2000E2. Three particles, A, B, and C, have equal positive charges Q and are held in place at the vertices of an
equilateral triangle with sides of length l, as shown in the figures below. The dotted lines represent the bisectors for each side. The base of the triangle lies on the x-axis, and the altitude of the triangle lies on the y-axis.
a.
i. Point P1, the intersection of the three bisectors, locates the geometric center of the triangle and is one point
where the electric field is zero. On Figure 1 above, draw the electric field vectors EA, EB, and EC at P, due to each of the three charges. Be sure your arrows are drawn to reflect the relative magnitude of the fields.
ii. Another point where the electric field is zero is point P2 at (0, y2). On Figure 2 above, draw electric field
vectors EA, EB, and EC at P2 due to each of the three point charges. Indicate below whether the magnitude of each of these vectors is greater than, less than, or the same as for point P1.
EA EB EC Greater than at P1 Less than at P1 The same as at P1 b. Explain why the x-component of the total electric field is zero at any point on the y-axis.
c. Write a general expression for the electric potential V at any point on the y-axis inside the triangle in terms of Q,
l, and y.
d. Describe how the answer to part (c) could be used to determine the y-coordinates of points P1 and P2 at which
the electric field is zero. (You do not need to actually determine these coordinates.)
Copyright ? 2000 by College Entrance Examination Board. All rights reserved.
College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board.
2000E3. A capacitor consists of two conducting, coaxial, cylindrical shells of radius a and b, respectively, and length L >> b. The space between the cylinders is filled with oil that has a dielectric constant κ. Initially both
cylinders are uncharged, but then a battery is used to charge the capacitor, leaving a charge +Q on the inner cylinder and -Q on the outer cylinder, as shown above. Let r be the radial distance from the axis of the capacitor.
a. Using Gauss's law, determine the electric field midway along the length of the cylinder for the following values
of r, in terms of the given quantities and fundamental constants. Assume end effects are negligible. i. a < r < b ii. b < r << L
b. Determine the following in terms of the given quantities and fundamental constants.
i. The potential difference across the capacitor ii. The capacitance of this capacitor
c. Now the capacitor is discharged and the oil is drained from it. As shown above, a battery of emf ? is connected
to opposite ends of the inner cylinder and a battery of emf 3? is connected to opposite ends of the outer cylinder. Each cylinder has resistance R. Assume that end effects and the contributions to the magnetic field from the wires are negligible. Using Ampere's law, determine the magnitude B of the magnetic field midway along the length of the cylinders due to the current in the cylinders for the following values of r. i. a < r < b ii. b Copyright ? 2000 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board.