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ENGINEEtING ELECTtOMAGNETICS 2

(ENG2031)

Thursday 6th May 2021

Release time: 9:30:00 AM (BST) for 4 hours

This exam should take you 2 hours to complete. However, you have a 4-hour window to download/complete/upload your submission.

Attempt ALL questions

Section A (50 marks): Q1 and Q2 (25 marks per question)

Section B (30 marks): Q3 and Q4 (15 marks per question)

Section c (20 marks): Multiple choice (Quiz) Q5-Q9 (4 marks per question)

The numbeTs in squaTe bTacKets in the Tight-hand maTgin indicate the maTKs allotted to the paTt of the question against which the maTK is shown. These maTKs aTe foT guidance onlg.

DATA SHEET IS PtOVIDED AT THE END OF PAPEt

A calculator may be used. show intermediate steps in calculations.

SECTION A

Q1 A hollow spherical conductor of internal radius R2 and external radius R3 surrounds a conductive sphere of radius R1 , which is charged with a charge Q, as shown in Figure Q1.

(a) Derive the expression for the magnitude of the electric ield E(T), for T between 0 and ∞. Note that T = 0 is the origin of the spherical reference system, as shown in Figure Q1. [8]

(b) Hence or otherwise, derive the expression for the scalar potential Ⅴ (T), for T between 0 and ∞. [6]

(c) Give a qualitative graphical representation for the functions E(T) (magnitude of the electric ield) and Ⅴ (T) (scalar potential), consid- ering R1 = 30 cm, R2 = 50 cm, R3 = 80 cm, and Q = 800 x 10-12 C. use the appropriate units on the graphs. write the values of the elec- tric ield in the dielectric side of the interface for each of the metallic surfaces. write the values for the potential at T = 0, T = R1 , T = R2 , and T = R3 . [6]

(d) Briely describe what happens to the scalar potential and electric ield in the region with R1 < T < R2 at the static equilibrium if a charged sphere of radius Rs = 20 cm and charge Qs = 10-6 C is positioned close to the hollow conductor, with its centre at R4 = 4 m. Include a brief explanation for your answer. [5]

Q2 consider a current I = 1 A lowing in the positive g direction through arectan- gular conductor as shown in Figure Q2. The conductor has length L = 4 cm） width a = 1 cm）and height b = 0.1 cm. suppose that a magnetic ieldB(y) = B permeates uniformly the conductor） with B = 10 T.

(a) Briefly explain why a potential difference ∆V ≡ Vl − Vr arises at the equilibrium between the left and right sides of the conductor） as shown in the igure. [6]

(b) starting from J(y) = nq q yud and considering the Lorentz force） derive the force acting on the carriers responsible for the current I function of the relevant parameters a）b）I）B and the carrier density nq . use such result to demonstrate that the carriers responsible for the current I in the metal are negatively charged if the potential diference mea- sured between the left and the right facets of the conductor is positive: ΔⅤ = Ⅴl - Ⅴr 持 0. To this end）consider that Ⅴl - Ⅴr is positive if the carriers accumulated on the right side have a negative charge. [6]

(c) suppose that the carriers are electrons (qe = -1.6x10-19 c) and that ΔⅤ = Ⅴl - Ⅴr = 22 x 10-6 V. calculate the carrier density nq in the metal） expressed in units of m-3 . [8]

(d) Hence or otherwise） suppose that the current I is induced by a generator keeping the potential diference between the input and output

facets (sin and sout ) at a constant value ΔⅤI/o = 0.002 V. what is the resistivity p of the conductor (in Ω m)？ [5]

SECTION B

Q3 Electric ields and circuits.

(a) Use macroscopic ohm,s and kirchhof,s laws to demonstrate that the total resistance of two resistors connected in series in a circuit equates the sum of their resistance. [6]

(b) Use macroscopic ohm,s and kirchhof,s laws to demonstrate that the total resistance R of two resistors of resistance R1 and R2 connected in parallel is R =(R 1(-)1 + R2(-)1 ,-1 . [5]

(c) Calculate the total capacitance C that two capacitors with capacitance C1 = 1 pF and C2 = 2 pF introduce in a circuit if they are connected in series. [2]

(d) Calculate the total capacitance C that two capacitors with capaci- tance C1 = 11 pF and C2 = 22 pF introduce in a circuit if they are connected in parallel. [2]

Q4 Electric and magnetic ields in vacuum.

(a) consider a region of space free from matter and from magnetic ields. show that it cannot be permeated by an electric ield of the form:

where A is a constant value diferent from zero and with units V/m2 . [5]

(b) Now, consider that a magnetic ield permeates the region of space in (a). write an expression representing a time-varying magnetic ield B(y (从,g, z, t) that renders the vector ield proposed in (a) suitable to describe an electric ield. check and demonstrate that the expression you found is a suitable representation of a magnetic ield. [8]