Energy Density of a Magnetic Dipole

R

α

function

u [ρ, z, R, α]

Aϕ [ρ, z, R, α]

Bρ [ρ, z, R, α]

Bz [ ρ, z, R, α ]

A circular conductor with the current

I

1

and the radius

R

lies in the

x-y

plane at

z=0

. The vector potential

Aϕ(ρ,z,R,α)

in the

ϕ

direction as a function of

ρ

and

z

has the same symmetry as the current density in cylindrical coordinates

ρ

,

ϕ

,

z

. According to the cylindrical symmetry the observation points in the

x-z

plane can be taken at

ϕ=0

. The source is described by the angle

α

, running from

0

to

2π

. The following computations are made: • the magnetic field

Bρ(ρ,z,R,α)

in the

ρ

direction• the magnetic field

Bz(ρ,z,R,α)

in the

z

direction• the magnetic energy density

u(ρ,z,R,α)

• the integrated magnetic field

Bρ(ρ,z,R)

in the

ρ

direction• the integrated magnetic field

Bz(ρ,z,R)

in the

z

direction• the integrated magnetic energy density

u(ρ,z,R)

• the integrated vector potential

Aϕ(ρ,z,R)

in the

ϕ

directionThe fields at

α=0

can be regarded as a good approximation of the integrated fields. The four field

A ϕ

,

B ρ

,

B z

,

u

are displayed for the four independent variables

ρ

,

z

,

R

,

α

. The observation points are described by

ρ

,

z

and the source by

R

,

α

.