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PROBLEM 6 (25 pts)
A sphere of radius 2a carries a positive volume charge density of ρ. A spherical
cavity of radius a is then scooped out and left empty, as shown in the figure
below.
1. A negative point charge -q is placed on the y axis at y = 4a. Based on the
superposition principle, find the magnitude and the direction of the force
acting on the charge. Explain (using 2-3 short sentences, max!) how you
apply the superposition principle to the physical situation in the present
problem (15 pts.);
2. The charge is now moved to the y = -4a point. Again, find the magnitude
and the direction of the force acting on it (10 pts.).
PROBLEM 7 (25 pts.)
The figure below shows a thin rod of length L The linear density of charge in
the rod is . The coordinates of the point P are: x = 0, y = h.
1. Find the expression for the vertical component Ey of the electric field
vector at point P (13 pts.);
2. Find the expression for the horizontal component Ex of the electric field
vector at point P (12 pts.)
Hint: It is not necessary to perform integration for finding the vertical component
Ey – there is an alternative procedure. Knowing the expression for the electric
field at a point located at distance d from the midpoint of a charged rod, one can
readily find the answer to Task 1 based on symmetry considerations. The point
credit for such solutions will be the same as for those obtained by performing
integration. However, such method works only for the Ey component, not for Ex!
PROBLEM 8 (25 pts.)
A simple pendulum consists of a small object of mass m = 10 g (the
"bob") suspended by a cord of length L = 1 m of negligible mass. When gravity
is the only force acting on the bob,
and the pendulum is in
equilibrium, the string is vertical (of course!), as shown in plot (a).
Then, the bob is charged with a positive charge of Q = 100 nC, and a
uniform horizontal electric field of strength E = 300 000 N/C is applied
to the pendulum. The pendulum moves to a new equilibrium position,
and now the string makes an angle (θ) with the vertical, as shown in
(b).
1. What is the tension force F1 in the string when only the force of
gravity is acting on the bob? (the situation in plot (a)) (1 pt.);
2. Find the value of θ in the “new” equilibrium position after the
electric field is applied (8 pts.);
3. Find the tension force F2 in the string in the “new” equilibrium
position, and express the result as the ratio of F2/F1 (8 pts.);
4. The known equation for the oscillation period of a simple “gravityonly driven” pendulum is T= 2π (L/g)1/2 . Express this equation in terms
of the tension force in the pendulum’s string (4 pts.);
5. Assuming that the result you get in Task 4 is generally valid, find
the pendulum’s oscillation period in the “new” equilibrium position
(i.e., corresponding to small oscillations of θ about its equilibrium
value) (4 pts.).