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T02 - TUTORIAL 2 1(SADIKU p 74 practice exercise 3.6...

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TUTORIAL 2 1. (SADIKU, p. 74, practice exercise 3.6) Determine the divergence of the following vector fields. Also, evaluate them at the specified points: (a) A = yz a x + 4 xy a y + y a z at (1, –2, 3); (b) B = ρ z sin φ a ρ + 3 ρ z 2 cos φ a φ at (5, π /2, 1); (c) C = 2 r cos θ cos φ a r + r 1/ 2 a φ at (1, π /6, π /3). 2. Gauss’s Law says that if we have a point charge outside some closed surface S, then there is no flux of E from S. Prove this assuming only Coulomb’s Law. 3. Consider a charged conductor bounded by a closed surface S as shown in (a). (a) (b) In Lectures, we applied Gauss’s Law to show that any charge on the conductor resides on the surface S. Now consider the surface indicated in (b). i) Where does the charge reside? Consider that the conductor is now uncharged and contains a hollow cavity as shown in (c). (c) (d) ii) If the cavity contains charge Q cav , what is the charge on the inner surface of the conductor? iii) If the conductor is then charged and contains charge Q con , what are the charges on the inner and outer surface, respectively? We now place a second, initially uncharged, conductor in the cavity, as shown in (d) with other charges as previously. iv) What is the charge on the surface of the inner conductor?
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The inner conductor is now charged and contains charge Q inner . v) What happens to the charge on the outer surface of the outer conductor? vi) The charge on the outer conductor is now doubled. What happens to the charge on the inner conductor? What does this mean with regards to electrostatic screening? vii) We now ground the outer conductor. How does this change the screening properties? 4. In Lectures, we calculated E for a uniformly charged sphere of charge density ρ v and radius a . Calculate the field E for a conducting sphere of the same radius and carrying the same total charge. 5. A sphere of radius 10 cm has ρ v r = 0 01 3 . C / m 3 . If D is to vanish for r > 10 cm , calculate the value of a point charge that must be placed at the centre of the sphere.
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