PHY 712 Spring 2014 -- Lecture 31 1

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 31:Special Topics in Electrodynamics:

Electromagnetic aspects of superconductivity

04/14/2014

PHY 712 Spring 2014 -- Lecture 31 204/14/2014

PHY 712 Spring 2014 -- Lecture 31 304/14/2014

PHY 712 Spring 2014 -- Lecture 31 404/14/2014

Comment on HW

2

23 3

2

The estimate of the energy loss per cycle is actually discussed in thebeginning of Chap. 14. Dividing that result by the total energy

:of the particle

43

mc

E qE mc

E

2

23 3

2

Note that measured in statcoulombs

and measured in

ha

cm

s the units of ergs with

.

0.43

77

q q

E qE mc

PHY 712 Spring 2014 -- Lecture 31 504/14/2014

Comment on HW

q

v

v’r

2

1

2

Suppose thatˆ

ˆ ˆ ˆ' sin cos sin sin cosˆ ˆsin cos

ˆˆ ˆcos sin

Cross section depends on ( )i

vv a b a b a

q q

q q

v zv x y zr x zε yε

vε v

x z

PHY 712 Spring 2014 -- Lecture 31 604/14/2014

Special topic: Electromagnetic properties of superconductors

Ref:D. Teplitz, editor, Electromagnetism – paths to research, Plenum Press (1982); Chapter 1 written by Brian Schwartz and Sonia Frota-PessoaHistory:

1908 H. Kamerlingh Onnes successfully liquified He1911 H. Kamerlingh Onnes discovered that Hg at 4.2 K has vanishing resistance1957 Theory of superconductivity by Bardeen, Cooper, and Schrieffer

PHY 712 Spring 2014 -- Lecture 31 704/14/2014

Some phenomenological theories < 1957

/0

2

Drude model of conductivity in "normal" materials

( )

fo r

t

dm e mdt

et em

nene tm

v vE

Ev v

J v J E E

2

London model of conductivity in superconducting materials

From Maxwell's equations:4 1

1

dm edt

d e d d nenedt m dt dt m

c

c t

c t

v E

v E

EJB

E

E

B

J v

PHY 712 Spring 2014 -- Lecture 31 804/14/2014

Some phenomenological theories < 1957

2

22

2 2

2

London model of conductivity in superconducting materials

From Maxwell's equations:4 1 1

4 1

4

d d nenedt dt m

c c c

c

t t

t

t

c

c

J v

E BB E

BB B

B

E

J

J

J 3

2

2 32

2

2 3

3

3

32

2 2

22 2

2

2 222

1

4 1

4 1

0 wi1 1 h4

t LL

c

mc c

mc cm

t tne

t tne

t t t

t tc

c ne

B

B BE

B B B

B

PHY 712 Spring 2014 -- Lecture 31 904/14/2014

London model – continued

2

22 2

2 2 2

2

2

2

2

0 with

0

London model of conductivity in superconducting materials

1 14

For slowly varying solution:

1 for

LL

L

d d nenedt dt m

t t nemc

c

t

B

B

J v E

B

/

/

( , )ˆ :

( , ) (0, )

( , ) (0, )London leap:

L

L

z

xz z

xz z

B x tt t

B x t B t et t

B x t B t e

z

2x/2

2

Consistent results for current densit

( ) ( ) B (0)e

y:

4 1 ˆ Ly y L z

L

J Jc

nex xmc

B JBJ y

PHY 712 Spring 2014 -- Lecture 31 1004/14/2014

London model – continued2

22

/

/

4

Vector po

Penetration len

tential for

gth for superconductor:

( , ) (0, )0 :

ˆ ( ) ( ) ( 0)

L

L

L

xz z

xy y L z

neB x t B t e

A

m

A x B e

c

x

AA y

2x/

2

( ) B (0Recall ) e

0 or =

form for current density:

0

Ly L z

nexmc

ne ne emmc c

J

m

J A v A

xL

PHY 712 Spring 2014 -- Lecture 31 1104/14/2014

Magnetization field

0

, Gibbs fr

Treating Lond

ee energy ass

on current in terms of cor

ociated with magnetization

responding magnetization

for superconductor

field :=

:1( ) ( 0) ( ) (0

4For

)8

4

a

L

H

S a S SG H G H dHM H G

x

MB H

HM

M

2

2

Gibbs free energy associated with magnetization for normal conductor:( ) ( 0)

Condition at phase boundary between normal and superconducting states:1( ) (0) ( ) (0)

8

(0) (0

N N

N N C C

N

a

a

C S S

S

H

G H G H

G H G G H G H

G G

2

2 2

1)8

1( ) ( )

for

0 otherwise8 a C

S a

C

C aN a

H

H H HH G H

HG

PHY 712 Spring 2014 -- Lecture 31 1204/14/2014

Magnetization field

H

B

HC-4M

HCGS-GN

HC

H

H