Physics 226 Fall 2013 #Problem Set 1

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Physics 226 Fall 2013 #Problem Set 1

NOTE: Show ALL work and ALL answers on a piece of separate loose leaf paper, not on this sheet.

Due on Thursday, August 29th

1) Skid and Mitch are pushing on a sofa in opposite directions with forces of 530 N and 370 N respectively. The mass of the sofa is 48 kg. The sofa is initially at rest before it accelerates. There is no friction acting on the sofa. (a) Calculate the acceleration of the sofa. (b) What velocity does the sofa have after it moves 2.5 m? (c) How long does it take to travel 2.5 m?

2) You have three force

vectors acting on a mass at the origin. Use the component method we covered in lecture to find the magnitude and direction of the re- sultant force acting on the mass.

3) You have three force

vectors acting on a mass at the origin. Use the component method we covered in lecture to find the magnitude and direction of the re- sultant force acting on the mass.

4) A bowling ball rolls off of a table that is 1.5 m tall. The

ball lands 2.5 m from the base of the table. At what speed did the ball leave the table?

5) Skid throws his guitar up

into the air with a velocity of 45 m/s. Calculate the maximum height that the guitar reaches from the point at which Skid lets go of the guitar. Use energy methods.

6) A beam of mass 12 kg and length 2 m is attached to a

hinge on the left. A box of 80 N is hung from the beam 50 cm from the left end. You hold the beam horizontally with your obviously powerful index finger. With what force do you push up on the beam?

Mitch Sofa Skid

7) The tennis ball of mass 57 g which

you have hung in your garage that lets you know where to stop your car so you don’t crush your garbage cans is entertaining you by swinging in a vertical circle of radius 75 cm. At the bottom of its swing it has a speed of 4 m/s. What is the tension in the string at this point?

y

8) Derivatives:

a) Given: y = (4x + L)(2×2 – L), find dx dy

.

b) Given: Lx2 Lx2lny , find dx dy

.

9) Integrals:

a) Given: 45 45 dr cosk, evaluate.

b) Given:  R0 2322 dr xr kxr2 , evaluate.

ANSWERS:

 

1) a) 3.33 m/s2 b) 4.08 m/s c) 1.23 s 2) 48.0 N, 61.0º N of W 3) 27.4 N, 16.1º S of E 4) 4.52 m/s 5) 103.3 m 6) 78.8 N

7) 1.78N 8) a) 24×2 + 4xL – 4L

b) 22 x4L L4

 

9) a) r k2

b)

22 xR

x1k2

F2 = 90 N

F1 = 40 N 35

45 x

F3 = 60 N

y

F1 = 45 N 60

F2 = 65 N

50 x

70

F3 = 85 N

Guitar

Skid

 

 

Physics 226 Fall 2013

Problem Set #2

1) A plastic rod has a charge of –2.0 C. How many

electrons must be removed so that the charge on the rod becomes +3.0C?

–

+

+

+

2)

Three identical metal spheres, A, B, and C initially have net charges as shown. The “q” is just any arbitrary amount of charge. Spheres A and B are now touched together and then separated. Sphere C is then touched to sphere A and separated from it. Lastly, sphere C is touched to sphere B and then separated from it. (a) How much charge ends up on sphere C? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched? (d) Explain the relevance of the answers to (b) and (c).

 

3)

Skid of 40 kg and Mitch of 60 kg are standing on ice on opposite sides of an infinite black pit. They are each carrying neutral massless spheres while standing 8 m apart. Suppose that 3.0 x 1015 electrons are removed from one sphere and placed on the other. (a) Calculate the magnitude of the electrostatic force on each sphere. Are the forces the same or different? Explain. (b) Calculate the magnitude of the accelerations for Skid and Mitch at the moment they are 8 m apart. Are they the same or different? Explain. (c) As Skid and Mitch move closer together do their accelerations increase, decrease, or remain the same? Explain.

4) An electron travels in a circular orbit around a stationary

proton (i.e. a hydrogen atom). In order to move in a circle there needs to be a centripetal force acting on the electron. This centripetal force is due to the electrostatic force between the electron and the proton. The electron has a kinetic energy of 2.18 x 10–18 J. (a) What is the speed of the electron? (b) What is the radius of orbit of the electron?

 

5)

Three charges are arranged as shown. From the left to the right the values of the charges are 6 μC, – 1.5 μC, and – 2 μC. Calculate the magnitude and direction of the net electrostatic force on the charge on the far left.

6) For the same charge distribution of Problem #5, calculate

the magnitude and direction of the net electrostatic force on the charge on the far right.

7)

Two charged spheres are connected to a spring as shown. The unstretched length of the spring is 14 cm. (a) With Qa = 6 μC and Qb = – 7 μC, the spring compresses to an equilibrium length of 10 cm. Calculate the spring constant. (b) Qb is now replaced with a different charge Qc. The spring now has an equilibrium length of 20 cm. What is the magnitude of the charge Qc? (c) What is the sign of Qc? How do you know this?

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

the location in between the charges that you could put a proton so that the proton would have a net force of zero.

9) Three charges are fixed to an xy coordinate system.

A charge of –12 C is on the y axis at y = +3.0 m. A charge of +18 C is at the origin. Lastly, a charge of + 45 C is on the x axis at x = +3.0 m. Calculate the magnitude and direction of the net electrostatic force on the charge x = +3.0 m.

10) Four charges are situated

at the corners of a square each side of length 18 cm. The charges have the same magnitude of q = 4 μC but different signs. See diagram. Find the magnitude and direction of the net force on lower right charge.

 

+5q – 1q Neutral

C B A

Skid Mitch

Infinite Black Pit

– –

3 cm 2 cm

+

– + Qa Qb

+

8 cm

+ 4 μC 12 μC

 

 

11) For the same charge distribution of problem #10, find the magnitude and direction of the net force on upper right charge.

 

20

12)

All the charges above are multiples of “q” where q = 1μC. The horizontal and vertical distances between the charges are 15 cm. Find the magnitude and direction of the net electric force on the center charge.

 

13) Use the same charge distribution as in problem #12 but change all even-multiple charges to the opposite sign. Find the magnitude and direction of the net electric force on center charge.

14) Two small metallic spheres, each

of mass 0.30 g, are suspended by light strings from a common point as shown. The spheres are given the same electric charge and it is found that the two come to equilibrium when the two strings have an angle of 20 between them. If each string is 20.0 cm long, what is the magnitude of the charge on each sphere?

– 4q +9q +4q

+3q +3q +8q

15)

+6q R2 – 4q 12 cm m A meter stick of 15 kg is suspended by a string at the

60 cm location. A mass, m, is hung at the 80 cm mark. A massless charged sphere of + 4 μC is attached to the meter stick at the left end. Below this charge is another charge that is fixed 12 cm from the other when the meter stick is horizontal. It has a charge of – 4 μC. Calculate the mass, m, so that the meter stick remains horizontal.

 

ANSWERS:

 

7) a) 945 N/m b) 4.2 x 10–5 C 8) 2.93 cm 9) 0.648 N, 17.2º 10) 4.06 N, 45º 11) 6.66 N, 64.5º 12) 19.69 N, 80.1º 13) 18.5 N, 23.4º 14) 1.67 x 10–8 C 15) 10.56 kg

1) 3.1 x 1013 e–

2) a) +1.5q b) +4q c) +4q 3) a) FE, Skid = 32.4 N b) aSkid = 0.81 m/s2 4) a) 2.19 x 106 m/s b) 5.27 x 10–11 m 5) FE = 133.2 N, → 6) FE = 24.3 N, →

 

 

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 to 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 traveled 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

M S

 

 

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)

A 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 is the magnitude of the torque on the 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) 1.67 x 105 N/C, WEST

b) 1.17 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) 0.297 mm 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 +

6 cm

– – 4 μC 12 μC

d

– + – q q 2q

+

 

 

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)

13 cm

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

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 by 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

+ + + +

+ +

– – –

3 μC/m

4 μC/m

P

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.

P

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 – –

L G

vo vo

P

45º

P – +

 

 

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

d 1

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) 1.79 x 104 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

10) a) R k2E 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)

A right circular cone of height 25 cm and radius 10 cm is enclosing an electron, centered 12 cm up from the base. See Figure G. (a) Using integration and showing all work, find the net flux through the cone’s surface. The electron is now centered in the base of the cone. See Figure L. (b) Calculate the net flux through the surface of the cone.

3) Using the cube in #1, you place a 4μC charge directly in the center of the cube. What is the flux through the top face? (Hint: Consider that this problem would be MUCH more difficult if the charge was not centered in the cube.)

4) Using the cube in #1, you place a 4μC charge at the lower,

left, front corner. What is the net flux through the cube? (Hint: Think symmetry.)

5) You have a thin spherical shell

of radius 10 cm with a uni- form surface charge density of – 42 μC/m2. Centered inside the sphere is a point charge of 4 μC. Find the magnitude and direction of the total electric field at: (a) r = 6 cm and (b) r = 12 cm.

6) You have a solid sphere of radius 6 cm and uniform volume charge density of – 6 mC/m3. Enclosing this is a thin spherical shell of radius 10 cm with a total charge of 7 μC that is uniformly spread over the surface. (a) What is the discontinuity of the E-field at the surface of the shell. (b) What is the discontinuity of the E-field at the surface of the solid sphere? Also, find the magnitude and direction of the total electric field at: (c) r = 4 cm, (d) r = 8 cm, and (e) r = 13 cm.

x 30º

y

7) Use the same set-up in #6 with the following exceptions:

The solid sphere has a total charge of 5 μC and the shell has uniform surface charge density of 60 μC/m2. Answer the same questions in #6, (a) – (e).

8) You have a thin infinite

cylindrical shell of radius 8 cm and a uniform surface charge density of – 12 μC/m2. Inside the shell is an infinite wire with a linear charge density of 15 μC/m. The wire is running along the central axis of the cylinder. (a) What is the discontinuity of the E-field at the surface of the shell? Also, find the magnitude and direction of the total electric field at: (b) r = 4 cm, and (c) r = 13 cm.

9) You have a thin infinite

cylindrical shell of radius 15 cm and a uniform surface charge density of 10 μC/m2. Inside the shell is an infinite solid cylinder of radius 5 cm with a volume charge density of 95 μC/m3. The solid cylinder is running along the central axis of the cylindrical shell. (a) What is the discontinuity of the E-field at the surface of the shell? (b) What is the discontinuity of the E-field at the surface of the solid cylinder. Also, find the magnitude and direction of the total electric field at: (c) r = 4 cm, (d) r = 11 cm, and (e) r = 20 cm.

 

–

–

G L

+

 

 

10) You have a thick spherical shell of outer diameter 20 cm and inner diameter 12 cm. The shell has a total charge of – 28 μC spread uniformly throughout the object. Find the magnitude and direction of the total electric field at: (a) r = 6 cm, (b) r = 15 cm, and (c) r = 24 cm.

11) You have a thick cylindrical shell

of outer diameter 20 cm and inner diameter 12 cm. The shell has a uniform volume charge density of 180 μC/m3. Find the magnitude and direction of the total electric field at: (a) r = 6 cm, (b) r = 15 cm, and (c) r = 24 cm.

12)

You have an thin infinite sheet of charge with surface charge density of 8 μC/m2. Parallel to this you have a slab of charge that is 3 cm thick and has a volume charge density of – 40 μC/m3. Find that magnitude and direction of the total electric field at: (a) point A which is 2.5 cm to the left of the sheet, (b) point B which is 4.5 cm to the right of the sheet, and (c) point C which is 1 cm to the left of the right edge of the slab.

13)

You have an infinite slab of charge that is 5 cm thick and has a volume charge density of 700 μC/m3. 10 cm to the right of this is a point charge of – 6 μC. Find that magnitude and direction of the total electric field at: (a) point A which is 2.5 cm to the left of the right edge of the slab, (b) point B which is 6 cm to the right of the slab, and (c) point C which is 4 cm to the right of the point charge.

 

14) You have two infinite sheets of charge with equal surface charge magnitudes of 11 μC/m2 but opposite signs. Find the magnitude and direction of the total electric field, (a) to the right of the sheets, (b) in between the sheets, and (c) to the left of the sheets.

15)

A hydrogen molecule (diatomic hydrogen) can be modeled incredibly accurately by placing two protons (each with charge +e) inside a spherical volume charge density which represents the “electron cloud” around the nuclei. Assume the “cloud” has a radius, R, and a net charge of –2e (one electron from each hydrogen atom) and is uniformly spread throughout the volume. Assume that the two protons are equidistant from the center of the sphere a distance, d. Calculate, d, so that the protons each have a net force of zero. The result is darn close to the real thing. [This is actually a lot easier than you think. Start with a Free-Body Diagram on one proton and then do F = ma.]

 

ANSWERS:

 

 

NOTE: Units for 1 – 4

are CmN 2 1) a) 0 for F/B,  0.375 for L/R,  0.65 for T/B

b) & c) 0 2) a) – 1.81 x 10–8

b) – 9.05 x 10–9 3) 7.54 x 104 4) 5.66 x 104 5) a) 9.99 x 106 N/C, OUTWARD [O] b) 7.99 x 105 N/C

INWARD [I] 6) a) 6.29 x 106 N/C

b) 0 c) 9.04 x 106 N/C, I d) 7.63 x 106 N/C, I e) 8.36 x 105 N/C, O 7) a) 6.78 x 106 N/C

b) 0 c) 4.99 x 105 N/C, O d) 7.03 x 106 N/C, O e) 6.67 x 106 N/C, O

8) a) 1.36 x 106 N/C b) 6.74 x 106 N/C, O c) 1.24 x 106 N/C, O 9) a) 1.13 x 106 N/C b) 0 c) 2.15 x 105 N/C, O d) 1.22 x 105 N/C, O e) 9.15 x 105 N/C, O 10) a) 0 b) 2.94 x 106 N/C, I c) 4.37 x 106 N/C, I 11) a) 0 b) 5.49 x 105 N/C, O c) 1.09 x 106 N/C, O 12) a) 3.84 x 105 N/C, L b) 5.20 x 105 N/C, R c) 4.30 x 105 N/C, R 13) a) 3.84 x 105 N/C, R b) 3.57 x 107 N/C, R c) 3.18 x 105 N/C, L 14) a) 0 b) 1.24 x 106 N/C, R c) 0 15) 0.794R

10 cm

A B C

10 cm

A B –

C

R

+ +

d d

 

 

Physics 226 Fall 2013

 

Problem Set #6

NOTE: Any answers of zero must have some kind of justification. 1) You have a cylindrical metal shell of

inner radius 6 cm and outer radius 9 cm. The shell has no net charge. Inside the shell is a line of charge of linear density of – 7 μC/m. Find the magnitude and direction of the electric field at (a) r = 3 cm, (b) r = 7 cm, and (c) r = 13 cm. Also, calculate the surface charge density of the shell on (d) the inner surface and (e) the outer surface.

2) You have a uniformly charged

sphere of radius 5 cm and volume charge density of – 7 mC/m3. It is surrounded by a metal spherical shell with inner radius of 10 cm and outer radius of 15 cm. The shell has a net charge 8 μC. (a) Calculate the total charge on the sphere. Find the magnitude and direction of the electric field at (b) r = 13 cm and (c) r = 18 cm. Also, calculate the surface charge density of the shell on (d) the inner surface and (e) the outer surface.

 

3) Two 2 cm thick infinite slabs of metal are positioned as

shown in the diagram. Slab B has no net charge but Slab A has an excess charge of 5 μC for each square meter. The infinite plane at the origin has a surface charge density of – 8 μC/m2. Find the magnitude and direction of the electric field at (a) x = 2 cm, and (b) x = 4 cm. Also, calculate the surface charge density on (c) the left edge of A, (d) the right edge of A, and (e) the left edge of B.

4) A positive charge of 16 nC is nailed down with a #6 brad.

Point M is located 7 mm away from the charge and point G is 18 mm away. (a) Calculate the electric potential at Point M. (b) If you put a proton at point M, what electric potential energy does it have? (c) You release the

proton from rest and it moves to Point G. Through what potential difference does it move? (d) Determine the velocity of the proton at point G.

5)

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 of the net electric potential at point P.

6) Use the same charge distribution as in problem #5 but

change all odd-multiple charges to the opposite sign. Find the magnitude of the net electric potential at point P.

7) A parallel plate setup has a distance

between the plates of 5 cm. An electron is place very near the negative plate and released from rest. By the time it reaches the positive plate it has a velocity of 8 x 106 m/s. (a) As the electron moves between the plates what is the net work done on the charge? (b) What is the potential difference that the electron moves through? (c) What is the magnitude and direction of the electric field in between the plates?

 

3 cm 5 cm 8 cm 0 10 cm

A B

– 8q

– 4q

+9q

+9q

– 5q

+6q +6q

+2q

P

+ M

G

–

 

 

 

8)

A uniform line of charge with density, λ, and length, L is positioned so that its left end is at the origin. See diagram above. (a) Determine an equation (using integration) for the magnitude of the total electric potential at point P a distance, d, away from the origin. (b) Calculate the magnitude of the electric potential at P if d = 2 m, L = 1 m, and λ = – 5 μC/m. c) Using the equation you derived in part a), calculate the equation for the electric field at point P. It should agree with the result we got in Lecture Example #19.

 

9) You have a thin spherical shell

of radius 10 cm with a uni- form surface charge density of 11 μC/m2. Centered inside the sphere is a point charge of – 4 μC. Using integration, find the magnitude of the total electric potential at: (a) r = 16 cm and (b) r = 7 cm.

10) You have a uniformly

charged sphere of radius 5 cm and volume charge density of 6 mC/m3. It is surrounded by a metal spherical shell with inner radius of 10 cm and outer radius of 15 cm. The shell has no net charge. Find the magnitude of the electric potential at (a) r = 20 cm, (b) r = 12 cm, and (c) r = 8 cm.

11) Use the same physical situation with the exception

of changing the inner sphere to a solid metal with a surface charge density of 9 μC/m2 and giving the shell a net charge of – 3 μC. Find magnitude of the electric potential at (a) r = 20 cm, (b) r = 12 cm, (c) r = 8 cm, and (d) r = 2 cm.

12) CSUF Staff Physicist & Sauvé Dude, Steve

Mahrley, designs a lab experiment that consists of a vertical rod with a fixed bead of charge Q = 1.25 x 10–6 C at the bottom. See diagram. Another bead that is free to slide on the rod without friction has a mass of 25 g and charge, q. Steve releases the movable bead from rest 95 cm above the fixed bead and it gets no closer than 12 cm to the fixed bead. (a) Calculate the charge, q, on the movable bead. Steve then pushes the movable bead down to 8 cm above Q. He releases it from rest. (b) What is the maximum height that the bead reaches?

 

 

13)

d

P

0 – – – – –

+

L 20 cm

You have two metal spheres each of diameter 30 cm that are space 20 cm apart. One sphere has a net charge of 15 μC and the other – 15 μC. A proton is placed very close to the surface of the positive sphere and is release from rest. With what speed does it hit the other sphere?

14) A thin spherical shell of radius, R, is centered at the

origin. It has a surface charge density of 2.6 C/m2. A point in space is a distance, r, from the origin. The point in space has an electric potential of 200 V and an electric field strength of 150 V/m, both because of the sphere. (a) Explain why it is impossible for r < R. (b) Determine the radius, R, of the sphere.

– 4 μC 12 μC 15) – – +

6 cm The two charges above are fixed and cannot move. Find a

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

 

 

ANSWERS:

1) a) 4.20 x 106 N/C, I

b) 0 c) 9.68 x 105 N/C, I d) 1.86 x 10–5 C/m2 e) – 1.24 x 10–5 C/m2 2) a) – 3.67 x 10–6 C b) 0 c) 1.20 x 106 N/C, O d) 2.92 x 10–5 C/m2 e) 1.73 x 10–6 C/m2 3) a) 7.35 x 105 N/C, L

b) 0 c) 6.5 x 10–6 C/m2 d) – 1.5 x 10–6 C/m2 e) 1.5 x 10–6 C/m2 f) – 1.5 x 10–6 C/m2 4) a) 2.06 x 104 V b) 3.29 x 10–15 J c) – 1.26 x 104 V d) 4.91 x 105 m/s 5) 5.02 x 105 V

6) – 7.87 x 104 V 7) a) 2.92 x 10-17 J b) 182.2 V c) 3644 N/C

8) a)   

   

d Ldlnk

b) – 1.83 x 104 V 9) a) – 1.47 x 105 V b) – 3.90 x 105 V 10) a) 1.41 x 105 V b) 1.88 x 105 V c) 2.59 x 105 V 11) a) – 8.37 x 104 V b) – 1.12 x 105 V c) – 8.62 x 104 V d) – 9900 V 12) a) 2.48 x 10–6 C b) 1.42 m 13) 1.4 x 107 m/s 14) 2.86 m 15) 1.5 cm

q

Q

 

 

Physics 226 Fall 2013

 

Problem Set #7 1) You have a parallel plate capacitor of plate separation

0.1 mm that is filled with a dielectric of neoprene rubber. The area of each plate is 1.8 cm2. (a) Calculate the capacitance of the capacitor. The capacitor is charged by taking electrons from one plate and depositing them on the other plate. You repeat this process until the potential difference between the plates is 350 V. (b) How many electrons have been transferred in order to accomplish this?

2) A capacitor with ruby mica has an effective electric field

between the plates of 4600 V/m. The plates of the capacitor are separated by a distance of 4 mm. 50 mJ of energy is stored in the electric field. (a) What is the capacitance of the capacitor? (b) Calculate the energy density in between the plates.

3) A capacitor with a dielectric of paper is charged to 0.5 mC.

The plates of the capacitor are separated by a distance of 8 mm. 40 mJ of energy is stored in the electric field. (a) What is the strength of the effective electric field? (b) Calculate the energy density in between the plates.

4) A capacitor of 10 μF is charged by connecting it to a

battery of 20 V. The battery is removed and you pull the plates apart so that you triple the distance between them. How much work do you do to pull the plates apart?

5) The flash on a disposable camera contains a capacitor

of 65 F. The capacitor has a charge of 0.6 m C stored on it. (a) Determine the energy that is used to produce a flash of light. (b) Assuming that the flash lasts for 6 ms, find the power of the flash. (Think back to 225.)

6) A spherical shell conductor of

radius B encloses another spherical shell conductor of radius A. They are charged with opposites signs but same magnitude, q. (a) Using integration, derive an equation for the capacitance of this spherical capacitor. (b) Calculate the capacitance if A = 45 mm and B = 50 mm. (c) If q = 40 μC, what is the energy density in between the shells?

 

7) You attach a battery of 15 V to an air-filled capacitor of 5 μF and let it charge up. (a) If the plate separation is 3 mm, what is the energy density in between the plates? You then remove the battery and attach the capacitor to a different uncharged capacitor of 2 μF. (b) What is the amount of charge on each capacitor after they come to equilibrium?

8) You attach a 100 pF capacitor to a battery of 10 V. You

attach a 250 pF battery to 7 V. You remove both of the batteries and attach the positive plate of one capacitor to the positive plate of the other. After they come to equilibrium, find the potential difference across each capacitor.

9) Do problem #8 but when you attach the capacitors

together attach the opposite sign plates instead of the same sign plates.

10)

Determine the equivalent capacitance between points A and B for the capacitors shown in the circuit above.

11)

Determine the equivalent capacitance between points A and B for the capacitors shown in the circuit above.

12) Design a circuit that has an equivalent capacitance of

1.50 μF using at least one of each of the follow capacitors: a 1 μF, a 2 μF, and a 6 μF. [You must also show where your A and B terminals are located.]

 

A

20 F

4 F

4 F

6 F

12 F

B

30 F

A 12 F

18 F 6 F

20 F

B

12 F 75 F

 

 

13) The two capacitors above both have plates that are

squares of sides 3 cm. The plate separation is 1.2 cm for both. Between each of the capacitor plates are two different dielectrics of neoprene rubber and Bakelite. Everything is drawn to scale. Find the capacitance of each capacitor. (HINT: Think series and parallel.)

14) The plates of an air-filled capacitor have area, A, and are

separated by a distance, d. The capacitor is charged by a battery of voltage, V. Three things are going to change: (1) The plates of the capacitor are pulled apart so that the distance between the plates triples. (2) The area of the plates increase by a factor of 6. (3) The voltage of the battery decreases by a factor of 4. Determine expressions in terms of A, d, and/or V for (a) the new capacitance, (b) the new charge, and (c) the new energy density.

 

15)

A massless bar of length, L, is hanging from a string that is attached 1/3 of the length of the bar from the right end. A block of mass, M, is hung from the right end. The left end of the bar has an air-filled massless capacitor of plate area, A, and plate separation, d. Find an expression for the potential difference between the plates so that this system is in equilibrium. (HINT: You will

need the equation dx dU

F  from 225.)

 

ANSWERS:

(a) (b)

1) a) 1.067 x 10–10 F

b) 2.34 x 1011 e–

2) a) 2.95 x 10–4 F b) 5.05 x 10–4 J/m3 3) a) 2 x 104 V/m

b) 6.7 x 10–3 J/m3 4) 4 x 10–3 J 5) a) 2.8 x 10–3 J b) 0.467 W

6) a)

AB AB4C o

b) 5.01 x 10–11 F c) 1.125 x 105 J/m3 7) a) 1.11 x 10–4 J/m3 b) 2.14 x 10–5 C, 5.36 x 10–5 C

8) 7.86 V 9) 2.14 V 10) 4 μF 11) 9 μF 13) a) 3.85 pF b) 3.76 pF

14) a) d

A2 C o

 

b) d2 AV

Q  o

c) 2 2

o

d288 Vu

15) A

Mg dV

o

M

 

 

Physics 226 Fall 2013

 

Problem Set #8 1) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 2) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 3) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 4)

An Oppo Digital Blu-Ray player [DMP-95] (Yes, I am an audiophile.) has a power cable which has a metal that allows 9 x 1019 electrons per cubic millimeter. On average, the cable passes 1 x 1022 electrons every hour. The electrons passing through the player have a drift velocity of 4.5 μm/s. (a) What current does the Oppo draw? (b) Calculate the diameter of the cable?

5) The Large Hadron Collider at CERN creates proton beams which collide together resulting in pictures like the one at the right. Some of these beams can have a radius of 1.1 mm with a current of 1.5 mA. The kinetic energy of each proton in this beam is 2.5 MeV. (a) Calculate the number density of the protons in the beam. (b) If the beam is aimed at a metal target, how many protons would strike the screen in 1 minute?

C1 = 8 μF C2 = 15 μF

20 V

C3 = 30 μF

6)

Two copper wires are soldered together. Wire #1 has a radius of 0.7 mm. Wire #2 has a radius of 1.2 mm. Copper has a number density of 8.47 x 1028 e–/m3. The drift velocity in Wire #1 is 0.72 mm/s. If you want the current to remain the same in both, what is the drift velocity in Wire #2?

7) A nichrome cable has a current of 140 A running through

it. Between two points on the cable that are 0.22 m apart, there is a potential difference of 0.036 V (a) Calculate the diameter of the cable. (b) How much heat energy does this part of the wire emit in 1 minute?

8) A “Rockstar” toaster uses a

tungsten heating element (wire). The wire has a diameter of 1.2 mm. When the toaster is turned on at 20 C, the initial current is 1.6 A. (a) What is the current density in the wire? (b) A few seconds later, the toaster heats up and the current is 1.20 A. What is the temperature of the wire? (c) If the toaster is plugged into a standard wall outlet in Kankakee, Illinois, what is the rate that energy is dissipated from the heating element?

9) Skid runs a 10 mile line of copper cable out to his shack in

the sticks so he can have electricity to play Lord of the Rings Online. At 20ºC the resistance of the cable is 12 . At 50ºC the cable emits 1.5 kJ every second. (a) What is the resistance of the cable at 50ºC? (b) What is the current running through the cable at 50ºC? (c) Calculate the current density at 50ºC.

 

C1 = 18 F

Wire #1 Wire #2 C2 = 6 μF

C3 = 4 μF

C4 = 30 μF 25 V

C1 = 5 F C2 = 4 μF

15 V C3 = 1 μF

C4 = 12 μF

 

 

10) A modern hair dryer uses a nichrome heating element that typically is 30-gauge wire around 40 cm in length. The gauge rating on a wire refers to its diameter. In this case, 30-gauge wire has a diameter of 0.254 mm. Nichrome has a number density of 7.94 x 1028 e–/m3. If the drift velocity of the electrons in the wire is 18.7 mm/s, what is the voltage that the hair dryer is plugged into?

 

11) Before LCD, LED, Plasma,

and (the latest) OLED TVs, there were CRT (Cathod-Ray Tube) TVs. Inside these TVs were electron guns that shot an electron beam of diameter 0.5 mm and current density of 244 A/m2 onto the inside of a glass screen which was coated with phosphor. How many electrons would hit the phosphor every minute?

12)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

 

13)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

14)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

15) Design a circuit that has an equivalent resistance of

1.00  using at least one of each of the follow resistors: a 1 , a 2 , and a 6 . [You must also show where your A and B terminals are located.]

 

ANSWERS:

NOTE: Some of these answers are minimal since there are checks that you can do to verify your answers.

 

A

27 

B 54 

8 

30 

16 

14 

10 

30 

B

18 

96 

6 

32  18 

60  A

A

20 

30 

B

30 

7 

50 

12 

45 

60 

1) CEQ = 18 μF 8) a) 1.415 x 106 A/m2 2) CEQ = 6 μF b) 94.1ºC 3) CEQ = 2 μF c) 144 W

9) a) 13.4  4) a) 0.444 A b) 2.96 mm b) 10.58 A

c) 5.14 x 105 A/m2 5) a) 1.13 x 1014 p+/m3 b) 5.63 x 1017 p+ 10) 95.0 V 6) 0.262 mm/s 11) 1.8 x 1016 e–

7) a) 0.033 m 12) 4  b) 302 J 13) 14  14) 22 

 

 

Physics 226 Fall 2013

 

Problem Set #9

NOTE: You can only use circuit tricks on 9 – 11 but not on any others. 1) Analyze the following circuit using a VIR chart. 2) Swap the location of the battery and R1 in the circuit from

problem #1. Analyze the circuit using a VIR chart. 3) Analyze the following circuit using a VIR chart. 4) The battery in this problem has an internal resistance of

0.15 . (a) Analyze the following circuit using a VIR chart. (b) Is this circuit well designed? Discuss, explain.

 

5) Analyze the following circuit using a VIR chart.

6) Analyze the following circuit using a VIR chart. 7) The battery in this problem has an internal resistance of

1 . (a) Analyze the following circuit using a VIR chart. (b) Is this circuit well designed? Discuss, explain.

8) A load of 3.5  is connected across a 12 V battery. You

measure a voltage of 9.5 V across the terminals of the battery. (a) Find the internal resistance of the battery. (b) Is this circuit well designed? Discuss, explain.

9) Analyze the circuit from problem

#5 using a VIR chart. You are using only the diagram in #5, not the values. New values are given at the right. You may use a circuit trick for this circuit, but only for ONE value.

10) Analyze the circuit from problem

#6 using a VIR chart. You are using only the diagram in #6, not the values. New values are given at the right. You may use a circuit trick for this circuit, but only for ONE value.

 

R1

20 V

R2 R3 R4

R5

Given: R1 = 12  R2 = 3  R3 = 8  R4 = 36  R5 = 15 

 

50 V

R1 Given: R1 = 28  R2 = 6  R3 = 84  R4 = 7  R5 = 54 

 

R3

R2

R4

R5

55 V

R1 Given: R1 = 18  R2 = 32  R3 = 15  R4 = 21  R5 = 42  R6 = 30  R7 = 52 

R3

R2

R4 R5

R6 R7

R1

VB

R2

R3 R4

Given: VB = 60 V V2 = 50 V

 

I1 = 2 A I4 = 3 A

 

R3 = 8 

R1

VB

R2 R3

R4

R5

Given: V5 = 32 V

 

I2 = 0.4 A I4 = 0.5 A

 

R1 = 36  R6 R3 = 60  R4 = 36  R6 = 32 

R1

VB

Given: VB = 32 V

 

R2 I1 = 4 A R3 R3 = 12 

R4 R4 = 8 

Given: VB = 63 V R1 = 8  R2 = 20  R3 = 35  R4 = 49 

 

Given: VB = 75 V R1 = 16  R2 = 40  R3 = 48  R4 = 24  R5 = 8  R6 = 24 

 

 

11) Analyze the following circuit using a VIR chart. 12) Using the information you are

given for the circuit at the right, answer the following. (a) Determine the magnitude and direction of the current in the circuit. (b) Determine which point, A or B, is at a higher potential.

13) Calculate the unknown currents I1, I2, and I3 for the circuit

below.

14) Calculate the unknown currents I1, I2, and I3 for the circuit below.

Given: 15) Calculate the unknown currents I1, I2, and I3 for the circuit

below.

ANSWERS:

NOTE: These answers are minimal since there are checks that you can do to verify your answers.

 

R1 R2

R3 R4

R5 R6

I1 8 V VB = 50 V

R1 = 9  R2 = 4  R3 = 18  R4 = 4  R5 = 7  R6 = 12 

B

A

17 V

13  7 

5 

11 

23 V

6 

1 

10 V

25 V

3 

5 

7 

I1

I2

I3

4 

9 

10 

4  7 

I2

6 

I3 22 V

3  10 V I1

4 

4  25 V

2  5 

I2 I3

20 V 4 

7) REQ = 8  1) REQ = 2  8) a) 0.923  2) REQ = 11.48  9) REQ = 21  3) REQ = 25  10) REQ = 25  4) REQ = 12.15  11) REQ = 20  5) REQ = 12  12) a) 1.11 A 6) REQ = 40 

 
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