UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For the following qualifications..- B. Sc. M. Sci. Physics 1B26: Electricity and Magnetism COURSE CODE : PHYSIB26 UNIT VALUE : 0.50 DATE : 07-MAY-02 TIME : 14.30 TIME ALLOWED : 2 hours 30 minutes 02-C1069-3-180 2002 University of London TURN OVER
Answer SIX questions from Section A and THREE questions from Section B. The numbers in square brackets in the right hand margin indicate the provisional allocation of maximum marks per sub-section of a question. Electron charge, e = 1.60 10-19 C Electron mass, me = 9.11 x 10-31 kg Proton mass, mp = 1.67 10-27 kg Permittivity of free space, eo = 8.85 x 10-12 C2N -1 m 2 Permeability of free space,/.to = 1.26 10-6 T ma 1 Velocity of light in vacuo, c - 3.00 x 108 m s -1 Acceleration due to gravity, g - 9.81 m s 2 Gravitational constant, G -- 6.67 x 10 1~ N m 2 kg 2 Section A. Write down expressions, defining the symbols used, for the inverse-square laws relating to the electrostatic force (Coulomb's law) and the gravitational force. Calculate the electrostatic and gravitational forces between the electron and proton in a hydrogen atom, given that the distance separating these particles is 5 10 "11 m, and comment on the relative magnitudes of the forces. Briefly describe which other force is present, as well as the electrostatic and gravitational forces, that is important in the nucleus of a helium atom.. Write down expressions for the electric potential V and the electric field E a distance x along the x-axis from a point charge +q, and define the vector operator V. Use Cartesian coordinates, and state the direction of E. Sketch the electric field lines and the lines of equal electric potential in a plane containing a single point charge +q. Sketch the electric field lines in a plane containing two points charges, one of size +2q, the other of size -q, separated by a distance d. Indicate the location of the 'neutral point' (the point where the net electric field is zero). [21 TURN OVER PHYS1B26/2002
. Define. Write down expressions, defining the symbols used, for: (i) (ii) the electrostatic energy of a set of n charges, ql, q2... qn at positions rl, r2... rn respectively, and the electrostatic energy of a continuous charge distribution p in terms of the electric potential V. Consider a straight row of N charges, alternately +q and -q, with a constant separation d. For N very large, calculate the electrostatic energy of each charge. [Hint: In (l+x) = x -xz/2 + x3/3 - x4/4...] Calculate the value of the electrostatic energy per charge if the charges are Na + and C1- ions of spacing 0.28 nm. the term 'dielectric'. Write down expressions, defining the symbols used, for the magnitude of the electric field between the charged plates of(i) an air-filled parallel plate capacitor, and (ii) a dielectric-filled parallel plate capacitor. (Assume uniform fields, neglecting edge effects.) Derive an expression for the induced charges on the surfaces of the dielectric adjacent to the metal plates of the capacitor. Sketch a cross-section of the dielectric-filled parallel-plate capacitor, indicating the signs of the charges on the contact surfaces of both the metal plates and the dielectric. [1] [1] CONTINUED PHYS 1 B26/2002
. For a positive point charge placed near an earthed flat metal surface, sketch and describe (i) the location of the image charge, and (ii) the locations of the real charges. Indicate the sign of the charges in both cases. An air-inflated spherical balloon of radius 10 cm and mass 5 g, made of a nonconducting material, is uniformly charged over its surface. What is the minimum charge required on the balloon so that, having been placed in contact with a ceiling, it will remain there? (Assume that the ceiling is an earthed fiat conducting plane, and ignore any buoyancy effects.) The balloon is now replaced by a solid polystyrene ball of the same radius and the same mass as the balloon. The polystyrene ball is uniformly charged over its volume. What is the minimum charge required on the polystyrene ball so that, having been placed in contact with a ceiling, it will remain there? [1]. State Amprre's law for steady currents in its integral form, defining the symbols used. A beam of protons from an accelerator has a circular cross section with radius a = 2 mm and constitutes a total current oflo = 200 ~A. Assuming a uniform charge distribution over the beam's cross section, calculate the magnetic field B at (i) 1 mm and (ii) 2 mm from the centre of the beam. [5]. Sketch an RL series circuit connected across a battery, and define the symbols used in the following equation: I = L(l_e -t/r ). R A coil has an inductance of 50 mh and a resistance of 0.3 fl Ifa 12 V battery is connected across the coil, how much energy U~q is stored in the magnetic field after the current has reached its equilibrium value? How long, in terms of z, would it take for the energy stored in the field to reach ½Ueq after the switch is closed to complete the circuit? TURN OVER PHYS 1 B26/2002
. Briefly describe (i) the source of energy, (ii) the two ways in which energy is stored, and (iii) the one way in which energy is dissipated, it, an alternating-current RLC series circuit. Write down an expression, defining the symbols used, for the impedance Z of an RLC series circuit. Give a definition of the resonant frequency o30 of the circuit. The average power consumption of an RLC series circuit is given by the following equation: R P av Z2 Show that Pay has a maximum at a certain frequency, and sketch the form of this resonance peak as a function of the frequency of the applied alternating voltage. CONTINUED PHYS1B26/2002
Section B 9. (a) State Gauss' Law both in words and in its integral form. A non-conducting sphere of radius R has a uniform charge density p(r) = p, with total charge Q - 4~zpR3/3. Derive an expression for the electric field as a function of the distance r from the centre of the sphere in terms of the total charge Q, (i) inside the sphere, and (ii) outside the sphere. Prove that the field is continuous at the surface of the sphere. Another non-conducting sphere of radius R has a non-uniform charge density p(r) = Ar 2, where A is a constant. Determine the total charge Q on the sphere in terms of A and R. Derive expressions for the electric field as a function of the distance r from the centre of the sphere (i) inside the sphere, and (ii) outside the sphere. [21 [1] [1] (b) A dipole consists of two charges -q and +q situated at (x, y) = (-d, 0) and (d, 0) respectively. Calculate the magnitude and direction of the electric field E at the point P~ = (4d, 0). Calculate the magnitude and direction of the electric field E at the point P2 = (0, 4d). On the same diagram, sketch (i) the electric field lines and (ii) the equipotential surfaces, around the dipole, in the (x,y) plane. TURN OVER PHYS1B26/2002
10. (a) A long coaxial cable of length I consists of an inner wire of radius a surrounded by a concentric thin metal sheath of radius b, as shown in the figure below. The inner wire is given a charge +Q and the outer sheath is earthed, and so has a potential of zero volts. Io In terms of the total charge Q on the inner wire, derive expressions for the electric field strength E as a function of the radial distance r from the central axis in the region a < r < b : (i) if the gap between the two wires is filled with air (dielectric constant 1 ~ 1), and (ii) if the gap between the two wires is filled with a plastic of dielectric constant 1 =2. What is the value of E in the regions r < a and r > b? For the air-filled cable, derive an expression for the potential V as a function of the radial distance r from the central axis in the region a < r < b. Hence show that the capacitance per unit length of the air-filled cable is given by C = 2rtao/In(b/a). [5] (b) Air suffers electrical breakdown for electric fields in excess of 3 10 6 Vm -1. Derive a general expression for the maximum sustainable potential difference across an airfilled capacitor of the type described in section (a) of this question, and evaluate it for the case of a cable with a - 3 mm and b = 5 mm. [5] CONTINUED PHYS1B26/2002
11. (a) State the Kirchhoff Rules for electrical networks, and briefly state which conservation law each of the Rules is derived from. Determine the currents 11, I2 and h in the circuit below: [5] 2. d'a. t ~ T" 4V ~" " " :l (b) Define the symbols used in the following expression of the Biot-Savart law: B = ;o_az fas x 4re d r 2 Show that the magnetic field B at the common centre of curvature O of two concentric quartercircles of radii R1 and R2 which form a circuit as shown at right, has magnitude, when a current 1 flows, of: 8 R2 " Indicate the direction of this field. The figure at right shows three circuits carrying the same current/, made of concentric arcs (of radii r, 2r and 3r) and radial lengths. Rank them according to magnitude of B produced at the centre of curvature (the dot), greatest first, and explain the reasons for your choice. "'-," P ~a 0 (a) (b) (c) [6] TURN OVER PHYS1B26/2002
12. (a) State Faraday's law in its integral form, for a surface Sbounded by a closed loop F, defining the symbols used. A smooth metal rod of negligible resistance is made to slide at a constant velocity v in the direction shown in the figure below over two horizontal parallel rails, also of negligible resistance, a distance d apart. The circuit is completed by a resistance R connecting the two rails, as shown. The whole system is in a constant magnetic field B, as shown..l Calculate the EMF e induced in the circuit in terms of d, B and v. Calculate the current that flows in the circuit and indicate its direction. Calculate the force on the rod and indicate its direction. Determine the power dissipated in the resistor. (b) Determine the EMF induced around a plane circular loop of wire of area A rotating at an angular velocity to about an axis which is a diameter of the circle in the presence of a uniform magnetic field B directed perpendicular to the axis of rotation. (c) An aeroplane with a wingspan w = 25 m is flying horizontally (with its wings in the horizontal plane) at a speed ofv = 750 km/hour in a region where the vertical component of the earth's magnetic field is B = 5 x 10-5 T. By considering the equilibrium situation in which the Lorentz force acting on the charges in the wing are balanced by the induced electric field, find the potential difference AVbetween the wingtips. [5] CONTINUED PHYS 1 B26/2002
13. (a) An RLC series circuit is connected to a voltage source of angular frequency o3 = 100n rad s -1. An AC voltmeter, which measures the root mean square voltage, is placed across R, L and C in turn and records 10, 20 and 30 V respectively. Draw a phasor diagram to show the amplitude and relative phase of each of these voltages. Determine the amplitude of the applied voltage. If the root-mean-square current taken from the source is 2 A, find the values of R, L and C. The frequency of the applied voltage is now altered, without altering the peak voltage, until a current resonance occurs at an angular frequency COo. Show that, at resonance, the impedance of the circuit is a minimum. Calculate values for o30 and for the root-mean-square current at resonance. [41 (b) The circuit shown below is a ladder &resistors with n rungs. ( t~ 0.~ (~x-,) (ex) Find the equivalent resistance between points P~ and P2 for n = 1, n = 2, and n = 3. [6] END OFPAPER PHYS 1B26/2002