HW-1 solution
1. For the two vectors A x 2 y z and B 2 x 3 y z , find (a) A B and A B , (b)
component of B along A , (c) angle between A and B , (d) A B , (e) A B A B .
Solution:
(a) A B ( x 2 y z ) (2 x 3 y z ) 3x y 2 z
| A B | 32 (1) 2 (2) 2 3.74
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University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 10 (compiled by Daniel Larson)
1. Griths 9.19
(a) For a poor conductor,
, we can expand the second square root in the formula for (Eqn 9.126):
1+
2
1+
1
2
2
+ .
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink)
SOLUTION TO MIDTERM EXAMINATION 1
Directions: Do all 3 problems, which have unequal weight. This is a closed-book closed-note exam
except for one 8 1 11 inch sheet containing a
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 8 (compiled by Daniel Larson)
1. Griths 7.21 The emf is the time-derivative of the ux due to the small loop that passes through the big
loop, namely E = d = M dI
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 9 (compiled by Daniel Larson)
qt
1
r
r
1. Griths 10.3 First well calculate the elds. E = V A = 0 4 0 r12 . B = A = 4 0 r = 0.
2
t
These elds should be familiar, b
Solutions Homework 7
Physics 110A
Spring 2017
(1) The volume between two concentric conducting spherical surfaces of radii a and b(a < b) is
filled with an inhomogeneous dielectric constant
=
0
1 + Kr
(1)
where 0 and K are constants and r is the radial co
Physics 110A
Problem Set 2
Spring 2017
Due by 5:30 PM on Friday, February 3. Turn in to the designated homework box in LeConte
Reading: Griffiths, Sections 1.4 1.6, 2.2 2.3
(1) Prove the following vector identities
(a)
A g( A) (g) A
=
g
g2
(b)
2 ( A) = (2
Physics 110A
Problem Set 6 Solutions
Spring 2017
(1) (a) The energy is given by
U = p2 E21 ,
(1)
where E21 is the electric field due to dipole p1 at p2 . Note that were just as fine
calculating the energy in terms of the first dipole and the electric fiel
Problem Set 1
Physics 110A
Spring 2017
(1) (a) We consider the vectors component-wise, i.e.
u = (u1 , u2 , , un )
(1)
v = (v1 , v2 , , vn ).
(2)
Thus, we can write
|u + v|2 + |u v|2 =
=
=
n
n
X
X
(ui + vi )2 +
(ui vi )2
i=1
n
X
i=1
n
X
(3)
i=1
(ui + vi )2
Physics 110A
Problem Set 3 Solutions
Spring 2017
(1) (a) Its pretty clear that the first vector field has a non-zero divergence, so lets look at the
second:
(y x) +
(z y) +
(2z + x)
x
y
z
= 1 + (1) + 2
(2)
= 0,
(3)
F2 =
(1)
so F2 is divergence-less. This
Problem Set 1
Physics 110A
Spring 2017
(1) Vectors.
(a) Consider two n-dimensional vectors u and v. Show that
|u + v|2 + |u v|2 = 2 |u|2 + |v|2 .
(1)
(b) Prove, using index notation, the following identity for the curl of a curl:
( v) = ( v) 2 v.
(2)
(2)
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 11 (by Daniel Larson)
1. Polarization
(a) We want to convert the real electric eld E into a complex electric eld.
E = E0 Re ei(kzt) x + E0 bRe ei(kzt+) y = Re E0
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 7 (compiled by Daniel Larson)
1. Griths 6.17 In a linear material, we know H is proportional to B: B = H = 0 (1 + m )H, so for a long
wire it should be circumfere
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 6 (compiled by Daniel Larson)
1. Griths 5.24 If B is uniform, then it is not a function of position, so any derivative of it vanish. In particular,
B = 0 and B =
Physics 110A, Electromagnetism and Optics
Discussion Section Problems, Set #1
1. Consider an n-sided polygon with point charges of magnitude q placed at each vertex. Prove
that the electric eld is zero in the center of the polygon.
2. A rod of length L ca
Physics 110A, Electromagnetism and Optics
Homework #3
Due at 5:30 PM on Friday, February 12. Turn in to the designated homework box in LeConte 251.
Reading: Griths, Sections 1.5 - 1.6, 2.2 - 2.3
Homework:
1. Consider a point charge q positioned at the ori
Physics, 110A, Electromagnetism and Optics
Homework #2
Due by 5:30 PM on Friday, February 5. Turn in to the designated homework box in LeConte 251.
Reading: Griths, Sections 1.4 1.6, 2.2 2.3
Homework:
1. Prove, using index notation or otherwise, the follo
Homework #4
1. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric
metal shell (inner radius a, outer radius b). The shell carries no net charge.
(a) Find the surface charge density at R, at a, and at b.
(b) Find the potenti
Homework #4
1. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric
metal shell (inner radius a, outer radius b). The shell carries no net charge.
(a) Find the surface charge density at R, at a, and at b.
(b) Find the potenti
HW-3
1. Find the electric field a distance z above the center of circular disk of radius R which
carries a uniform surface charge density . What does your formula give in the limit
R and R < z .
Solution:
Obviously the electric field is in the z direction
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 1
1. Griths 1.14 Under a rotation, the coordinates y and z transform into y = y cos + z sin and z =
y sin + z cos , so we can invert these equations to nd y = y c
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 5 (compiled by Daniel Larson)
1. Griths 5.1 Since the eld is pointing into the page, a positive charge would feel a force in the direction
v B, which is up. So th
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 4 (compiled by Daniel Larson)
1. Griths 3.33 To get a general formula for the electric eld from an electric dipole, lets start with the general
formula for the po
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink )
Solution Set 3 (compiled by Daniel Larson)
1. Griths 3.1 We want to calculate the average potential on the surface of a sphere due to a point charge q
located somewhere within
University of California, Berkeley
Physics 110A Fall 2001 Section 1 (Strovink)
Problem Set 2
1. Griths 2.18
everywhere within the outer sphere. Construct
a Gaussian surface consisting of a third sphere
at radius r, where R1 < r < R2 . Consider the
volume
Physics 110A
Problem Set 5 Solutions
Spring 2017
(1) (a) The potential being constant inside might satisfy the boundary condition, but the
uniqueness theorem only applies if the potential we choose satisfies Poissons equation
in the first place, which a c
Physics 110A
Solutions Homework 4
Spring 2017
(1) First for r < R, using Gauss Law with a spherical Gaussian surface with radius r we have,
E(r)4r2 =
=
=
=
= E(r) =
Qen
0
Z
1 r k
4r02 dr0
0 r0 =0 r0
Z
4k r
r0 dr0
0 r0 =0
2kr2
0
k
20
If r > R, then by by a
Physics 110A, Electromagnetism and Optics
Midterm #1
Please read the questions carefully and show all work. You are allowed to use your own notes. If you
find a root or fraction that you cannot evaluate, either estimate it or leave it unsimplified. There
Physics 110A
Problem Set 3
Spring 2017
(1) Let F1 = x2 x
+ y2y
+ z 2
z and F2 = (y x)
x + (z y)
y + (2z + x)
z.
(a) Which vector field is divergence-less? Calculate a vector potential for this field.
(b) Which vector field is curl-less? Calculate a scal
Physics 110A
Problem Set 5
Spring 2017
(1) The Return of Maxwells Demon. Maxwells Demon, embarrassed a couple problem sets
ago, is back with a vengeance.
(a) Ive got a point charge q lying off-center in a conducting spherical shell, he says. Since
its a c
Physics 110A
Problem Set 7
Spring 2017
Due by 5:30 PM on Friday, March 17. Turn in to the designated homework box in LeConte
(1) The volume between two concentric conducting spherical surfaces of radii a and b(a < b) is
filled with an inhomogeneous dielec
Physics 110A
Problem Set 6
Spring 2017
(1) Two dipoles p1 = p1 y
and p2 = p2 x
lie at the points (0, a) and (a, 0) where a > 0.
(a) Calculate the interaction energy between the two dipoles.
(b) Calculate the force that p2 exerts on p1 .
(2) A square pyr
Physics 110A
Solutions Homework 2
Spring 2017
(1)
(2) There are three forces on the ball, gravitational force acting downwards, electrastatic force
to the right and tension along the string. Note that the tension T makes an angle with the
vertical. Given