PHYS 251

STUDY GUIDE FOR PART I:

ELECTRIC FIELDS

Dr. Holmes

Electric Fields
Gauss' Law
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Electric Fields

Outline
Supplementary Homework Problems
Answers to Problems
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Outline

  1. Coulomb's law: basic electric force
    1. new force: Fe = k q1 q2 /r²
    2. similar to gravity: Fgr = G m1 m2 /r² S-1
    3. direction: can be either attractive or repulsive
    4. constant: k = 1/(4peo) = 9 x 109Nt m²/Coul² (like G = 6.67 x 10-11Nt m2/kg2)
    5. recall: force is a vector! (work in components) S-2
  2. Electric field
    1. definition: E = F(on qtest)/qtest
    2. for a point charge: magnitude: E = k q /r² ;
      direction: away from + charge; toward - charge.
    3. for several point charges: add as vectors! S-3,4
      Ex = S kqi cos(q i) / ri² ; Ey = S kqi sin(q i) / ri²
    4. for an extended charged object: add or integrate:
      Ex = ò k dq cos(q )/r² ; Ey = ò k dq sin(q )/r²
      1. line (dq = l dx) S-5
      2. on axis of a ring (dq = l dl )

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Supplementary Problems (S- ):

1) At what point between the earth and moon will an object neither fall toward the earth or toward the moon? (Neglect the motions of the earth and the moon, and neglect the gravity of the sun. The earth-moon distance is 384,000 km; the radius of the earth is 6400 km; the radius of the moon is 1750 km; the mass of the earth is 6.0 x 1024kg; the mass of the moon is 1.23% that of the earth. [Extra credit (+4): Include the motion of the moon orbiting the earth. Hint: the object will also orbit the earth with the same period as the moon.]

2) At a particular instant a particle of mass mA = 5 mg and charge of qA = 5 m Coul is located at the origin. A second particle of mass mb = 2 mg and charge of qB = -6 m Coul is located at that same instant at a position (-5 m, +3 m) relative to the origin. a) What is the force on qA due to the presence of qB ? b) Is the magnitude of the force on qB due to the presence of qA the same, greater or smaller than the magnitude of the force on qA due to the presence of qB ? c) Assuming the two charges have no other forces on them, what is the magnitude of the resulting acceleration of particle A due to the electrical interaction with particle A? d) Is the acceleration of particle B due to the presence of particle A the same, greater or smaller than the magnitude of the acceleration of particle A?

3) Consider two charges: qA has a charge of 5 nCoul and is located at (5 m, 3 m); qB has a charge of -4 nCoul and is located at (7 m, -2 m).
a) What is the electric field at the origin due to qA alone?
b) What is the electric field at the origin due to BOTH charges?

4) A horizontal wire of length l = 20 cm has a total charge Q = 2.0 m C uniformly distributed on the wire. Calculate the x and y components of the electric field at a field point located a distance a = 5 cm above the center of the wire. Do this by breaking the wire into five pieces; then finding the charge of each piece; then, assuming that this charge is a point charge located at the center of each piece, finding the x and y components of the electric field at the field point from each piece; and then adding up the fields due to each of the five pieces to get the total x and y components of the electric field. [Extra credit (+3): Write a computer program to do the above procedure, but break the wire into 20 (or more) segments instead of 5, and find a better approximation to the electric field at the indicated field point.]

5) For the charged wire of problem 4 above: (a) What is the charge density, l, on the wire? (b) What is the angle q L that the line from the left end of the wire to the field point located a distance a above the center of the wire makes with the wire? (c) What is the electric field strength E at this field point? (d) What is the direction of the electic field at this field point? (e) Compare your answer to the approximate answer from the numerical method of problem C above.

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Answers to Supplementary Problems:

1) 345,660 km from earth.

2) (a) 7.94x10-3 Nt; (b) same; (c) 1.59 m/s²; (d) acceleration of B is larger.

3) (a) E = 1.32 Nt/Coul at 211°; (b) E = .99 Nt/Coul at 241°

4) Ex = 0 Nt/Coul; Ey = 3.240 x 106 Nt/Coul

5) (a) 10.0 mC/m; (b) 26.2°; (c) 3.220x106 Nt/Coul; (d) straight up (away from the wire).

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Gauss' Law

Outline
Supplementary Homework Problems
Answers to Problems
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Outline

  1. Flux - definition
    1. example: volume flow and velocity vector: dV/dt = v dA
    2. electric flux = EdA
  2. Gauss' law: E· dA = 4p kQenclosed = Qenclosed/e o
    1. statement of the law
    2. using the law to find E (need symmetry to do integral)
      1. hollow sphere S-6
      2. solid sphere
      3. cylinder S-7
      4. plane S-8

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Supplementary Problems (S- ):

6) A hollow conducting sphere of radius 12 cm has a charge of 7 nCoul placed on it. Assume there are no other charges in the area. a) What is the magnitude and direction of the electric field 1 cm below the top of the sphere (inside the sphere) ? b) What is the magnitude and direction of the electric field 1 cm above the top of the sphere (outside the sphere) ?

7) A coaxial cable (assume a long cable) has an inner wire of radius 2 mm and an outer cylinder of radius 3 cm. A charge of -40 nCoul/m is placed on the inner wire and a charge of +50 nCoul/m is placed on the outer cylinder. Assume vacuum between the cylinder and the wire.
a) What is the electric field 1 cm from the axis (center) of the wire (8 mm from the surface of the wire, between the wire and the cylinder) ? b) What is the electric field at a radius of 4 cm (1 cm from the surface of the cylinder outside the cable) ?

8) Two plates are separated by 2 cm and are 200 cm by 200 cm in size.
(a) If a charge of +5 m Coul is transferred from an originally neutral bottom plate to a top plate (also originally neutral) which is parallel to the bottom plate, what is the electric field 0.5 cm below the top plate directly underneath the center of the top plate? (Assume a uniform distribution of the ±5 m Coul on the top/bottom plate.)
(b) Does the distance from the top plate to the field point affect the value of the field?
(c) If an electron were placed at the point 0.5 cm below the top plate, what would be the force on the electron?
(d) What would the acceleration of the electron be?
(e) Would this acceleration increase, stay the same, or decrease as the electron moved from this point in the direction of the acceleration?
(f) If the electron started at the negative plate from rest, how fast would it be going when it reached the positive plate?

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Answers to Supplementary Problems:

6) (a) 0 Nt/Coul (no direction);
(b) 3.73x103 Nt/Coul directed up.

7) (a) 7.2x104 Nt/Coul directed toward the axis;
(b) 4.5x103 Nt/Coul directed away from the axis.

8) (a) 1.4x105 Nt/Coul directed from top to bottom plate;
(b) the 0.5 cm distance does not affect the field unless it puts it outside the plates instead of between the plates;
(c) 2.24 x 10-14 Nt directed from the bottom plate to the top plate;
(d) 2.46 x 1016 m/s² directed from the bottom plate to the top plate;
(e) stay the same;
(f) 3.14 x 107 m/s.

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