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EE361

Instructor: Evan Goldstein

Assignment: 5

Due: Saturday, 25 May

Name: ____________________________________

Score: ____________________________________

Do this assignment on your own. You are free to discuss lectures and reading and general approaches to problems with

other students. But you are expected to turn in work that you have done yourself, without checking answers withothers. To receive full credit, the answers must be correct and it must be apparent how you arrived at them.

1. The fundamental theorem of calculus describes two ways of determining the total change of afunction. (a) State the fundamental theorem accurately in symbols. (b) Provide a geometrical

interpretation for it.

2. The fundamental theorem for gradients also describes two ways of determining the total

change of a function. (a) Does it apply to vector-valued functions or scalar-valued functions?

(b) State the fundamental theorem for gradients accurately in symbols. (c) Provide a

geometrical interpretation for it.

3. (a) State in words (no equations) the Divergence Theorem. (b) Using words and diagrams,

explain intuitively why it is true.

4. The fundamental theorems of vector calculus have a strong kinship with the fundamentaltheorem of ordinary single-variable calculus, both in structure and in underlying intuition.

(a) State clearly the structural resemblance between the FTCs for vector calculus and the

FTC for ordinary single-variable calculus. (b) The underlying intuitive reasons why the

FTCs for vector calculus are true bears a strong general resemblance to the underlying

reason why the FTC for single-variable calculus is true. Describe the resemblance clearly.

5.

a. Draw, in the (x, y) plane, a vector function of (x, y) that has zero divergence

everywhere.

b. Draw, in the (x, y) plane, a vector function of (x, y) that has positive divergence

everywhere.

c. Draw, in the (x, y) plane, a vector function of (x, y) that has negative divergenceeverywhere.

d. Draw, in the (x, y) plane, a vector function of (x, y) that has infinitely large

divergence at some point. Label that point.

e. Draw, in the (x, y) plane, a vector function of (x, y) that has positive divergence at

points x>0 and negative divergence at points x

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g. Thinking of V as the water mass flow density under a jacuzzi, draw cartoons at some

representative place in the water mass showing that these units make sense.

h. Draw the cartoon in particular at the jacuzzis drain. Show that the sign of the

divergence there makes sense.

i. State in words the reason why, intuitively, the divergence of Vthe limiting flux ofV out of a volume as the volume gets smallexactly equals the sum of the three

partial space-derivatives of V.

6. The aim of this problem is to probe a fundamental theorem of vector integral calculus and to

probe the equations that describe, from a fundamental physical point of view, virtually

everything in a given region of space. We know that the electromagnetic force is

overwhelmingly the dominant force in everyday life and over ordinary human length scales.

Gravitation and the strong and weak nuclear forces here are completely negligible by

comparison.

A given cubical region of space is two meters on a side and contains an electric field

E(x,y,z) = xxy yyz2 zxyV/m.

a. Find the total outward flux of the electric field flowing through the surface of a cube

centered at the origin with 2-meter-long sides parallel to the x, y, and z axes.

b. Find the integral of the divergence of the electric field E over the volume of the cube

of (a).

c. What fundamental theorem of vector calculus has relevance to this problem?d. What does that theorem say about the problem?

7. (a) State in words (no equations) Stokes Theorem. (b) Using words and diagrams, explain

intuitively why it is true. Do not cheat and just start regurgitating whatever inane stuff some

ignorant teacher might have told you about this.

8. One of the weaker forces in the universe is characterized by a force field F= RGMm/ R3,exerted by mass M on mass m, where G is a constant, M is the source mass, m is the test

mass, and Ris the vector pointing from M to m, with magnitude R.

a. Starting with this force vector, construct a gravitational field vector that serves

precisely the same purpose as the electric and magnetic fields weve been dealing

with in class. What is the resulting gravitational field?b. Sketch this field.

c. By inspecting your sketch, what is the curl of this field?

d. Re-express the field in Cartesian coordinates.

e. What is the divergence of this field?

f. What physical quantity in space serves as a faucet, or source, for this field?

g. Does superposition apply to this field? Why?h. What does superposition mean for this field?

i. Give a real-world physical example of the application of superposition to this field.

j. Modify the above force law so as to make superposition no longer hold. In your

example, why does superposition no longer hold?

k. Similarly, what does superposition mean for the electric field E, and why does

superposition hold forE?

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9. Two charged particles in free space electrostatically repel one another with a force of 1

microNewton. They are now placed in water, with relative permittivity of 80, holding their

separation constant.

a. Whats the force between them in water?

b. Whats happening physically that causes the force in water to be different from theforce in vacuum? Include a sketch to make your thinking clear.

10.

a. Using Maxwells Equations, explain clearly the conditions under which the electric

field has zero curl.

b. Using Maxwells Equations and fundamental theorems of vector calculus, explain

clearly the conditions under which the circulation of the electric field around any

closed path must be zero.

c. Using Maxwells Equations and diagrams, explain clearly what inductance is.

d. Using Maxwells Equations and diagrams, explain clearly how a moving-magnet

electric generator works.

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