(Old code ELEC104P) ELEC0005 2019 Page 1 of 7 TURN OVER List of Physical Constants: Electronic charge: e = 1.60 × 10 19 C Permittivity of vacuum or air: o = 8.85 × 10-12 F m-1 Permeability of vacuum or air: o = 4 × 10-7 H m-1 Answer ALL the questions in Section A. Answer TWO questions from Section B There is a formula sheet on page 7. SECTION A (There are EIGHT questions in this section) 1. Identify the quantities in the following list that are (a) vectors and (b) vector fields. [6 marks] (i) Velocity of a ball (ii) Acceleration of a train (iii) Gravity around the earth (iv) Magnetic Field around a solenoid (v) Position of a point relative to another point (vi) Force on an electron in an electric field. 2. Sketch the electric field, E, inside and outside a thin, spherical shell with a charge Q evenly distributed over the surface of the shell. Both inside and outside the shell is a vacuum, so that εr = 1. Explain your answer. [6 marks] 3. What happens to an insulator when it is placed in an electric field [6 marks] 4. By using Gauss’ law (magnetic), equation [5] on the formula sheet, derive the conditions on the normal (perpendicular) component of the magnetic flux density, B, at the surface of a magnetic material with a relative permeability of μr. Explain the steps in your answer. [7 marks] (Old code ELEC104P) ELEC0005 2019 Page 2 of 7 CONTINUED 5. By considering the potential difference between two points close together on an equipotential surface in a region of an electric field, E, show that the electric field must be at right angles to the equipotential surface. The potential difference is given by equation [3] on the formula sheet. [6 marks] 6. A solenoid has a time varying current flowing through the coil. Three different loops of wire are wound around the solenoid. The wires are made from copper, a resistive alloy and a graphite loaded filament; they have resistance values of Rc, Ra, Rg respectively. The values are such that Rc < Ra < Rg. What are the relative values of the electromotive force (e.m.f) in each loop Justify your answer. [6 marks] 7. An n-channel MOSFET is made using an oxide that is 2 nm thick and has a relative permittivity of 4. The channel is 90 nm long and 900 nm wide. The electron and hole mobilities in the channel are 1500 cm2 V-1 s-1 and 450 cm2 V-1 s-1, respectively and the threshold voltage is 1 V. Calculate the device transconductance parameter gn, and suggest how it could be doubled without changing the fabrication process technology. [7 marks] 8. An NPN bipolar junction transistor is operated in the active state. Use a diagram to indicate the flow of electrons and holes between the base, collector and emitter (making sure to label each part) and write down the bias state (i.e. forward or reverse biased) of each of the PN junctions. [6 marks] (Old code ELEC104P) ELEC0005 2019 Page 3 of 7 TURN OVER SECTION B Answer TWO out of FOUR questions from this section. 9. (a) State Gauss’ law (electric) in words. [3 marks] (b) A conductor can have a surface charge density of σ C m-2 at the surface of the conductor. (i) Using Gauss’ Law (equation [6] on the formula sheet at the end of the paper) derive the boundary condition on the normal component of the electric flux density D at the surface of a conductor. Assume a surface charge density of σ C m-2 on the surface. Explain the steps carefully. [6 marks] (ii) Sketch the electric flux density around a charged disc, showing the field both close to the disc and at a distance. (You may prefer to use two sketches.) [4 marks] (iii)A thin disc is uniformly charged with 5.0 μC distributed over both surfaces. The radius of the disc is 0.3 m. What is the approximate magnitude of the electric flux density at a distance of (A) 1mm from the centre of the flat surface and (B) 25 m from the disc (εr = 1) [5 marks] (c) A thundercloud can be approximated as a dipole, oriented with the axis at right angles to the surface of the earth. The earth’s surface can be taken, over a region, to be a flat conducting surface. (i) Explain briefly how the method of images is used. (ii) A thundercloud forms between the heights of 5 km and 8 km above the ground and can be considered to be a dipole of charges arranged as follows: +10 C at 8 km and -10 C at 5km, at either end of the cloud. Using the method of images, what is the electric flux density (D) on the ground, just below the thundercloud [7 marks] (Old code ELEC104P) ELEC0005 2019 Page 4 of 7 CONTINUED 10. (a) Sketch the electric flux density around a line charge of charge density q C m-1, explaining the key features of the diagram. [6 marks] (b) T
he electric flux density, D, at a distance R away from a line charge (line charge density of q C m-1) is given by the equation: 2 R q R =D a equation 10.1 where Ra is a unit vector in the radial direction. Two identical line charges with charge density 20 μC m-1, are placed along the x axis and the y axis. What is the electric flux density at position given by coordinate (3, 3, 3) m [7 marks] (c) A single line charge is placed along the x-axis. (i) What is the potential difference between two points, R1 = 1 cm and R2 = 3 cm along a radial direction away from a single line charge of 10 μC m-1 The electric flux density is given by equation 10.1, the co-ordinates of the two points are R1 = (1,0,0) and R2 = (3,0,0). The points are in a vacuum, so εr = 1. [5 marks] (ii) What is the potential difference if the line charge is placed inside and along the axis of a cylinder of dielectric of value ε = 3 and of radius 2 cm [3 marks] (iii)Sketch the electric field distribution in the cylinder and dielectric and outside the cylinder. [4 marks] (Old code ELEC104P) ELEC0005 2019 Page 5 of 7 TURN OVER 11. (a) The mutual inductance, M, of two single loops of conducting wire, 1 and 2, is given by 12 1 M I = , where current I1 is flowing in loop 1 and 12 is the magnetic flux from coil 1 linking into loop 2. (i) How is the definition altered for the self-inductance of a loop (ii) What is the self-inductance of a solenoid, with N loops per unit length of radius a The magnetic field inside a solenoid with current I flowing in it is NI. [9 marks] (b) A magnetic core material has two coils (numbered 1 and 2) wound around it (figure 11.1). The lower coil (coil 1) has a battery connected to through switch S1, which is closed at t = 0. Then at t = t1 the second switch S2 is closed, giving a short circuit across the coil. Figure 11.1 (i) The current I1 in coil 1, does not increase instantaneously to its maximum value when the switch S1 is closed at t = 0, nor decrease to zero when S2 is closed at t = t1. Explain why. (ii) Sketch the current I2 in the second coil as a function of time. (iii) Which of the quantities L1, L2 and M , does the current I2 depend on and why (iv) What is the effect of the magnetic core material [16 marks] R2 R 1 I 1 S1 S2 t=0 t=t1 I 2 L1 self-inductance of loop 1 L 2 self-inductance of loop 2 M mutual inductance of the loops 1 and 2 (Old code ELEC104P) ELEC0005 2019 Page 6 of 7 CONTINUED 12. (a) Define mobility in the context of electrical conduction, giving two physical contributions to electron mobility in materials. [3 marks] (b) Estimate the drift velocity of electrons in a 10 μm long strip of graphene across which a voltage of 100 mV has been applied, giving units. Assume the mobility of electrons in the graphene to be 100,000 cm2 V-1 s-1. [5 marks] (c) Show that the conductivity, σ, of an n-type semiconductor can be expressed as: = ( + 2 ) , where μe is the electron mobility, μh the hole mobility, ND is the donor concentration and ni is the intrinsic carrier concentration. [4 marks] (d) For donor concentrations greater than 1011 cm-3 explain why the contribution of holes to the conductivity of n-type silicon can be neglected (ni(Si) ≈ 1010 cm-3 at room temperature). [2 marks] (e) Figure 12.1 shows the conductivity of a variety of n-type Si samples as a function of donor concentration, at two different temperatures, 300 K and 100 K. (i) From the data, estimate the electron mobility in intrinsic Si material at 300 K. [5 marks] (ii) Estimate the electron mobility in intrinsic Si material at 100 K, and comment on the answer with respect to the mobility at 300 K, giving a physical reason. [4 marks] (iii) Give a reason why the conductivity plot shown in Figure 12.1 is not a linear function of doping concentration. [2 marks] Figure 12.1 (Old code ELEC104P) ELEC0005 2019 Page 7 of 7 END OF PAPER FORMULA SHEET The symbols have the usual meaning. [1] Permittivity D = 0E + P = 0 rE [2] Permeability B = μr μo H and B = 0 (H + M) [3] Electric potential between A and B = A B BA dVV cE [4] Ohm’s law J= E [5] Gauss’ law (magnetic) 0= S dSB [6] Gauss’ law (electric) = VS dvd SD [7] Continuity of charge = + V’SS dv dt d d SJ [8] From the Biot-Savart law = 2 0 4 r I d r udl B [9] The Lorentz force: BvEF += QQ [10] Maxwell via Ampère’s law: + = SSc d dt d dd SDSJcH [11] Faraday’s law = Sc M d dt d d SBcE [12] MOSFET transconductance parameters = ( ), = ( ) [13] MOSFET I-V relationship IDS = gn VOV – 1 2 VDS è ÷VDS