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Generalized Ampere’s Law Tested I I Consider a parallel plate capacitor that is being charged Try Ampere’s modified Law on two nearly identical surfaces/loops
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Generalized Ampere’s Law Tested
I I
•Consider a parallel plate capacitor that is being charged•Try Ampere’s modified Law on two nearly identical surfaces/loops
11 0 1 0 0
EdB ds Idt
22 0 2 0 0 0
EdB ds I Idt
0 00
d Qdt
0I
Comparision of Induction
BdE dsdt
0 0 0EdB ds I
dt
•No magnetic monopole, hence no magnetic current•Electric fields and magnetic fields induce in opposite fashions
Maxwell’s EquationsIf we combine all the laws we know about electromagnetism, thenwe obtain Maxwell’s equations.
These four equations plus a force law form the basis for all of electromagnetism!
Thesbe laws predict that accelerating charges will radiate electromagnetic waves!
The fact that classical models of the atom contradicted Maxwell’s equations motivated quantum mechanics.
Maxwell’s Equations
BdE dsdt
Integral Form
0 0 0EdB ds I
dt
0
S
B dA
0
in
S
qE dA
Gauss’s laws, Ampere’s law and Faraday’s law all combined!
They are nearly symmetric with respect to magnetism and electricity.
The lack of magnetic monopoles is the main reasonwhy they are not completely symmetric.
Maxwell’s Equations
BdE dsdt
Integral Form
0 0 0EdB ds I
dt
0
S
B dA
0
in
S
qE dA
dBEdt
0 0 0
dEB Jdt
0
E
0B
Differential Form
Maxwell and Lorentz Force Law
dBEdt
0 0 0
dEB Jdt
0
E
0B
Differential Form
BvqEqF ~~~~
FYI, These are connected to the integral equations via the generalized stokes equation
Derivatives and Partial Derivatives•When you have multiple variables, and you need to take the derivative, you use a partial derivative•Partial derivatives are like ordinary derivatives, but all other variables are treated as constants•{We have done this before; remember the gradient}
sinf kx t cosf k kx tx
Vector Derivatives: Dot products in Cartesian Coordinates
ˆ ˆ ˆx y zx y z
ˆ ˆ ˆ ˆˆ ˆx y zB x y z B x B y B zx y z
Nambla: a vector derivative
0yx zBB B
x y z
is the divergence of B.B
Vector Derivatives: Cross products in Cartesian Coordinates
ˆ ˆ ˆx y zx y z
ˆ ˆ ˆ ˆˆ ˆx y zB x y z B x B y B zx y z
Nambla: a vector derivative
ˆ ˆ ˆy y xz z zB B BB B Bx y z
y z z x x y
is the curl of B.B
0 0 0dEJdt
Vector Derivatives: In other coordinates
ˆ ˆ ˆx y zx y z
1 1ˆ ˆ ˆ ˆˆ ˆsin rB r B r B B
r r r
Nambla needs to be convertedif we change coordinates
1 1 ˆ 0sin
rBBB
r r r
Spherical:1 1ˆ ˆˆ
sinrr r r
Two of Maxwell’s Equations
dBEdt
0 0 0
dEB Jdt
ˆ ˆ ˆx y z
x y z
Nambla: a vector derivative
ˆ ˆ ˆ ˆˆ ˆx y zB x y z B x B y B zx y z
ˆ ˆ ˆy yx xz zB BB BB Bx y z
y z z x x y
y x zE E Bx y t
yx z BE Ez x t
y xz E BEy z t
0 0y x zB B Ex y t
0 0yx z EB B
z x t
0 0y xzB EB
y z t
Using Maxwell’s Equations
Waves from Electromagnetism•Consider electric fields (pointing in the y-direction) that depend only on x and t•Consider magnetic fields (pointing in the z-direction) that depend only on x and t•Consider vacuum , aka free space, so J=0
Plane waves
(We could be more general)
y x zE E Bx y t
yx z BE Ez x t
y xz E BEy z t
0 0y x zB B Ex y t
0 0yx z EB B
z x t
0 0y xzB EB
y z t
Using Maxwell’s Equations
y zE Bx t
0 0
yz EBx t
Electromagnetic Waves
•These equations look like sin functions will solve them.
0 cosyE E kx t 0 coszB B kx t
0 0
0 0 0 0
sin sin
sin sin
kE kx t B kx t
kB kx t E kx t
0 0 0 0 0 0 kE B kB E
Electromagnetic Waves0 0 0 0 0 0 kE B kB E
•These equations imply
2 20 0 0 0 0 0k E B B E
2
20 0
1k
0 0
1k
82.998 10 m/sc
•The speed of light (in vacuum)
