124 Physics Spring 2011 Document #11: Homework #06
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PHYS 124: Homework #06
Feb 27, 2012
Homework #06 is due in Box outside of Rock 207:
4:00 PM Sharp, Monday, March 5, 2012
Reading assignment:
Read text by Schey, Div, Grad, Curl and All That Chapter III. Try to get through this
reading before March 2. However:
Skip the section from pages 82 through 86 involving the curl in non-Cartesian Coordinates.
Skip sections on pages 101 through 104 on simply-connected regions.
Physics 124 Spring 2011 Document #11: Homework #06
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This homework due in Box outside of Rock 207:
4:00 PM Sharp, Monday February March 5, 2012
Problem 1:
This problem corresponds to Schey, Problem II-4, part (a) only (page 52):
Evaluate the surface integral S G(x, y, z ) dS for the function G(x, y, z ) = z on the surface S
which is the plane dened by x + y + z = 1 in the rst octant (where all coordinates are positive).
Problem 2:
This problem corresponds to Schey, Problem II-5, part (a) only (page 53):
Evaluate the surface integral S F n dS for the function F (x, y, z ) = x kz on the surface S
which is the plane dened by x + y + 2z = 2 in the rst octant (where all coordinates are positive).
Problem 3:
This problem corresponds to Schey, Problem II-8 (page 53):
An electrostatic eld is given by
E (x, y, z ) = (yz + xz + kxy )
where is a constant. Use Gauss Law to nd the total charge enclosed by a closed surface
consisting of two parts:
1
S1 , the hemisphere dened by z = (R2 x2 y 2 ) 2
S0 , the circular base of this hemisphere in the xy -plane.
Problem 4:
This problem corresponds to Schey, Problem II-14 (page 56):
Calculate the divergence of each of the following functions in accordance with:
divF =
Fx Fy Fz
+
+
x
y
z
F = (x2 + y 2 + kz 2 )
F = ( 3 + kz 2 )
Problem 5:
This problem corresponds to Schey, Problem II-23 (page 58):
Verify the Divergence Theorem for the function F (x + y + kz ) for a cube with each side
of length b positioned with one corner at (0, 0, 0) and the other corner at (b, b, b) as shown in the
gure at the top of Page 59.
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Physics 124 Spring 2011 Document #11: Homework #06
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Problem 6:
Suppose within some nite region a putative electric potential function is given by:
(x, y, z ) 0 xyz
Calculate the associated electric eld. Determine whether this eld can possibly correspond
to a physical electrostatic eld in accordance with Conservation of Energy. Can the total charge
contained within the region be non-zero?
Problem 7:
Same as Problem 6 but now use this function:
(x, y, z ) 0 x2 y 2 z 2
Problem 8:
Consider the = function:
F y z + kx
Demonstrate that Stokes Theorem is true for a surface on the xy -plane corresponding to a
square with one corner at the origin and the other corner at (x, y, z ) = (1, 1, 0)
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Physics 124 Spring 2011 Document #11: Homework #06
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Problem 9: (from a previous years exam:)
Suppose we are given scaler electric potential (also known as voltage) which varies in accordance with one given Cartesian component z as follows:
0 z 2
2L2
where 0 and L represent positive constants in appropriate units. Assume this denition applies
over all space for both positive and negative values of z . Suppose we consider a closed surface
S that is dened to be the outer surface of a cube of dimension L L L with one corner at
(x, y, z ) = (0, 0, 0) and the other corner at (x, y, z ) = (L, L, L)
(z ) =
Part (a): Calculate the associated electric eld within the cube as a function of position. Explain
how you know this.
Part (b): Determine whether this eld can possibly correspond to a physical electrostatic eld in
accordance with Conservation of Energy. Explain your work.
Part (c): Find the total enclosed charge contained within cube. Explain your work.
Problem 10: (from a previous years exam:)
Consider the function:
F = x + y
Demonstrate that Stokes Theorem is true for this function for a loop in the xy -plane corresponding to a unit circle (radius = 1) centered at the origin and the at surface bounded by this
loop.
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Physics 124 Spring 2011 Document #11: Homework #06
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Problem 11: (from a previous years exam:)
A xed electric potential is dened over a region of space as follows:
V (x, y, z ) = Ay 2 + By + C
where A, B , and C are given positive coefcients with proper units.
Suppose that we place a point charge particle with given charge +q0 and a given mass m is
at the origin (x, y, z ) = (0, 0, 0) with given initial velocity v0 = v0. Suppose that at some time
later we nd the particle at a given position (xf , yf , zf ). Suppose that the only force on the charged
particle is the force due to the applied electric eld.
Part (a) What is the electric eld as the origin? (x, y, z ) = (0, 0, 0). Ignore the point charge
particle. Explain your work.
Part (b) What is the force on the point charge particle when it arrives at position (xf , yf , zf )?
Explain your work.
Part (c) What is the volume charge density at some arbitrary point in space (x, y, z )? Ignore the
point charge particle. Explain your work.
Part (d) What is the speed of the particle when it arrives at position (xf , yf , zf )? Explain your
work.