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PH504/10/Test UNIVERSITY OF KENT SCHOOL OF PHYSICAL SCIENCES CLASS TEST PH504-A Friday 12th November 2010 These questions will be marked each out of 25. Answering the four questions should take 40 minutes). Question S4 is printed overleaf. S1. In some region of space, the scalar electrostatic potential is the following function of Cartesian coordinates x, y, and z: V(x,y,z) = x2 y + xy2, where the potential is measured in volts and the distances in meters. Find the electric field at the point x = 3 m, y = 3 m, z = 3 m. . [10 marks] S2. State how capacitance C, charge Q and potential V are related. Find the potential energy of a capacitor with C = 1 pF (pico Farad) and charge Q = 1 mC (milli-Coulomb). [20 marks] S3 A polythene sheet with relative permittivity r = 2.3 is placed transverse to an electric field of 1 x 103 V/m. Find the magnitudes of the electric field E, displacement D and polarisation P within the polythene. What is the surface charge density? (Note the formula D = r E + P). S1. The electric field is the gradient of -V The gradient can be expressed in vector form e.g. V (V /x,V /y,V /dz) Since V(x,y,z) = x2 y + xy2 V (2xy y 2, x 2 2xy,0) Therefore E V E (27,27,0) S2. V/m C = Q/V where V is the electric potential. Potential energy is U = QV/2 = Q2 / 2C = 0.5 x 10-6 / 10-12 = 5 x 10 5 J S3. E0 = 1 x 103 V/m The dielectric generates a field which opposes the external field, reducing it by the factor r: E = E0 / r = 4.35 x102 V/m, The displacement is the same inside and outside the polythene: D = D o = o E0 = 8.85 x 10-9 Cm-2, P = D - o E =5 x 10-9 Cm-2 = , the surface charge density.