Midterm #1 - Department of Physics



Exam 1 Solution

Closed book exam. A calculator is allowed.

Exam is worth 100 points, 25% of your total grade.

[pic]

1. Consider 3 point particles with electrical charge arranged in the form of an equilateral triangle as shown. The side length is a = 5 cm, and the top particle has charge

q1 = +2(C while the bottom two particles have charge q2 = q3 = +1 (C ([pic]).

a) [4 points] What is the direction of the force acting on particle 1

(the one at x = 0 and y > 0)?

The force must be repulsive since all charges are positive. By symmetry, since the bottom 2 charges have the same magnitude, the force could only point in the [pic]direction.

b) [8 points] Calculate the magnitude of the force acting on particle 1.

[pic] = force on particle one from particle 2

[pic]

1. Continued:

c) [6 points] Calculate the electric field at the location of particle 1 arising from particles 2 and 3 ?

[pic]

d) [6 points] Does a position exist where a fourth charge can be added to put the triangle in electrostatic equilibrium? If so, sketch where it would lie approximately, and write down an equation for its charge in terms of:

the distance d from particle 1, the side length a, and the magnitude of the other charges (in other words, you don’t have to solve for d).

Yes, add a negative charge inside triangle to pull charges back together. By symmetry, it will lie along y axis. Let’s solve for the total force acting on particle 1 to be zero, letting d be the distance from particle 1 on the y axis:

[pic]

[pic]

2. Consider electric charge distributed along a one-dimensional path in the form of a semi-circle with radius R = 5 m in the upper two quadrants of the x-y plane as shown. The semicircle has a center at (x, y) = (0,0) and has a linear charge density of ( = +2(C/m.

a) [6 points] Find component of the electric field along the x-axis (Ex) at the origin (0,0).

By symmetry, Ex = 0 (each side of semi-circle cancels out)

Otherwise, try explicitly [pic] instead of [pic]below.

b) [6 points] Find component of the electric field along the y-axis (Ey) at the origin (0,0).

Field points away from the positive line charge, so in [pic] direction.

[pic]

2. Continued:

c) [6 points] If a semi-circle is added to the lower two quadrants (y < 0, with the center at the origin again) with a linear charge density of equal magnitude but opposite sign to the upper semicircle (i.e. (2(C/m), calculate the total Ex and Ey at the origin again arising from both semicircles.

Field lines point toward negative charge, so in same direction as from upper semi-circle of positive charge. By symmetry, field strength doubles. Otherwise, try explicitly integrating ( from ( to 2( in previous problem.

[pic]

[pic]

3. Consider a cube with side length s = 3.0 m and one corner at the origin (0,0,0) as shown.

a) [6 points] What is the total electric flux emanating out through the surface of the cube if the electric field is [pic]? ([pic] is a unit vector pointing in the z-direction.)

The electric field points in the z direction. The only faces that will possibly contribute flux are the ones at z=0 and z=s, since [pic] for the other faces.

[pic]

In other words, the net flux through the cube’s surface for a constant field is zero. You could also use the differential form of Gauss’ Law:

[pic] since the electric field is a constant.

Thus, the total charge enclosed is zero, and so by the integral form of Gauss’ Law the flux must sum to zero.

b) [6 points] What is the total electric flux emanating out through the surface of the cube if the electric field is [pic]?

Now we have two components of the field, so 4 faces contribute:

[pic]

3. Continued:

c) [6 points] What is the total charge enclosed by the cube if the electric field is the same as in part (b)?

[pic]

d) [6 points] What is the electric charge density at the center of the right face at x = 3.0 m if the electric field is the same as in part (b)?

[pic]

[pic]

4. A very long cylinder (length L = 5 m) carrying a total electric charge of +5(C distributed uniformly along its length is surrounded by a concentric cylindrical shell with a total charge of (2.5(C distributed uniformly along its length as shown. The radius of the inner cylinder is a = 0.005 m and the radius of the outer cylinder is b = 0.01 m.

a) [6 points] What is the electric field strength and direction between the two cylinders (a ................
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