Exam 1 -- Fall 2003 solutions

# Exam 1 -- Fall 2003 solutions - Physics 240 Fall 2003...

• Notes
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1: Imagine a uniform sphere of charge with a radius R and total charge Q. A point charge q placed on the surface of this sphere feels a repulsive force F=kqQ/R 2 . I now remove a small sphere of charge, centered on a point R/2 from the center of the sphere, with radius R/2, on the side opposite our test charge q. What is the magnitude of the force repelling q now? a) kqQ/2R 2 b) *17kqQ/18R 2 c) 3kqQ/4R 2 d) 7kqQ/8R 2 e) kqQ/R 2 2: Three identical objects, each with charge Q, sit on corners of a square with edge length L. What is the magnitude of the electrostatic force on the charge in the upper right corner? a) 2.8kQ 2 /L 2 b) kQ 2 /L 2 c) *1.4kQ 2 /L 2 d) 3.4kQ 2 /L 2 e) 2.0kQ 2 /L 2 Q q R q R Empty region Q Q Q L To find the force in the situation on the right we have to calculate how much force the now missing sphere was applying before it was removed. This force is: kqQ little /(3/2R) 2 =4kqQ little /9R 2 What is the charge in the now empty region? 4/3 π (R/2) 3 * ρ , where ρ is the charge density in the original sphere Q/(4/3 π R 3 ). Combining these we have: Q little = Q*((R/2) 3 / R 3 ) = Q/8 Putting this in the above force calculation we get: F little = kqQ/18R 2 This means that the total force in the second picture is kqQ/R 2 – kqQ/18R 2 = 17kqQ/18R 2 Each of the other two charges produces a force, with magnitude kQ 2 /r 2 , acting on the charge at the corner. The one from the upper left acts horizontally, the one from the lower right acts vertically. Adding these together produces a total force with magnitude: 2kQ 2 /r 2 1.4kQ 2 /r 2
3: Consider the electric dipole shown in the figure below. What is the electric field at a distance x along the perpendicular bisector of the dipole? a) y x kQ ˆ 2 b) * ( ) y d x kQd ˆ 2 2 3 2 2 + c) ( ) y d x kQ ˆ 2 2 + d) x d x kQ ˆ 4 2 2 + e) Zero 4: An infinite plane of charge creates an electric field which is uniform in space, and has a magnitude σ /2 ε 0 . A finite disk of charge (radius R) creates a field which, along its central axis, has the value: Now, imagine an infinite plane of charge with charge density σ , from which a hole of radius R has been removed. What is the magnitude of the electric field at a point P a distance R directly above the center of the hole?

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