 mark]

(c) What is the self inductance of the winding?    [2 marks]

Q4 (8 marks) To make a transformer, an insulated secondary copper winding with 2N turns is added

to the inner core in Figure 2.  Since there is negligible leakage flux, winding conduction losses dominate.  The corresponding sinusoidal steady-state circuit model is shown in the top part of Figure 3 (on page 4), where the boldface symbols are phasor representations of the corresponding currents, voltages, and impedances (i.e., these are complex numbers.) In the model, Rs  = 2Rp .    In Gradescope, upload no more than 2 A4 pages with clearly labelled responses to the following:

(c) Draw a phasor diagram relating Vp , Il , and Vl for the bottom circuit in Figure 3.   [2 marks]

(d) Taking Z = lZlejφ to be the full load impedance, show that the transformer voltage regulation performance is

(lVs,nIl _ lVs,ll)/lVs,ll =╱′1 + 12 cos(φ)Rp /lZl + 36(Rp /lZl)2  _ 1、

where ‘l’ denotes full load, and ‘nI ’ no load, with the same lVp l in both cases.      [3 marks]