- (3%) Problem 27: Consider an electric dipole where the positive charge, +q, is located on...

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- (3%) Problem 27: Consider an electric dipole where the positive charge, +q, is located on the positive z axis a distance d/2 above the x - y plane, and the negative charge, -q, is located on the negative z axis a distance d/2 below the x y plane, as shown. If the evaluation point is at a distance which is very large as compared to the separation between the charges, then the form of the expression for the net electric field has a very simplified form in terms of the electric dipole moment (EDM). 2 +q d/2 d/2 -9 13% Part (a) Enter an exact expression for the magnitude of the vector E+, the contribution of the positive charge to the electric field at the point (0,0,z) where z> d/2. = = (1/4) (q/( z - d/2) 8 9 HOME Grade Summary Deductions 0% Potential 100% Late Work % 100% Late Potential 100% 63 52 d ()7 ^^4 g h m * 1 n q u + - y ) BACKSPACE Submit Hint Submissions Attempts remaining: 993 (0% per attempt) detailed view END 1 0% DEL CLEAR 2 0% 3 0% 4 0% Feedback I give up! Hints: for a 0% deduction. Hints remaining: 0 -The magnitude of the electric field is given by an inverse-square law. Look at the figure in the problem statement, and consider carefully what you use for the distance between the charge and the evaluation point. Feedback: 0% deduction per feedback. 13% Part (b) Enter an exact expression for the magnitude of the vector E_, the contribution of the negative charge to the electric field at the point (0,0,z) where z> d/2. A 13% Part (c) The function f(x) = (1 + x)- may be rewritten using a Taylor expansion, and when x is very small, the approximate form simplifies to f(x) = a + x + where a and are simple numeric factors, and the higher-order terms are negligibly small. Input the form of the approximate expression with the correct numeric factors. ... A 13% Part (d) Use the approximate expression for f(x) = (1 + x)- previously entered to obtain and enter a simplified expression for (1 - d/(2z))-2 in the limit where z >>> d. A 13% Part (e) Use the approximate expression for f(x) = (1 + x)- previously entered to obtain and enter a simplified expression for (1 + d/(2z))- in the limit where z d. 13% Part (f) Apply the approximations obtained in the previous steps to the exact expressions obtained earlier for + and . Combine these expressions to obtain the net electric field, Enet, at the point (0,0,z) in the limit where z d. Enter your vector result. A 13% Part (g) The EDM (electric dipole moment) is represented by a vector, p. The magnitude of the vector is the product of the positive charge and the separation between the charges. The direction is from the negative and towards the positive charge. Rewrite your vector expression for the electric field of the dipole in terms of the EDM, p. 13% Part (h) Which statement best compares the asymptotic behavior of the electric field of a dipole and the electric field of a point charge? - (3%) Problem 27: Consider an electric dipole where the positive charge, +q, is located on the positive z axis a distance d/2 above the x - y plane, and the negative charge, -q, is located on the negative z axis a distance d/2 below the x y plane, as shown. If the evaluation point is at a distance which is very large as compared to the separation between the charges, then the form of the expression for the net electric field has a very simplified form in terms of the electric dipole moment (EDM). 2 +q d/2 d/2 -9 13% Part (a) Enter an exact expression for the magnitude of the vector E+, the contribution of the positive charge to the electric field at the point (0,0,z) where z> d/2. = = (1/4) (q/( z - d/2) 8 9 HOME Grade Summary Deductions 0% Potential 100% Late Work % 100% Late Potential 100% 63 52 d ()7 ^^4 g h m * 1 n q u + - y ) BACKSPACE Submit Hint Submissions Attempts remaining: 993 (0% per attempt) detailed view END 1 0% DEL CLEAR 2 0% 3 0% 4 0% Feedback I give up! Hints: for a 0% deduction. Hints remaining: 0 -The magnitude of the electric field is given by an inverse-square law. Look at the figure in the problem statement, and consider carefully what you use for the distance between the charge and the evaluation point. Feedback: 0% deduction per feedback. 13% Part (b) Enter an exact expression for the magnitude of the vector E_, the contribution of the negative charge to the electric field at the point (0,0,z) where z> d/2. A 13% Part (c) The function f(x) = (1 + x)- may be rewritten using a Taylor expansion, and when x is very small, the approximate form simplifies to f(x) = a + x + where a and are simple numeric factors, and the higher-order terms are negligibly small. Input the form of the approximate expression with the correct numeric factors. ... A 13% Part (d) Use the approximate expression for f(x) = (1 + x)- previously entered to obtain and enter a simplified expression for (1 - d/(2z))-2 in the limit where z >>> d. A 13% Part (e) Use the approximate expression for f(x) = (1 + x)- previously entered to obtain and enter a simplified expression for (1 + d/(2z))- in the limit where z d. 13% Part (f) Apply the approximations obtained in the previous steps to the exact expressions obtained earlier for + and . Combine these expressions to obtain the net electric field, Enet, at the point (0,0,z) in the limit where z d. Enter your vector result. A 13% Part (g) The EDM (electric dipole moment) is represented by a vector, p. The magnitude of the vector is the product of the positive charge and the separation between the charges. The direction is from the negative and towards the positive charge. Rewrite your vector expression for the electric field of the dipole in terms of the EDM, p. 13% Part (h) Which statement best compares the asymptotic behavior of the electric field of a dipole and the electric field of a point charge?