VII. Electrodynamics •  Dynamics •  Ohm’s law •  Conductivity, resistivity and resistance

Today’s lecture

Magnetostatics Electrostatics

Magnetostatics and electrostatics

nb

b

uP

P

P

⋅=

⋅∇−=

:

:

σ

ρ

nb

b

uMK

MJM

×=

×∇=

aIm ⋅=: dqp

⋅=:

0ερ

=⋅∇ E

0

0

=⋅

=×∇

∫ ldE

E

∫∫ =⋅S

enclosedQadE0ε

0

0

=⋅

=⋅∇

∫∫ adBB

S

( )rJB ⋅=×∇ 0µ

enclosedP

IldB 0µ=⋅∫

A

B

VE

PED

+= 0ε

( )∫∫∫=V

d dudrrE ''

41)( 20

τρ

πε

VE ∇−=

AB

×∇=

( ) ∫×

= ')'(4 20 dl

durIrB d

πµ

MBH

−=0

1:µ

( ) HHB m

⋅=⋅+= µχµ 10 ( ) EED e

εχε =⋅+= 10

MHJH free

⋅∇−=⋅∇=×∇ freeDPD ρ=⋅∇×∇=×∇

Reminder: What are static fields ?

Electrostatics: Stationary charges ⇒constant electric fields ⇒electrostatics Magnetostatics: Steady currents⇒constant magnetic fields⇒magnetostatics Steady current (I=const) produces time-independent B field For a steady current (no charges piling up) Continuity equation in magnetostatics: Electrostatics Magnetostatics Stationary charges stationary currents In conductor: In Electrodynamics (as we will see):

0=−=⋅∇t

Jδρδ

0

≠=σJE

( ) .consttE =

( ) .consttB =

0

=J 0

≠J

0=tδρδ

0=tδρδ

0=−=⋅∇t

Jδρδ

( ) ( )tBBtEE

== , ( ) 0

≠= tJJ

0

=E

What does the current density depend on? For most materials J is proportional to the force (on each charge). The proportionality factor is the conductivity σ : Not to confuse with volume and surface charge densities ρ,σ For electromagnetic forces: When v is small or the contribution of B is small this becomes

Ohm’s law: Remember: Electrostatics (stationary charges): inside conductor. Generally inside conductor:

Ohm’s Law in terms of J and E

fqFJ

⋅=⋅= σσ

( )BvEJ

×+⋅=σ

EJ⋅=σ

σρ

1= resistivity

conductivity σ

0

=E

0

≠=σJE

For steady currents: For uniform conductivity: Laplace equation holds: Example: wire carrying current I: What is its potential and field? Boundary conditions: V=0 at z=0 V=V0 at z=L Outer surface:

Ohm’s Law

( ) 0=∇⋅+∇⋅=⋅⋅∇=⋅∇ σσσ

EEEJ

0=⋅∇ J

RIALIVA

LVAEAJI ⋅=⎟

⎠⎞

⎜⎝⎛

⋅⋅=⇔⋅⋅=⋅⋅=⋅=σ

σσ 00

0=∇σ

0=∇⇒ E

02 =∇⇒ V z

A L

σ

000 =⇒=⇒=⋅ ⊥

nVEuJ n δ

δ

LzVzV ⋅

=⇒ 0)( zuLVVE 0−=∇−=⇒

familiar version of Ohm’s law: R is resistance, depends usually on geometry and conductivity.

Ohm’s “law” (better Ohm’s experimental observation):

• Conductivity and resistivity are inherent material properities. • Resistance depends on the geometry of the arrangement. • Conductivity and resistivity are among the physical properties that can vary the largest amount from Conductor (Cu ρ=1.7·10-8 Ωm) to insulator (Quartz ρ≈1016 Ωm).

Conductivity, resistivity and resistance

EJ⋅=σ

σρ

1= resistivity

conductivity σ

RIV ⋅=IVR = resistance

Two concentric metal spherical shells, of radius a and b, respectively are separated by weakly conducting material of conductivity σ. (Problem 7.1) (a) If they are maintained at a potential difference V, what current flows from one to the other? (b) What is the resistance between the shells? (c) Note that if b>>a the outer radius b is irrelevant. How do you explain that? Exploit this observation to determine the current flowing between to metal spheres (radius a) Immersed deep in the sea and held quite far apart with potential difference V between them. (d) What is R for two cylinders?

Problem

a b