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Unformatted text preview: 1 TUTORIAL 4 1. (SADIKU, p. 67, practice exercise 3.3) Determine the gradient of the following scalar fields: (a) U = x 2 y + xyz ; (b) V = " z sin # + z 2 cos 2 # + " 2 ; (c) f = cos " sin # ln r + r 2 # . 2. Sketch the gradients to the above surface equipotentials. 3. (SADIKU, p. 81, practice exercise 3.8) Determine the curl of the following vector fields, and 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 = " z sin # a " + 3 " z 2 cos # a # at (5, /2, 1); (c) C = 2 r cos " cos # a r + r a # at (1, /6, /3). 4. Recall that the circulation of the electrostatic field is zero. Show that for any scalar field that " #" $ = . 2 5. (SADIKU p. 140-142, example 4.12) Given the potential V = 10 r 2 sin " cos # , (a) Find the electric flux density D at (2, /2, 0). (b) Calculate the work done in moving a 10 C charge from point A (1, 30 , 120 ) to B (4, 90 , 60 ). 6. Consider a system of N point charges Q i . Evaluate the potential at a point P distant from the system of charges by proceeding as follows: (i) write the general expression for the potential using the Principle of Superposition. (ii) using the cosine rule express the displacement R i in terms of r and r i (iii) power series expand the denominator, keeping terms to order 1/ r 3 . You have just completed to third order a general multipole expansion - interpret the terms in your results! 7. (SADIKU problem 4.36, p. 160) (a) Prove that when a particle of constant mass and charge is accelerated from rest in an electrostatic field, its final velocity is proportional to the square root of the potential difference through which it is accelerated. (b) Find the magnitude of the constant of proportionality if the particle is an electron....
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- Spring '07