Maxwell's EquationsMaxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships. These basic equations of electricity and magnetism can be used as a starting point for advanced courses, but are usually first encountered as unifying equations after the study of electrical and magnetic phenomena. Symbols Used| E = Electric field| ρ = charge density| i = electric current| B = Magnetic field| ε0 = permittivity| J = current density| D = Electric displacement| μ0 = permeability| c = speed of light| H = Magnetic field strength| M = Magnetization| P = Polarization| Integral form| Differential form|

| Index

Maxwell's equations concepts|
HyperPhysics***** Electricity and Magnetism | R Nave|
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Maxwell's EquationsIntegral form in the absence of magnetic or polarizable media: I. Gauss' law for electricity | | II. Gauss' law for magnetism | |
III. Faraday's law of induction | |
IV. Ampere's law | |
Differential form| Discussion|
| Index

Maxwell's equations concepts|
HyperPhysics***** Electricity and Magnetism | R Nave|
| Go Back|

Maxwell's EquationsDifferential form in the absence of magnetic or polarizable media: I. Gauss' law for electricity | | II. Gauss' law for magnetism | |
III. Faraday's law of induction | |
IV. Ampere's law | |
Integral form| Discussion|
Differential form with magnetic and polarizable media|
| Index

Maxwell's equations concepts|
HyperPhysics***** Electricity and Magnetism | R Nave|
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Maxwell's EquationsDifferential form with magnetic and/or polarizable media: I....

...The Maxwell equations
Introduction:-
One of Newton's great achievements was to show that all of the phenomena of classical mechanics
can be deduced as consequences of three basic, fundamental laws, namely Newton's laws of
motion. It was likewise one of Maxwell's great achievements to show that all of the phenomena of
classical electricity and magnetism – all of the phenomena discovered by Oersted, Ampère, Henry,
Faraday and others whose names are commemorated in several electrical units – can be deduced as
consequences of four basic, fundamental equations. We describe these four equations in this
chapter, and, in passing, we also mention Poisson's and Laplace's equations. We also show how
Maxwell'sequations predict the existence of electromagnetic waves that travel at a speed of 3 % 108
m s−1. This is the speed at which light is measured to move, and one of the most important bases of
our belief that light is an electromagnetic wave.
Before embarking upon this, we may need a reminder of two mathematical theorems, as well as a
reminder of the differential equation that describes wave motion.
The two mathematical theorems that we need to remind ourselves of are:
The surface integral of a vector field over a closed surface is equal to the volume integral of its
divergence.
The line integral of a vector field around a closed plane...

...Applications of Differential Forms
Maxwell Faraday and Maxwell Ampere Equations
R. M. Kiehn (in preparation - last update 10/31/97) Physics Department, University of Houston, Houston, Texas
Abstract: The topological universality of the Maxwell Faraday and Maxwell Ampere equations is an artifact of C2 differential forms on a domain of dimension n * 4. Starting with a 1-form of (electromagnetic) Action, the Maxwell Faraday equations become a consequence of the Poincare lemma. Starting from an N-1 form density, the Maxwell Ampere equations become a consequence of the topological constraint that the N-1 form density is exact. The conservation of charge current is a consequence of the Poincare lemma. Geometrical structure constraining the deduced 2-form and the induced N-2 form establish equivalence classes of constitutive equations. Evolutionary processes acting on Maxwell-Faraday systems can be classified into reversible and irreversible categories, depending upon the Pfaff dimension of the Action 1-form. The perfect plasma equations are equivalent to the unique Hamiltonian dynamical systems on spaces of Pfaff dimension 3, and the Master equations describe reversible processes on the symplectic manifold of Pfaff dimension 4. Irreversible processes generate dynamical systems proportional to vector fields ÝExA + Bd, A 6 BÞ, on symplectic domains of Pfaff dimension 4.
THIS...

...2/20/2014
Frequently Used Equations - The Physics Hypertextbook
Frequently Used Equations
Mechanics
velocity
Δ
s
v=
Δ
t
ds
v=
dt
acceleration
Δ
v
a=
Δ
t
dv
a=
dt
equations of motion
v = 0+at
v
x =x0+v 0 +½ 2
t
at
weight
W =m g
momentum
p =m v
dry friction
ƒ μ
=N
centrip. accel.
v2
ac =
r
2
ac =−ω r
impulse
J =F Δ
t
impulse–momentum
F Δ= Δ
t m v
J =⌠ dt
F
⌠ dt =Δ
F
p
⌡
kinetic energy
potential energy
⌡
K =½
mv
2
gravitational p.e.
Δ g =mgΔ
U
h
Δ =−⌠ ·
U
F ds
⌡
F =−∇U
v 2= 0 +2 x − 0
v 2 a( x )
v =½ v + 0
( v)
work
W = s cs
FΔ o θ
W =⌠ ·
F ds
newton's 2nd law
∑F =m a
dp
∑F =
dt
efficiency
Wout
ℰ=
Ein
work–energy
FΔ cs =Δ
s oθ E
⌡
⌠ · =Δ
F ds
E
⌡
power
Δ
W
P=
Δ
t
P = cs
Fv o θ
dW
P=
dt
P =F ·
v
angular velocity
Δ
θ
ω=
Δ
t
dθ
ω=
dt
v =ω ×r
angular acceleration
Δ
ω
α=
Δ
t
dω
α=
dt
2
a = × −ω r
α r
equations of rotation
ω=ω +α
t
0
θ θ +ωt +½ t2
=0 0
α
torque
τ rF s θ
=
i
n
τ =r ×F
moment of inertia
I =∑mr2
rotational work
W =τ θ
Δ
I = r2dm
W= τ·
dθ
rotational k.e.
K =½ 2
Iω
angular momentum
L =mrv s θ
i
n
L =r ×p
hooke's law
F =− Δ
k x
L= ω
I
frequency
1
ƒ
=
T
ω=2ƒ
π
universal gravitation
Gm m
Fg =− 1 2r̂
r2
gravitational field
Gm
g =− 2 r̂
r
gravitational...

...There are now two separate equations: 60 = 6b - 6c and 60 = 3b + 3c
Solve both equations for b: b = 10 + c b = 10 - c
Now make both equations equal each other and solve for c: 10 + c = 10 - c 2c = 0 c = 0
The speed of the current was 0 mph Now, plug the numbers into one of either the original equations to find the speed of the boat in still water.
I chose the first equation: b = 10 + c or b = 10 + 0 b = 10
The speed of the boat in still water must remain a consistent 10 mph or more in order for Wayne and his daughter to make it home in time or dinner.
My Solution: c = current of river b = rate of boat d = s(t) will represent (distance = speed X time) Upstream: 60 = 6(b-c)
Downstream: 60 = 3(b+c)
There are now two separate equations: 60 = 6b - 6c and 60 = 3b + 3c
Solve both equations for b: b = 10 + c b = 10 - c
Now make both equations equal each other and solve for c: 10 + c = 10 - c 2c = 0 c = 0
The speed of the current was 0 mph Now, plug the numbers into one of either the original equations to find the speed of the boat in still water.
I chose the first equation: b = 10 + c or b = 10 + 0 b = 10
The speed of the boat in still water must remain a consistent 10 mph or more in order for Wayne and his daughter to make it home in time or dinner.
My Solution: c = current of river b = rate of boat d = s(t) will...

...Hall
Differential Equations
March 2013
Differential Equations in Mechanical Engineering
Often times college students question the courses they are required to take and the relevance they have to their intended career. As engineers and scientists we are taught, and even “wired” in a way, to question things through-out our lives. We question the way things work, such as the way the shocks in our car work to give us a smooth ride back and forth to school, or what really happens to an object as it falls through the air, even how that people can predict an approximate future population. These questions, and many more, can be answered and explained through different variations of differential equations. By explaining and answering even just one of these questions through different differential equations I will also be answering two other important questions. Why is differential equations required for many students and how does it apply in the career of a mechanical engineer?
First some background. What is a differential equation?
A differential equation is a mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. They are used whenever a rate of change is known but the process
giving rise to it is not. The solution of a differential equation...

...Kinematic Derivation of the Wave Equation
http://prism.texarkanacollege.edu/physicsjournal/wave-e...
KINEMATIC DERIVATION OF THE HARMONIC WAVE EQUATION AND RELATED TOPICS An extremely important type of wave in physics is the harmonic wave. This is a wave consisting of propagating simple harmonic oscillations or linear combinations thereof. Attach a weight to a spring and hang the spring so the weight is free to move. Then lift the weight straight up and release it; it will oscillate up and down in a close approximation to simple harmonic motion. Now imagine that you have a whole array of such mass-spring systems, all oscillating independently. If you could add all their oscillations together, you would get linear combination of simple harmonic motions. Whether a harmonic wave involves propagating oscillations of a solid, liquid, gas, or electromagnetic ﬁeld, the oscillations will consist, in general, of a linear combination of simple harmonic motions as exempliﬁed by the sum of the oscillations of the multiple mass-spring system.
The One-Dimensional Wave Equation
Mathematically, a harmonic wave can represented by a sinusoidal function of time and position or a linear superposition of such functions (that is, you just add them together as in the case of sum of the oscillations of the spring and weight system described above), since a single such sinusoidal function describes the propagation of a single simple harmonic...

...
The short story Cold Equations by Tom Godwin takes place on a ship called EDS. The space cruiser is piloted by a man named Barton. He has an order of killing the stowaway who snuck onto the ship because the weight on the EDS is too much for the ship to handle. In the process of hunting down the stowaway, he realizes it was a young innocent girl named Marilyn. Once Barton understands what kind of person Marilyn is, he doesn’t kill her immediately because he knows her reasons were pure. Marilyn only wanted to see her brother, Gerry, again after ten years of being apart and was ignorant to the fact that her life can end with the decision of sneaking onto the ship. Barton begins to feel compassion after being with her and tries to comfort her, but knows what her fate is. He lets Marilyn live long enough to let her speak with Gerry once more before he follows through with the command. After Gerry and Marilyn speak he ejects her out into space. The ending was logical and no other endings would be possible because one the equation that was calibrated delicately, and two Barton could not throw the out the fever serums because that is the main reason for going on the trip to Woden.
A theoretical ending of Cold Equations could have been that Barton sacrifices himself for Marilyn, but since she is lighter than him, the fragile calibrated equation would be disrupted due to the change in weight. On EDS everything on ship is...

...CHAPTER 2
FIRST ORDER DIFFERENTIAL EQUATIONS
2.1 Separable Variables
2.2 Exact Equations
2.2.1 Equations Reducible to Exact Form.
2.3 Linear Equations
4. Solutions by Substitutions
2.4.1 Homogenous Equations
2.4.2 Bernoulli’s Equation
2.5 Exercises
In this chapter we describe procedures for solving 4 types of differential equations of first order, namely, the class of differential equations of first order where variables x and y can be separated, the class of exact equations (equation (2.3) is to be satisfied by the coefficients of dx and dy, the class of linear differential equations having a standard form (2.7) and the class of those differential equations of first order which can be reduced to separable differential equations or standard linear form by appropriate.
2.1 Separable Variables
Definition 2.1: A first order differential equation of the form
[pic]
is called separable or to have separable variables.
Method or Procedure for solving separable differential equations
(i) If h(y) = 1, then
[pic]
or dy = g(x) dx
Integrating both sides we get
[pic]
or [pic]
where c is the constant of integral
We can write
[pic]
where G(x) is an anti-derivative...