Dipole

:This article is about the electromagnetic phenomenon. From the point of view of the mathematics of distributions, a dipole can be taken to be the directional derivative of a Dirac delta function. A dipole is also a type of radio antenna.

Field of a point dipole

The strength, B, of a dipole magnetic field is given by:

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:mathbf{B}(mathbf{r}, lambda) = (mu_0/4pi)(mathbf{M}/mathbf{r}^3)(1+3sin^2lambda)^{1over2}

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where:

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:B is the strength of the field, measured in teslas

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:r is the distance from the center, measured in metres

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:λ is the magnetic latitude (90°-θ) where θ = magnetic colatitude, measured in radians or degrees from the dipole axis (magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis)

Related Topics:
Radian - Degree

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:M is the dipole moment, measured in ampere square-metres

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:μ0 is the permeability of free space, measured in henrys per metre.

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That's the magnitude of the field; the field itself is a vector quantity:

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:mathbf{B}(mathbf{r}) = (mu_0/4pi r^3)left(3(mathbf{m}cdothat{mathbf{r}})hat{mathbf{r}}-mathbf{m} ight)

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where

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:B is the field

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:r is the vector from the position of the dipole to the position where the field is being measured

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:r is the absolute value of r: the distance from the dipole

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:hat{mathbf{r}} = mathbf{r}/r is the unit vector parallel to r

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:m is the (vector) dipole moment

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:μ0 is the permeability of free space

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This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.

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The vector potential A is

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:mathbf{A}(mathbf{r}) = (mu_0/4pi r^2)(mathbf{m} imeshat{mathbf{r}})

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with the same definitions as above.

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The electric field of an electric point dipole is

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:mathbf{E}(mathbf{r}) = (1/4piepsilon_0 r^3)left(3(mathbf{p}cdothat{mathbf{r}})hat{mathbf{r}}-mathbf{p} ight)

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where

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:E is the field

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:r, r, hat{mathbf{r}} are as above

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:p is the (vector) dipole moment

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:ε0 is the permittivity of free space.

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Notice that this is formally identical to the magnetic field of a point magnetic dipole; only a few names have changed.

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The (scalar) potential is

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:V(mathbf{r}) = (1/4piepsilon_0 r^2)(mathbf{p}cdothat{mathbf{r}}).

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