Displacement Current and EM waves

Lecture 28 (the last!)

Inconsistent (or not)?

Earlier: Why can’t I use the Ampère Law?

I

A problem with the Ampère Law…

• RHS of the Ampère Law is equal for surfaces with the same boundary!

• No problems here!

M1

@M1 @M2

M2

@M ⌘ @M1 = @M2

I

@M

~d` · ~B = µ0Ienc.(M)

• RHS of the Ampère Law is equal for surfaces with the same boundary!

• “Houston, we have a problem!”

• Steady state is fine but time dependent Q is a problem!

A problem with the Ampère Law…

M1

I

@M

~d` · ~B = µ0Ienc.(M)

Similar problem with the Faraday Law:

• Time independent equation (E is conservative):

• Faraday-Maxwell Equation:

• Symmetry Ampère-Maxwell Equation?

I

@M

~d` · ~E = 0

I

@M

~d` · ~E = �Z

Md2A n̂ · @

@t~B

I

@M

~d` · ~B = µ0Ienc + C0

Z

Md2A n̂ · @

@t~E

• Ampère-Maxwell Equation:

Consider the capacitor: Displacement Current

Q = CV =A✏0`

`E

ID ⌘ dQ

dt= A✏0n̂ · @

~E

@t

ID = I

IDI I

I

@M

~d` · ~B = µ0Ienc + ✏0µ0

Z

Md2A n̂ · @

@t~E

~|tot

= ~|+ ✏0

@t ~E

Electrodynamics & the (Integral) Maxwell Equations• Gauss Law (E):

• Gauss Law (B):

• Ampère Law:

• Faraday Law:

I

Md2A n̂ · ~E = Qinside/✏0

I

Md2A n̂ · ~B = 0

I

@M~d` · ~B = µ0I

I

@M~d` · ~E = � d

dt

Z

Md2A n̂ · ~B

+µ0✏0d

dt

Z

Md2A n̂ · ~E

Ampère-Maxwell Law: More than meets the eye!

• In vacuum:

• Coupling between time-dependent equations could lead to propagation of perturbations in the fields!

• Waves!

I

@M

~d` · ~B = ✏0µ0

Z

Md2A n̂ · @

@t~E

I

@M

~d` · ~E = �Z

Md2A n̂ · @

@t~B

Source

Remember… (review?)

Do Maxwell Eqn’s lead to a wave equation?

• Consider a very small loop..

• Keep only the lowest order terms…

• Identical derivation for Ampère Law

Mathematical aside: Stokes’ Theorem

Z

@⌃↵ =

Z

⌃d↵

Zx2

x1

dx ddxf(x) = f(x2)� f(x1)

Simplest realization:

Integral of total d

Evaluate on boundary

Aside… Maxwell Equations Differential Form:

• Gauss Laws:

• Faraday-Maxwell Law:

• Ampère-Maxwell Law:

~r⇥ ~E = �@t ~B

~r⇥ ~B = µ0~|+ µ0✏0@t ~E

~r · ~B = 0

~r · ~E = ⇢✏0

Do Maxwell Eqn’s lead to a wave equation?

r2 ~E = µ0✏0@2t~E

Maxwell Equation to describe a wave eqn!

• Wave equation for EM waves:

• Identify the propagation speed of the wave:

• The speed of the wave matches the speed of light!

v = c =1

p✏0µ0

⇡ 3⇥ 108 m/s

@2

@z2h = 1v2

@2

@t2h

@2

@z2~E = ✏0µ0

@2

@t2~E

Faraday’s Hypothesis:• Light is an electromagnetic wave!

• Unified theories of Electricity, Magnetism and Light!

Heinrich HertzJames Clerk Maxwell Michael Faraday(Proposal) (Theoretical Motivation) (Experiment)

Light is not the only type of EM wave!

Demo on EM waves

More features of EM waves• Wave equation:

• Plug in sinusoidal solution

• Dispersion relation:

E = E0 sin(kz � !t)

k = !/c

@2

@z2~E = ✏0µ0

@2

@t2~E

�⌫ = c

wave number: � = 2⇡/k

More features of EM waves• Plane wave solution:

1c~kk ⇥ ~E0 = ~B0

~kk · ~E0 = 0