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10. (15 points) Electric field and electric potential = As shown below, a square of side 2 contains a positive charge +Q fixed at the lower left corner and negative point charges -Q fixed at the other three corners of the square. Point P is located at the center of the square. 0 .P a + R a. On the diagram above, indicate with an arrow the direction of the net electric field at point P. (1 point) b. Derive expressions for each of the following in terms of the given quantities and fundamental constants, (6 points) i. The magnitude of the electric field at point P. Express your answer in terms of Q, a, and k. Hints: (i) first find r in terms of a and (ii) realize that the charges in the upper left corner and in the lower right corner cancel each other. (3 points) ii. The electric potential at point P. Remember that V = k 29/). Express your answer in terms of Q,r, and K.(3 points) C. A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint of the bottom side of the square as shown above. As the charge is moved, is the work done by the electric field positive, negative, or zero? Explain your reasoning. (3 points) Positive Negative Zero

d. (5 points) i. Describe one way to replace a single charge in this configuration that would make the electric field at the center of the square equal to zero. Justify your answer. (2 points) ii. Describe one way to replace a single charge in this configuration such that the electric potential at the center of the square is zero but the electric field is not zero. Justify your answer. (3 points)