# Physics

Physics 226 Fall 2013

Problem Set #3 1) A charge of –1.5 C is placed on the x axis at

x = +0.55 m, while a charge of +3.5 C is placed at the origin. (a) Calculate the magnitude and direction of the net electric field on the x-axis at x = +0.8 m. (b) Determine the magnitude and direction of the force that would act on a charge of –7.0 C if it was placed on the x axis at x = +0.8 m.

2) For the same charge distribution of problem #1, do the

following. (a) Calculate the magnitude and direction of the net electric field on the x-axis at x = +0.4 m. (b) Determine the magnitude and direction of the force that would act on a charge of –7.0 C if it was placed on the x axis at x = +0.4 m.

3)

Charges are placed at the three corners of a rectangle as shown. The charge values are q1 = 6 nC, q2 = – 4 nC, and q3 = 2.5 nC. Calculate the magnitude and direction of the electric field at the fourth corner.

4) For the same charge distribution of problem #3, with the

exception that you change both q1 and q2 to the opposite sign, calculate the magnitude and direction of the electric field at the fourth corner.

5) A drop of oil has a mass of 7.5 x 10–8 kg and a charge of

– 4.8 nC. The drop is floating close the Earth’s surface because it is in an electric field. (a) Calculate the magnitude and direction of the electric field. (b) If the sign of the charge is changed to positive, then what is the acceleration of the oil drop? (c) If the oil drop starts from rest, then calculate the speed of the oil drop after it has travels 25 cm.

6) A proton accelerates from rest in a uniform electric field

of magnitude 700 N/C. At a later time, its speed is 1.8 x 106 m/s. (a) Calculate the acceleration of the proton. (b) How much time is needed for the proton to reach this speed? (c) How far has the proton traveled during this time? (d) What is the proton’s kinetic energy at this time?

7) All the charges above are multiples of “q” where

q = 1μC. The horizontal and vertical distances between the charges are 25 cm. Find the magnitude and direction of the net electric field at point P.

8) Use the same charge distribution as in problem #7 but

change all even-multiple charges to the opposite sign. Find the magnitude and direction of the net electric field at point P.

9) In the above two diagrams, M & S, an electron is given an

initial velocity, vo, of 7.3 x 106 m/s in an electric field of 50 N/C. Ignore gravitation effects. (a) In diagram M, how far does the electron travel before it stops? (b) In diagram S, how far does the electron move vertically after it has traveled 6 cm horizontally? (Hint: Think projectile motion)

– +

+ P

q3 q2

q1

35 cm

20 cm

– 8q

– 4q

+9q

+9q

– 5q

+6q +6q

+2q

P

– vo – vo

S M

10) A 2 g plastic sphere is suspended by a 25 cm long piece of string. Do not ignore gravity. The sphere is hanging in a uniform electric field of magnitude 1100 N/C. See diagram. If the sphere is in equilibrium when the string makes a 20 angle with the vertical, what is the magnitude and sign of the net charge on the sphere?

11) You have an electric dipole of

opposite charges q and distance 2a apart. (a) Find an equation in terms of q, a, and y for the magnitude of the total electric field for an electric dipole at any distance y away from it. (b) Find an equation in terms of q, a, and y for the magnitude of the total electric field for an electric dipole at a distance y away from it for when y >> a.

12)

An dipole has an electric dipole moment of magnitude 4 μC·m. Another charge, 2q, is located a distance, d, away from the center of the dipole. In the diagram all variables of q = 20 μC and d = 80 cm. Calculate the net force on the 2q charge.

13) An electric dipole of charge 30 μC and separation 60 mm is put in a uniform electric field of strength 4 x 106 N/C. What the magnitude of the torque on the dipole torque on dipole in a uniform field when (a) the dipole is parallel to the field, (b) the dipole is perpendicular to the field, and (c) the dipole makes an angle of 30º to the field. 20º

14) An electron of charge, – e, and mass, m, and a positron of

charge, e, and mass, m, are in orbit around each other. They are a distance, d, apart. The center of their orbit is halfway between them. (a) Name the force that is acting as the centripetal force making them move in a circle. (b) Calculate the speed, v, of each charge in terms of e, m, k (Coulomb’s Constant), and d.

15) A ball of mass, m, and positive charge, q, is dropped from

rest in a uniform electric field, E, that points downward. If the ball falls through a height, h, and has a velocity of

gh2v , find its mass in terms of q, g, and E.

16) The two charges above are fixed and cannot move. Find a

point in space where the total electric field will equal zero.

ANSWERS:

1) a) 6.08 x 104 N/C, WEST

b) 0.426 N, EAST 2) a) 7.97 x 105 N/C, EAST b) 5.6 N, EAST 3) 516 N/C, 61.3º 4) 717 N/C, 69.8º 5) a) 153.1 N/C, SOUTH b) 19.6 m/s2

c) 3.13 m/s 6) a) 6.71 x 1010 m/s2

b) 2.68 x 10–5 s c) 24.1 m d) 2.71 x 10–15 J 7) 1.23 x 106 N/C, 80.5º 8) 3.06 x 105 N/C, 48.4º

9) a) 3.04 m b) 2.97 m 10) 6.49 x 10–6 C

11) a) 222 ay kqay4

b) 3y kqa4

12) 5.81 N 13) a) 0 b) 7.2 N·m c) 3.6 N·m

14) md2 kev

15) g

Eq m

16) 8.2 cm

–

+ y q

a

a

–q – 4 μC 12 μC

– +

– + – q

d

q q +

6 cm

Physics 226 Fall 2013

Problem Set #4

NOTE: Any answers of zero must have some kind of justification. 1) You have a thin straight wire of

charge and a solid sphere of charge. The amount of charge on each object is 8 mC and it is uniformly spread over each object. The length of the wire and the diameter of the sphere are both 13 cm. (a) Find the amount of charge on 3.5 cm of the wire. (b) For the sphere, how much charge is located within a radius of 3.5 cm from its center?

2) A uniform line of charge with density, λ, and length, L

is positioned so that its center is at the origin. See diagram above. (a) Determine an equation (using integration) for the magnitude of the total electric field at point P a distance, d, away from the origin. (b) Calculate the magnitude and direction of the electric field at P if d = 2 m, L = 1 m, and λ = 5 μC/m. (c) Show that if d >> L then you get an equation for the E-field that is equivalent to what you would get for a point charge. (We did this kind of thing in lecture.)

3)

A uniform line of charge with charge, Q, and length, L, is positioned so that its center is at the left end of the line. See diagram above. (a) Determine an equation (using integration) for the magnitude of the x-component of the total electric field at point P a distance, d, above the left end of the line. (b) Calculate the magnitude and direction of the x-component of the total electric field at point P if d = 1.5 m, L = 2.5 m, and Q = – 8 μC. (c) What happens to your equation from part (a) if d >> L? Conceptually explain why this is true.

4)

You have a semi-infinite line of charge with a uniform linear density 8 μC/m. (a) Calculate the magnitude of the total electric field a distance of 7 cm above the left end of line. (You can use modified results from lecture and this homework if you like … no integration necessary.) (b) At what angle will this total E-field act? (c) Explain why this angle doesn’t change as you move far away from the wire. Can you wrap your brain around why this would be so?

d

5)

A uniform line of charge with charge, Q, and length, D, is positioned so that its center is directly below point P which is a distance, d, above. See diagram above. (a) Determine the magnitude of the x-component of the total electric field at point P. You must explain your answer or show calculations. (b) Calculate the magnitude and direction of the y-component of the total electric field at P if d = 2 m, D = 4.5 m, and Q = –12 μC. HINT: You can use integration to do this OR you can use one of the results (equations) we got in lecture and adapt it to this problem.

6) You have an infinite line of charge of constant linear

density, λ. (a) Determine an equation for the magnitude of the total electric field at point P a distance, d, away from the origin. Use any method you wish (except Gauss’ Law) to determine the equation. There’s at least three different ways you could approach this. You can use the diagram in #5 where D → if you want a visual. (b) Calculate the electric field at d = 4 cm with λ = 3 μC/m.

P + + + + + +

0 2 L

2 L

13 cm

P

0

d

– – – – – – –

L

P

0

7 cm

+ + + + +

P

d

– – – – – – – D

7)

You have three lines of charge each with a length of 50 cm. The uniform charge densities are shown. The horizontal distance between the left plate and right ones is 120 cm. Find the magnitude and direction of the TOTAL E-field at P which is in the middle of the left plate and the right ones.

8) For the same charge distribution of problem #7, with the

exception that you change the sign of the 4 μC plate and you change the distance between the plates to 160 cm, find the magnitude and direction of the TOTAL E-field at P which is in the middle of the left plate and the right ones.

9) You have 3 arcs of charge, two ¼ arcs and one ½ arc.

The arcs form of circle of radius 5 cm. The uniform linear densities are shown in the diagram. (a) Using an integral and showing your work, determine the equation for the electric field at point P due to the ½ arc. (b) Calculate the magnitude and direction of the total electric field at point P.

10) For this problem use the same charge distribution as

problem #9, with the exception of changing all even charges to the opposite sign. (a) Using an integral and showing your work, determine the equation for the electric field at point P due to the ½ arc. (b) Calculate the magnitude and direction of the total electric field at point P.

11) You have two thin discs both

of diameter 26 cm. They also have the same magnitude surface charge density of, 20 μC/m2, but opposite sign. The charge is uniformly distributed on the discs. The discs are parallel to each

other and are separated buy a distance of 30 cm. (a) Calculate the magnitude and direction of the total electric field at a point halfway between the discs along their central axes. (b) Calculate the magnitude and direction of the total electric field at a point halfway between the discs along their central axes if the diameter of the discs goes to infinity. (c) Determine the total electric field at a point halfway between the discs along their central axes if discs have charge of the same sign.

+ + +

–

– 5 μC/m – 4 μC/m –

+ + +

P

3 μC/m 12) You have two concentric thin rings of

charge. The outer ring has a dia- meter of 50 cm with a uniformly spread charge of – 15 μC. The inner ring has a diameter of 22 cm with a uniform linear charge density of 15 μC/m. Calculate the magnitude and direction of the total E-field at point P which lies 40 cm away from the rings along their central axes.

13) A proton is released from rest 5 cm away from an infinite

disc with uniform surface charge density of 0.4 pC/m2. (a) What is the acceleration of the proton once it’s released? (b) Calculate the kinetic energy of the proton after 2.5 s. [See Conversion Sheet for metric prefixes.]

2 μC/m 14) In the above two diagrams, G & L, an electron is given

an initial velocity, vo, of 7.3 x 106 m/s above infinite discs with uniform surface charge density of –0.15 fC/m2. (a) In diagram G, how much time passes before the electron stops? (b) In diagram L, how far does the electron move horizontally after it has traveled 20 m vertically? (Hint: Think projectile motion)

15) Two thin infinite planes

of surface charge density 6 nC/cm2 intersect at 45º to each other. See the diagram in which the planes are coming out of the page (edge on view). Point P lies 15 cm from each plane. Calculate the magnitude and direction of the total electric field at P.

–

+

– – 2 μC/m 5 μC/m

+

+ +

+

+

+

P

– + P

P

– –

– –

L

vo vo

G

P

45º

ANSWERS:

1) a) 2.15 mC b) 1.25 mC

2) a) 22 Ld4 Lk4

b) 1.2 x 104 N/C, EAST

3) a)

22x Ld

d1 dL Qk

E

b) 9322 N/C, EAST c) 0 4) a) 1.46 x 106 N/C b) 45º c) Because Ex = Ey 5) a) 0 b) 3.63 x 105 N/C, SOUTH 6) 1.35 x 106 N/C, NORTH

7) 5.93 x 104 N/C, 13.6º 8) 2.37 x 104 N/C, 59.8º

9) a) R k2Ey

b) 4.85 x 105 N/C, 22.0º

R k210) a) E y

b) 2.05 x 106 N/C, 74.8º 11) a) 5.53 x 105 N/C, WEST b) 2.26 x 106 N/C, WEST c) 0 12) 1.01 x 105 N/C, WEST 13) a) 2.17 x 106 m/s2 b) 2.45 x 10–14 J 14) a) 4.9 s b) 3780 m 15) 2.6 x 106 N/C, 22.5º

Physics 226 Fall 2013

Problem Set #5

NOTE: Any answers of zero must have some kind of justification. 1)

A uniform electric field of strength 300 N/C at an angle of 30º with respect to the x-axis goes through a cube of sides 5 cm. (a) Calculate the flux through each cube face: Front, Back, Left, Right, Top, and Bottom. (b) Calculate the net flux through the entire surface. (c) An electron is placed centered 10 cm from the left surface. What is the net flux through the entire surface? Explain your answer.

2)