# Physics

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.]

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

2 μC/m

5 μC/m

+

+

+ +

+

+

+

P

– + P

P

– –

– –

L G

vo vo

P

45º

ANSWERS:

7) 5.93 x 104 N/C, 13.6º1) a) 2.15 mC b) 1.25 mC

2) a) 22 Ld4 Lk4

− λ

8) 2.37 x 104 N/C, 59.8º

9) a) R k2Ey

λ=

b) 4.85 x 105 N/C, 22.0º b) 1.2 x 104 N/C,

10) a) R k2E y

λ= EAST

3) a)

+ −=

22x Ld

d 1

dL Qk

E

b) 9322 N/C, EAST

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

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

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.

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.

x 30º

y

–

–

L G

+

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 an infinite 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) R

+ +

d d

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 ΣF = ma.]

10 cm

C B A

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

Physics 226 Fall 2013