Problems on Chapter 31 1. A 18.0F capacitor is placed across a 22.5V battery for a few seconds and is then connected across a 12.0mH inductor that has no appreciable resistance. a) After connecting the capacitor and inductor together, find the maximum c
1. You have a pure (24carat) gold ring with mass 17.7 g. Gold has an atomic mass of 197 g/mol and an atomic number of 79. i. How many protons are there in the ring, and what is their total positive charge? ii. If the ring carries no net charge, how many
Problems on Chapter 25 1. Three capacitors C1 = 3 C, C2 = 5 C, and C3 = 8 C are each charged by a 24V battery and then connected as shown. + C1 + C2 
X
+ C3

Y
a) Determine the final charge on each capacitor b) Determine the potential between points X
Problems on Chapter 26 (due at 9 am on Tuesday 29 June) 1. A lead wire of resistance R is drawn through a die so that its length is doubled, while its volume remains unchanged. What will be its new resistance in terms of its initial resistance? 2. A hair
Problems on Chapter 28 1. A particle of mass 0.195 g carries a charge of 2.50 x 10 8 C. The particle is given an initial horizontal velocity that is due north and has magnitude 4.00 x 104 m/s. What are the magnitude and direction of the minimum magnetic
Problems on Chapter 29 1. A long straight wire lies along the zaxis and carries a current of 4.00A in the +zdirection. Find the magnetic field (magnitude and direction) produced at the following points by a 0.500mm segment of the wire centered at the o
Problems on Chapter 30 1. a) The area of an elastic loop decreases at a constant rate, dA/dt =  3.50 x 102 m2/s. The loop is in a magnetic field B = 0.28 T whose direction is perpendicular to the plane of the loop. At t = 0, the loop has area A = 0.285
1. An electromagnetic wave has an electric field given by: = (225 /) sin[(0.077m1 ) 7 (2.3 10 rad/s)] a) What are the wavelength and frequency of the wave? b) Write down an expression for the magnetic field. Problems on Chapter 33 2. Unpolarized light of
DEFINITION
A function of two variables has a local
maximum at (a, b) if
f (x,
(x y) f (a,
(a b) when (x
(x, y) is near (a,
(a b).
b)
[This means that
f (x, y) f (a, b) for all points (x, y) in some
disk with center (a, b).]
The number f (a, b) is
called a
Chapter
p
12: Partial
Derivatives
Section 12.3
Partial
P
ti l derivatives
d i ti s are d
defined
fi d ass d
derivatives
i ti s of
fa
function of multiple variables when all but the variable of
interest are held fixed during the differentiation.
Definition
TANGENT PLANES
Suppose a surface S has equation z = f (x, y),
)
where f has continuous first partial derivatives,
and let P(x0, y0, z0) be a p
point on S. Let C1 and C2
be the two curves obtained by intersection the
vertical planes y = y0 and x = x0 with
DEFINITION If f is a function of two variables
x and
d y , then
th
the
th gradient
di t off f iis the
th vector
t
function f defined by
f
f
f ( x, y) f x ( x, y),
) f y ( x, y) i
j
x y
Du f ( x, y) f ( x, y) u
DEFINITION The directional derivative of f
a
THE CHAIN RULE (CASE 1) Suppose that z=f (x, y)
is a differentiable function of x and y, where x=g (t)
and y=h (t) and are both differentiable functions of t.
Then z is a differentiable function of t and
dz f dx f dy
dt x dt y dt
dz z dx
d
d
z ddy
dt x dt
Chapter 12: Functions of Several
Variables
Section 12.1
Introduction to Functions of Several Variables
Notation for Functions of Several Variables
Previously we have studied functions of one variable, y = f(x) in
which x was the independent variable and y
Problems on Chapter 24 1. A charge Q is placed at one corner of a square of side L, and charges + Q are placed at each of the other corners. What is the potential at the center of the square? 2. A nonconducting sphere of radius R has uniform volume charg
Problems on Chapter 23 1. A point charge q1 = 4.00 nC is located on the xaxis at x = 2.00 m, and a second point charge q2 = 6.00 nC is located on the yaxis at y = 1.00 m. What is the total electric flux due to these two point charges through a spherica
Chapter 22: Electric Fields
22  1
What is Physics?
The concept of the electric field allows us to give a description of how a charged particle can exert a force on another particle even though the two particle are not in contact action at a distance. The
Chapter 23 Gausss Law
231
What is Physics?
Gausss Law relates the electric fields at points on a (closed) surface to the net charge enclosed by that surface
Gausss Law is particularly useful (and powerful) when it uses the symmetry of the charge distribu
Chapter 24 Electric Potential
241
What is Physics? A conservative force has potential energy associated with it Hence we can apply the principle of conservation of energy to closed systems involving the force The electric force is a conservative force; i
Chapter 25 Capacitance
251
A capacitor is a device in which electrical energy can be stored
17 June 2010
1
252
Capacitance
Basic elements of a capacitor
Parallelplate capacitor
17 June 2010
2
Examples of capacitors
Circuit symbols for capacitor
17 June
Chapter 26: Current and Resistance
261
What is Physics?
We consider charges in motion (electric current) in a conductor due to potential difference between the ends of the conductor. Examples of electric current are everywhere: Lightning Nerves currents
Chapter 27  Circuits
271
What is Physics?
Every electrical device you come in contact with as well as the power grid supplying electricity for the country depends on electrical/electronic engineering (and electric circuits in order to operate)
24 June 2
Chapter 28 Magnetic Fields
281
What is Physics?
Applications of magnetic fields and magnetic forces are countless:
Magnetic recording of music and video images Control of CD and DVD players and computer hard
drives
Drive for speaker cones in headphone
 Chapter 29 Magnetic Fields Due to Currents
291
What is Physics? A moving charged particle (i.e., a current) produces a magnetic field. In this chapter we determine the magnetic field due to current in a very small section of currentcarrying wire. We t
 Chapter 30 Induction and Inductance
30  1
What is Physics?
Current
Magnetic Field
Faraday suggested the possibility of obtaining an electric field from magnetic field which will, in turn, give a current:
Magnetic Field
Electric Field
Current
Faradays l
311
Electromagnetic Oscillations & Alternating Currents
In this chapter we consider: 1. Oscillations in LC circuits 2. Damped oscillations in LRC circuits 3. Alternating currents (ac), and 4. Transformers
13 July 2010
1
Recall from mechanics conservation
Electromagnetic Waves Chapter 33
331
Whats Physics?
We are immersed in electromagnetic waves TV and radio signals; telephones; Satellite communications Todays high technology is largely based on electromagnetic waves
1
332
The Electromagnetic Spectrum M
Chapter 34: IMAGES
341
What is Physics?
Image formation is important in many areas including: TV Computers Satellite imaging Cameras, telescopes, microscopes The human eye etc 7/13/2010
1
342
Two Types of Image
To form a virtual image the rays of light
RIGHTHAND RULES
1
Righthand coordinate system
z
y
x
2
Vector cross product
z
C
C = Ax B
x
A
B
y
3
Force on moving charge in magnetic field
FB = qv x B
4
Force on currentcarrying conductor
FB = i L B sin
FB = i L x B
5
Direction of magnetic dipole