2018S1

Friday, 2 November 2018

9:48 PM

Machine generated alternative text: Question 1. (13 Marks) (a) (b) (c) (d) (e) (O (g) Define the electric field E at a point. If your definition is an equation, define all terms in the equation. Define the electrical potential difference Vab between two points, a and b. If your definition is an equation, define all terms in the equation. For a field E in the x direction, write an equation expressing V(x) in tenns ofE(x). (Note that E(x) is not necessarily uniform.) A charge —q is at x = a and a charge 4q is at the origin, as sketched. Is there a point on the x axis where the electric field is zero? If so, where is it? Explain clearly whether it can be in r<0, or a<x. Give any value Of x to 2 significant figures. Show your working. A thin rod of length L has a uniform linear charge density i.. Showing all steps, derive an expression for the electrical potential at a point P at a distance x from one end, on the axis of symmetry. Draw a diagram illustrating elements used in your calculation. If you use a spatial variable such as r, show it on the diagram. x a Using part (e) (or otherwise), derive an expression for the magnitude ofthe electric field at P. Comment briefly on the answer to (O in the limit
Machine generated alternative text: Question 2. Part (i) [23 Marksl 1+ (a) (b) (c) (d) -æc2 Two capacitors have capacitances of Cl 100 PF and C2 200 gF. They are initially connected in series across a battery with a potential difference of Vo = 100 V (all 2 significant figures). Stating your argument, calculate the charges and potential differences for each capacitor. Determine the total energy stored on the two capacitors in this series configuration. The two capacitors, still charged, are then connected in parallel, positive terminal to positive and negative to negative. Showing each step in your argument, calculate the potential difTerence on the two capacitors in this state. Determine the total energy stored in this parallel configuration. Ifyour answers for (b) and (d) are the same, explain why in one or two short clear sentences. Or, ifyour answers for (b) and (d) are the different, explain why in one or two short clear sentences. Part (ii) (a) (b) (c) (d) State Gauss' law for electricity. Ifyour statement is an equation, define each ofthe terms in the equation. Specify clearly the meaning of any symbols from calculus. Each of two conducting plates has area A. They are separated by a distance d (where so you may neglect edge effects). The upper plate carries a total charge Q and the lower plate a total charge of—Q, respectively, as indicated. Due to other distributions of charge that lie outside ofthe sketch, the external field immediately above the upper plate and immediately below the lower plate is E,) downwards, as sketched. Use Gauss' law explicitly to determine an expression for the surface charge densities 01 on surface 1 (the upper surface of the upper plate, see sketch). Draw a diagram and show clearly your Gaussian surface. State carefully each step of the argument. Using your answer to (b), determine the charge density 02 on surface 2 (the lower surface of the upper plate). State your argument carefully. The field E2 between the plates is omitted in the sketch. Determine the magnitude E2 of the electric field between the two plates. Use Gauss' law and state your argument carefully.
Machine generated alternative text: Question 3. 124 Marksl (b) (i) (ii) (iii) (iv) (v) (vi) (i) (ii) (iii) (iv) (v) State Faraday's Law of induction, and define all terms. Explain clearly the meaning Of the sign in your answer to (a) (i). State Ampere's Law. If your statement is an equation, define all terms and specify the meaning of any symbols from calculus. State the Law of Biot-Savart. If your statement is an equation, define all terms using a clearly labelled sketch. Use one of the laws above to derive an expression for the magnitude of the magnetic field at a perpendicular distance r from a very long, straight wire carrying a current i. Make sure you state which law you are using, and state each step of the argument. A long, straight cable transmits a charge of 25 coulombs over 050 ms. During that time interval, calculate the average magnetic field at a distance of O .10 metre from the cable. Above the equator, the magnetic field B is North. Well above the atmosphere, a proton has a velocity v in the East direction. Give an expression for the magnitude of the force on it and state its direction. (Neglect gravitational and electric forces.) What is the shape of the path of that proton? Briefly explain your answer. Derive an expression for the frequency of the motion of the proton that includes its dependence on B. Above Indonesia, B is North and has a magnitude of 40 HT. Determine the frequency of the proton motion. If the proton had a component of velocity in the Northerly direction, as well as one in the Easterly direction, what shape would its path be? Briefly explain your answer.
Machine generated alternative text: Question 4. [14 Marksl — = —Ego (a) (b) (c) (d) Faraday's law and the Maxwell's law can be rewritten in the following form: dB dt Derive the wave equation of electromagnetic waves by substituting equation (2) into d2 E (x,t) dx2 dt2 Prove that E(x, t) = Eo sin(kx — cot) is a valid solution of the wave equation. Calculate the speed of light in terms of and go. Show your working. Give a brief explanation of the terms linear polarized light and circular polarized light. What are the relative orientations between the electric field, magnetic field and the propagation direction of light in both cases? Use a sketch for your explanation. What is the frequency of an electromagnetic wave with a wavelength of 400 nm (blue light), 550 nm (green light) and 660 nm (red light), respectively? Show your working.
Machine generated alternative text: Question 5. 124 Marksl Interference and Diffraction (i) (ii) (iii) Explain the Huygens Principle of wavefronts. Use a sketch for your explanation. The red light of a laser (wavelength = 633 nm) is dispersed through a double slit (separation between centres of slits d = 0.65 mm). A screen is placed at a distance of 85cm behind the slit. At which locations from the central position of the screen do the first and the second order maxima of the double slit appear? One of the two slits is closed. How does the diffraction pattern change (width of the individual slits a = 0.15 mm)? Calculate the new positions of the first and second maxima. Show your working. Explain in a few words: how does the interference pattern change if the double slit experiment is performed in water (index of lefraction: n = 1.33). (b) A diffraction experiment can be performed either by using a X-ray beam or by using particle waves (electron or neutron diffraction). (i) (ii) (iii) (iv) (v) The kinetic energy of neutrons emitted from a neutron reactor is 1.00 MeV. Calculate the corresponding De-Broglie wavelength of the neutrons. Can these 'fast' neutrons be used for a single crystal Bragg diffraction experiment? Give a brief explanation. The neutrons can be slowed down by using a moderator which is kept at a temperature of 27 (z 300 K). The kinetic energy of the neutrons will now correspond to Ekin. = - kB T. Calculate their velocity and their De-Broglie wavelength. Give the answer to three significant figures. Show your working. Polycrystalline silicon with a lattice parameter of 5.431 Å is illuminated with with this neutron beam. A planar detector is placed at a distance of I .20 m behind the silicon sample. What is the radius of the first and second diffraction ring? Show your working. What are the energies of an electron and a photon with this wavelength? Show your working.
Machine generated alternative text: Question 6. [22 Marksl Compton Effect (a) What is the Compton effect? In a few clear sentences: give a brief explanation using a sketch of the involved particles. (b) Compton used the Ka radiation of a Molybdenum X-ray tube (Ao = 0.7093 Å). What are the final wavelengths for scattering angles of 600 , 900 and 1200 degree. Calculate the energy lost by the photon for each of these three angles. Show your working. Photoelectric Effect (c) (d) (e) (t) Sketch the experimental setup used to measure the photoelectric effect and label the different parts. Give a brief explanation of the photoelectric effect. How can you increase the number of expelled electrons? How can you increase the kinetic energy of expelled electrons? Does the kinetic energy of the electrons change if you change the material of the cathode? During the experiment the following values for the stopping voltage Vstop measured at a given frequency f of the incident light: F uen 1014 Hz Voltage Vstop [VI 5.19 0.20 6.88 0.89 Calculate the Planck constant h and the work function. Show your working. What energy and velocity have the emitted photoelectrons when Sodium (Na) is irradiated with UV-light with a wavelength of A = 100 nm (work function of Sodium: 2.28 eV). How do the energy and velocity of the photoelectrons change if the wavelength is doubled, ic.k=200nm.

 

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