cubes of Cs+ ions (side length a ~ 0.4 nm) surround Cl- ions, as pictured at right. (a) What is the magnitude of the net electrostatic force on the

Cl- ion due to its eight nearest neighbor Cs+ ions?

(b) Occasionally defects arise in which one of the Cs+ ions is missing. We call this a vacancy. In this case, what is the magnitude and direction (relative to the vacancy) of the net electrostatic force on the Cl- ion due to its remaining seven?

Problem 1: Cesium Chloride is a salt with a crystal structure in which

Problem 2: A positively charged wire is bent into a semicircle of radius R , as shown in the figure below.

The total charge on the semicircle is . However, the charge per unit length along the semicircle is non-uniform and given by

Q

0 cosλ λ θ= .

a) What is the relationship between 0λ , R and ? Q

b) If a particle with a charge q is placed at the origin, what is the total force on the particle? Show all your work including setting up and integrating any necessary integrals.

Problem 3: Careful measurements reveal an electric field

3

2 3ˆ1 ;

( )

0 ;

a r r Rr Rr

r R

⎧ ⎛ ⎞− ≤⎪ ⎜ ⎟= ⎨ ⎝ ⎠

⎪ ≥⎩

rE

where and a R are constants. Which of the following best describes the charge distribution giving rise to this electric field?

a) A negative point charge at the origin with charge 04q aπε= and a uniformly positive charged spherical shell of radius R with surface charge density

2/ 4q Rσ π= − .

b) A positive point charge at the origin with charge 04q aπε= and a uniformly negative charged spherical shell of radius R with surface charge density

2/ 4q Rσ π= − .

c) A positive point charge at the origin with charge 04q aπε= and a uniformly negative charged sphere of radius R with charge density . 3/(4 / 3)q Rρ π= −

d) A negative point charge at the origin with charge 04q aπε− = − and a uniformly

positive charged sphere of radius R with charge density . 3/(4 / 3)q Rρ π=

e) Impossible to determine from the given information.

When two slabs of N-type and P-type semiconductors are put in contact, the relative affinities of the materials cause electrons to migrate out of the N-type material across the junction to the P-type material. This leaves behind a volume in the N-type material that is positively charged and creates a negatively charged volume in the P-type material. Let us model this as two infinite slabs of charge, both of thickness with the junction lying on the plane . The N-type material lies in the range

az = 0 0 < z< a and has uniform

charge density +ρ0 . The adjacent P-type material lies in the range −a < z< 0 and has uniform charge density −ρ0 . Thus:

ρ(x, y, z) = ρ(z) =

+ρ0 0 < z< a

−ρ0 − a< z< 0

0 z >a

⎧

⎨⎪

⎩⎪

Find the electric field everywhere.

Problem 4:

(a) Using Gauss’s Law, find an expression for the direction and magnitude of the electric field E for the region r < a. (b) Similarly, find an expression for the direction and magnitude of the electric field E for the region . a r b< <

Problem 5 A very long conducting cylinder (length L and radius a) carrying a total charge +q is surrounded by a thin conducting cylindrical shell (length L and radius b) with total charge –q, as shown in the figure.

arranged such that the lines joining the center of the O atom with each H atom make an angle of 1040. A distance separates the centers of each hydrogen and oxygen pair. Each pair of hydrogen and oxygen atoms have dipole moments

119.6 10d −= × m1p and 2p . The entire molecule has a

dipole moment p , which is the sum of the dipole moments 1 2= +p p p . The magnitude of the total dipole moment is 306.1 10 C m−= × −p .

a) What is the effective charge on each hydrogen atom? b) Set the electric potential at infinity to zero, ( ) 0V ∞ = . Find an expression for the electric

potential at a point , a distance , from the center of the water molecule, due to each dipole, and . The line connecting the origin at the center of the water molecule with the point makes an angle

P r >> d1p 2pP θ with the direction of the total dipole moment p .

Problem 6: The water molecule, (see figure below), consists of two hydrogen atoms and one oxygen

A thin rod extends along the z-axis from z = −d to z = d . The rod carries a charge uniformly distributed along its length with linear charge density

Q 2d λ =Q / 2d .

a) Find the electric potential at a point z > d along the z-axis. Indicate clearly where

you have chosen your zero reference point for your potential. b) Use the result that to find the electric field at a point VE = −∇ z > d along the z-

axis.

c) How much work is done to move a particle of mass and positive charge from the point

m qz = 4d to the point z = 3d ?

Problem 7:

( )

2

0

0

0

0 for

1 for 0, ,

1 2 for 0

3 for

x d

xV ddV x y z

xV xd

V x d

< −⎧⎪

⎛ ⎞⎪− + − ≤ <⎜ ⎟⎪⎪ ⎝ ⎠= ⎨⎛ ⎞⎪− + ≤ <⎜ ⎟⎪ ⎝ ⎠

⎪− >⎪⎩

x

d

0

where -V is the potential at the origin and d is a distance.

0

This function is plotted to the right, with the x-axis in units of d and the y-axis in units of V0. (a) What is the electric field ( )xE for this problem?

<

Region I: -d > x Region II: − ≤ 0d x Region III: 0 x d≤ < Region IV: x d> (b) Plot the electric field that you just calculated on the graph below. Be sure to properly

label the y-axis (top and bottom) to indicate the limits of the magnitude of the E field!

Problem 8: The electric potential V(x,y,z) for a planar charge distribution is given by:

a) What is the electric field in each of the four regions I-IV labeled in the sketch? Clearly present your reasoning, relevant figures, and any accompanying calculations. Plot the x component of the electric field , xE , on the graph on the bottom of the next page. Clearly indicate on the vertical axis the values of xE for the different regions. b) Find the electric potential in each of the four regions I-IV labeled above, with the choice that the potential is zero at x = +∞ i.e. ( ) 0V +∞ = . Show your calculations. Plot the electric potential as a function of x on the graph on the bottom of the next page. Indicate units on the vertical axis. c) How much work must you do to bring a point-like object with charge in from infinity to the origin ?

Q+0x =

Problem 9: Three infinite sheets of charge are located at x = −d , x = 0 , and x = d , as shown in the sketch. The sheet at x = 0 has a charge per unit area of 2σ , and the other two sheets have charge per unit area of −σ .

A conducting wire is attached to an initially charged spherical conducting shell of radius

. The other end of the wire is attached to the outer surface of a neutral conducting

spherical shell of radius that is located a very large distance away (at infinity). When

electrostatic equilibrium is reached, the charge on the shell of radius is equal to

2a

a

2a

a) one fourth the charge on the shell of radius a .

b) half the charge on the shell of radius a .

c) twice the charge on the shell of radius . a

d) four time the charge on the shell of radius . a

e) None of the above.

Problem 10:

(a) Calculate the electric field between the two cylinders (a < r < b). (b) Calculate the potential difference between the two cylinders: (c) Calculate the capacitance of this system, C = Q/ΔV

(d) Numerically evaluate the capacitance for your experimental setup, given:

h ≅ 15 cm, a ≅ 4.75 cm and b ≅ 7.25 cm

(e) Find the electric field energy density at any point between the conducting cylinders. How much energy resides in a cylindrical shell between the conductors of radius (with ra r b< < ), height , thickness , and volume h dr 2 rh drπ ? Integrate your expression to find the total energy stored in the capacitor and compare your result with that obtained using . 2(1/ 2) ( )EU C Δ= V

Problem 11: Consider two nested cylindrical conductors of height h and radii a & b respectively. A charge +Q is evenly distributed on the outer surface of the pail (the inner cylinder), -Q on the inner surface of the shield (the outer cylinder).

a. The charges on the plates of a parallel-plate capacitor are of opposite sign, and they attract each

other. To increase the plate separation, is the external work done positive or negative? What happens

to the external work done in this process?

b. How does the stored energy change if the potential difference across a capacitor is tripled?

c. Does the presence of a dielectric increase or decrease the maximum operating voltage of a

capacitor? Explain.

d. If a dielectric-filled capacitor is cooled down, what happens to its capacitance?

Problem 12:

Explain your answer. (b) Is it possible for a constant magnetic field to alter the speed of a charged particle? What is the role of a magnetic field in a cyclotron? (c) How can a current loop be used to determine the presence of a magnetic field in a given region of space? (d) If a charged particle is moving in a straight line through some region of space, can you conclude that the magnetic field in that region is zero? Why or why not? (e) List some similarities and differences between electric and magnetic forces.

Problem 13: (a) Can a constant magnetic field set into motion an electron which is initially at rest?

at the earth’s equator where the earth’s magnetic field is horizontal, points north, and has magnitude -40.5 10 Tearth = ×B . The wire lies in a plane that is parallel to the surface of the earth and is oriented in the east-west direction. The density of copper is 38.9 10 kg mCuρ -3= × ⋅ . The resistivity of copper is . 81.7 10 Ω mrρ

−= × ⋅

a) How large must be, and which direction must it flow in order to levitate the wire? Use

J-29.8m sg = ⋅

b) When the wire is floating how much power will be dissipated per cubic

centimeter?

Problem 14: A copper wire of diameter d carries a current density J

(b) If this conductor is formed into a single circular turn and carries the same current, what is the value of the magnetic field at the center?

Problem 15: A conductor in the shape of a square loop of edge length ℓ = 0.400 m carries a current I = 10.0 A as in the figure. (a) Calculate the magnitude and direction of the magnetic field at the center of the square.

K=K i and 2ˆK= −K i in the opposite directions, as

shown in the figure below (The extent of the sheets in the y direction is infinite.) Note that K is the current per unit width perpendicular to the flow.

a) Find the magnetic field everywhere due to 1K .

b) Find the magnetic field everywhere due to 2K . c) Applying superposition principle, find the magnetic field everywhere due to both

current sheets. d) How would your answer in (c) change if both currents were running in the same

direction, with ? 1 2ˆK= =K K i

Problem 16: Consider two infinitely large sheets lying in the xy-plane separated by a distance d carrying surface current densities 1

ˆ

An infinitely long wire of radius carries a current density which is uniform and constant. The current points "out of" the page, as shown in the figure.

a J0

(a) Calculate the magnitude of the magnetic field for (i) B(r ) r < a and (ii) r > a . For both cases show your Amperian loop and indicate (with arrows) the direction of the magnetic field.

r

(b) What happens to the answers above if the direction of the current is reversed so that it flows "into" the page ? (c) Consider now the same wire but with a hole bored throughout. The hole has radius

(with ) and is shown in the figure. We have also indicated four special points: O, L, M, and N. The point O is at the center of the original wire and the point M is at the center of the hole. In this new wire, the current density exists and remains equal to over the remainder of the cross section of the wire. Calculate the magnitude of the magnetic field at (i) the point M, (ii) at the point L, and (iii) at the point N. Show your work.

b 2b < a

J0

Hint: Try to represent the configuration as the "superposition" of two types of wires.

Problem 17:

An inductor consists of two very thin conducting cylindrical shells, one of radius a and one of radius b, both of length h. Assume that the inner shell carries current I out of the page, and that the outer shell carries current I into the page, distributed uniformly around the circumference in both cases. The z-axis is out of the page along the common axis of the cylinders and the r-axis is the radial cylindrical axis perpendicular to the z-axis.

a) Use Ampere’s Law to find the magnetic field between the cylindrical shells. Indicate the direction of the magnetic field on the sketch. What is the magnetic energy density as a function of r for a < r < b?

b) Calculate the inductance of this long inductor recalling that 212BU L= I and using

your results for the magnetic energy density in (a). c) Calculate the inductance of this long inductor by using the formula

and your results for the magnetic field in (a). To do this you

must choose an appropriate open surface over which to evaluate the magnetic flux. Does your result calculated in this way agree with your result in (b)?

open surface

LIΦ = = ⋅∫ B dA

Problem 18: