Download 3318 Homework 7

ECE 3318
Applied Electricity and Magnetism
Spring 2017
Homework #7
Date assigned: March 9, 2017
Date due: March 23, 2017
Do Probs. 1, 2, and 6-11. (You are welcome to do the other problems as well for extra practice,
but only these should be turned in.)
1) Find the curl of the following vector functions:
V  xˆ  5 x 2 yz   yˆ  e  y x   zˆ  3xyz 2 
V  ˆ  cos      3z sin    zˆ  z  
V  rˆ  sin  cos    ˆ  3r sin    ˆ  r 3 cos  .
2) A simple “curl meter” is a paddle wheel that consists of a set of vanes attached to a spindle
(the axis of rotation) as shown below. It is used to determine the curl of a vector function that
describes the velocity vector of water. In order to determine the component of the curl of a
vector in the direction ˆ , at a given point in space, the curl meter is placed at that point in
space and the spindle axis is pointed in direction ˆ . A positive torque T indicates that the
meter wants to rotate in the direction defined by the right-hand rule, as shown in the figure
below.
Assume that our curl meter has the property that the magnitude of the torque is equal to the
square of the curl component. (This is because when water hits an object, the force is
proportional to the square of the velocity.) The torque is therefore given by

T   ˆ  curlV
,
2
 if ˆ  curlV  0 ,
 if ˆ  curlV  0.
1
ˆ
T
Part 1)
Find the torque T on the curl meter for the vector function

 

V  xˆ 2 x 2 y  yˆ 3zy 3  zˆ  2 xyz 
at the following points (in rectangular coordinates) and directions:
b) r  (1,1,1);
ˆ  xˆ
ˆ  yˆ
c) r  (1,1,1);
ˆ  zˆ
d) r  (1,1,1);
ˆ  zˆ + 2 xˆ + 4 yˆ / 21
a) r  (1,1,1);
e) r  (1, 2,3);


ˆ   xˆ + yˆ + zˆ  /
3 .
Part 2)
Find a unit vector ˆ that gives the direction to point the curl meter so that the torque will be
maximum in magnitude and positive, assuming the same vector function V as above and
assuming that we are at the point (1,1,1) in rectangular coordinates. Repeat for the case
where we want the torque to be maximum in magnitude but negative at this same point.
3) Consider the vector function

 

V  xˆ 2 x 2 y  yˆ 3zy 3  zˆ  2 xyz  .
2
Calculate an approximate expression for the integral
I   V dr ,
C
where the path C is a circular path of radius a = 0.0001 that is centered at the point (1, 2, 3)
and lies in a plane perpendicular to the vector direction xˆ  yˆ . The direction of integration
around the contour is counter-clockwise, as seen from an observer who is looking towards
the loop from far away in the first octant, as shown below.


(Note: You may assume that the curl of the vector function V is approximately constant over
the surface of the circular region inside the path C, since the radius is so small.)
Hine: Use Stokes’s theorem.
z
C
y
x
4) Assume a vector function F  ˆ (5 sin  )  ˆ (  2 cos  ) .
a) Evaluate
 F  d r around the contour ABCDA in the direction indicated in the figure.
b) Find  F .
c) Evaluate
   F   nˆ dS
over the surface defined by the contour ABCDA and compare
S
the result with that obtained in part (a), to see if you can verify Stokes’s theorem.
3
y
C
3 [m]
B
1 [m]
x
A
D
5) Green’s theorem states that if P ( x, y ) and Q ( x, y ) are two differentiable functions in a planar
region S of the xy plane bounded by a closed curve C , then
 Q
P 
  x  y  dS  
S
C
Pdx  Qdy ,
where the integral around C is taken in a counterclockwise sense. Show that this is a special
case of Stokes’s theorem for a planar surface bounded by a closed contour lying in the xy
plane,
  A  zˆ dS   A  dr ,
S
C
in which A  x, y   P  x, y  xˆ  Q  x, y  yˆ .
6) Calculate the voltage drop
B
VAB   E  d r
A
along the path C as shown below, for the electric field


E  xˆ  3x   yˆ 4 y 3  zˆ  4 z  .
4
You are free to choose a different path than the one specified, if you can prove that it makes
no difference (explain clearly why). The point A is at the origin, and point B is at (1, e, 0).
(Note: This may not be an electrostatic field – so the curl may or may not be zero!)
y
B
C : y  x2  e x 1
A
x
7) The height of a mountain is described by

z  103 1  a 2 x 2  b 2 y 2

for z greater than zero, where a = 0.001 and b = 0.002, and all dimensions are in meters.
Note that the peak of the mountain is at x = y = 0. Assume that a mountain climber is on the
side of the mountain at x = 500, y = 200.
Find a unit vector parallel to the xy plane (i.e., a horizontal unit vector) that points in the
direction that the climber located at (x, y, z) should head towards, in order to ascend the
mountain as rapidly as possible from her present location. (This corresponds physically to the
compass heading that she must head towards.) Also, find the horizontal unit vector that
points from where the climber is towards the top of the mountain. Are the two unit vectors
the same?
8) Consider a circular disk of uniform surface charge density s0 of radius a, lying in the z = 0
plane with the center of the disk at the origin. Calculate the potential at any point on the z
axis by using the potential-integral formula. Then use the gradient to find the electric field on
the z axis, for both z < 0 and z > 0.
9) Two infinite line charges (running in the z direction) are located at x  h as shown below.
Calculate the potential at any point (x, y), assuming zero volts on the z axis. When
calculating the potential, you may start with the potential of a single infinite line charge and
use superposition. (The potential of a single infinite line charge was derived in class in Notes
14.)
5
y
h
h
x
 l 0
l 0
10) Use the gradient to calculate the electric field vector at any point (x, y) in space, for the
problem of the two infinite line charges in the previous problem.
11) Consider a sphere of uniform charge density v 0 [C/m3] having a radius a, centered at the
origin. Perform the following steps:
a) Calculate the electric field both inside and outside the sphere using Gauss’ law.
b) Calculate the potential inside and outside the sphere by integrating the electric field to
get the potential, assuming zero volts at infinity.
c) Find the electric field inside and outside the sphere by taking the gradient of the
potential. You should be able to verify that your answer is the same as the electric
field that you started with from Gauss’s law.
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