Notes for Lecture

Feb 23, 2002

Notes for Lecture Notes for Lecture #15

Vector potentials in magnetostatics

The vector potential corresponding to a current density distribution J(r) is given by

A(r) =

m₀

4 p

ó
õ

d³r^¢

J(r^¢)

|r - r^¢|

(1)

This expression is useful if the current density J(r) is confined within a finite region of space. Consider the following example corresponding to a rotating charged sphere of radius a, with r₀ denoting the uniform charge density within the sphere and w denoting the angular rotation of the sphere:

J(r^¢) =

ì
í
î

r₀ w× r^¢

for r^¢ £ a

otherwise

(2)

In order to evaluate the vector potential (1) for this problem, we can make use of the expansion:

|r - r^¢|

å
lm

4 p

2 l +1

r_<^l

r_>^l+1

Y_lm(

^
r

)Y_lm^*(

^
r^¢

(3)

Noting that

r^¢ = r^¢

æ
ú
Ö

æ
ç
ç
ç
è

Y_1-1(

^
r^¢

)

^
x

^
y

Ö2

+ Y₁₁(

^
r^¢

)

^
x

^
y

Ö2

+ Y₁₀(

^
r^¢

)

^
z

ö
÷
÷
÷
ø

(4)

we see that the angular integral result takes the simple form:

ó
õ

dW^¢

å
m

Y_lm(

^
r

)Y_lm^*(

^
r^¢

) r^¢ =

r^¢

r d_l1.

(5)

Therefore the vector potential for this system is:

A(r) =

m₀ r₀ w×r

3 r

ó
õ

a

0

dr^¢ r^¢³

r_<

r_>²

(6)

which can be evaluated as:

A(r) =

ì
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ï
î

m₀ r₀ w× r

æ
ç
è

a²

3 r²

ö
÷
ø

for r £ a

m₀ r₀ w× r

3 r³

a⁵

for r ³ a

(7)

As another example, consider the current associated with an electron in a spherical atom. In this case, we assume that the current density is due to an electron in a bound atomic state with quantum numbers |n l m_l ñ, as described by a wavefunction y_{nlm_l}(r), where the azimuthal quantum number m_l is associated with a factor of the form e^{i m_l f}. For such a wavefunction the quantum mechanical current density operator can be evaluated:

J(r) =

-e (^h/_2p)

2 m_e i

(y_{nlm_l}^*Ñy_{nlm_l}-y_{nlm_l} Ñy_{nlm_l}^* ).

(8)

Since the only complex part of this wavefunction is associated with the azimuthal quantum number, this can be written:

J(r) =

-e (^h/_2p)

2 m_e i rsinq

æ
ç
è

y_{nlm_l}^*

¶f

y_{nlm_l}-y_{nlm_l}

¶f

y_{nlm_l}^*

ö
÷
ø

^
f

-e (^h/_2p) m_l

^
f

m_e r sinq

|y_{nlm_l}|².

(9)

where m_e denotes the electron mass and e denotes the magnitude of the electron charge.

For example, consider the |nlm = 211ñ state of a H atom:

J(r^¢) =

-e(^h/_2p)

64 m pa⁵

e^{-r^¢/a}

^
z

×r^¢,

(10)

where a here denotes the Bohr radius. Using arguments similar to those above, we find that

A(r) =

-e(^h/_2p) m₀

^
z

×r

192 mpa⁵ r

ó
õ

¥

0

dr^¢ r^¢³ e^{-r^¢/a}

r_<

r_>²

(11)

This expression can be integrated to give:

A(r) =

-e (^h/_2p) m₀

^
z

×r

8 mpr³

é
ê
ë

1 - e^-r/a

æ
ç
è

1 +

r²

2 a²

r³

8 a³

ö
÷
ø

ù
ú
û

(12)

File translated from T_EX by T_TH, version 2.20.
On 23 Feb 2002, 14:03.