MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2014 Exam One Solutions Problem 1(25 points):answers without work shown will not be given any credit. Four point-like objects of charge-20,+,-O,and +20 respectively are placed at the corners of a square of side a as shown in the figure below. +0 -0 a/2 a/2 +x + -2Q +2Q Part(a) Determine the electric field at the point P,which has coordinates()=(a/2,a) Express your answer in terms of k=1/4re,,a,i,and j as needed. Part(b) Choose infinity to be the zero point for the electric potential (V()=0).Determine the electric potential at the point P,which has coordinates (x)=(a/2,a).Express your answer in terms of k=1/4,a,i,and j as needed. Part(c) Determine the direction and magnitude of the electric dipole moment of this charge configuration.Express your answer in terms of k=1/4re,,a,i,and j as needed. 1
1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2014 Exam One Solutions Problem 1 (25 points): answers without work shown will not be given any credit. Four point-like objects of charge −2Q , +Q , −Q , and +2Q respectively are placed at the corners of a square of side a as shown in the figure below. Part (a) Determine the electric field at the point P , which has coordinates (xP , yP ) = (a / 2,a). Express your answer in terms of k = 1/ 4πε 0 , Q, a , ˆ i , and ˆ j as needed. Part (b) Choose infinity to be the zero point for the electric potential (V (∞) = 0 ). Determine the electric potential at the point P , which has coordinates (xP , yP ) = (a / 2,a). Express your answer in terms of k = 1/ 4πε 0 , Q , a , ˆ i , and ˆ j as needed. Part (c) Determine the direction and magnitude of the electric dipole moment of this charge configuration. Express your answer in terms of k = 1/ 4πε 0 , Q , a , ˆ i , and ˆ j as needed
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Problem 2(25 points):answers without work shown will not be given any credit. A stick with length L has a positive charge O uniformly distributed on it.It lies along the x-axis between the points x=0 and x=L.A point-like object with identical positive charge O lies on the x-axis at the point x=2L. uniformly charged stick point-like object of length L and charge O with charge O x=0 x=L x=a x=2L You may find the following integrals useful: 人 =-ln(a-x) x dx J (a-x)(a-x) jeya严 dx 1 xdx 1 62+y Part(a) Let x=a be the point on the x-axis between the two objects where the electric field is zero.Find a.Express your answer in terms of k=1/4,,and L as needed. Part(b) Choose the zero point for the electric potential to be at infinity V()=0.Determine the electric potential at the point x=a.You may express your answer in terms of any of the quantities mentioned in part (a),whether or not you answered part (a);in particular in terms of k=1/4n,,L,and a as needed. 7
7 Problem 2 (25 points): answers without work shown will not be given any credit. A stick with length L has a positive charge Q uniformly distributed on it. It lies along the x -axis between the points x = 0 and x = L . A point-like object with identical positive charge Q lies on the x -axis at the point x = 2L . You may find the following integrals useful: dx a − x = −ln(a − x) ∫ dx (a − x) 2 = 1 (a − x) ∫ dx (x 2 + y 2 ) 3/2 = 1 y 2 x (x 2 + y 2 ) ∫ 1/2 xdx (x 2 + y 2 ) 3/2 = − 1 (x 2 + y 2 ) ∫ 1/2 Part (a) Let x = a be the point on the x -axis between the two objects where the electric field is zero. Find a . Express your answer in terms of k = 1/ 4πε 0 , Q , and L as needed. Part (b) Choose the zero point for the electric potential to be at infinity V (∞) = 0 . Determine the electric potential at the point x = a . You may express your answer in terms of any of the quantities mentioned in part (a), whether or not you answered part (a); in particular in terms of k = 1/ 4πε 0 , Q , L , and a as needed
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