§1-1)The case only current

§1.THE MAGNETIC FIELD INCLUDING MAGNETIZED MATERIAL

1)The case only current existing

Vector potential by current

Fig.1 showing when current i(x',y',z') existing in the small volume element dV' at the volume region V, magnetic flux density is described using vector potential $\overset{⃑}{A}$

$\overset{⃑}{A} = \frac{μ_{0}}{4 π} ∭_{V}^{} \frac{\overset{⃑}{ｉ}}{r} d V'$ Eq.(1-1)

$\overset{⃑}{B} = \overset{⃑}{\nabla} \times \overset{⃑}{A}$ Eq.(1-2)

From Fig.1 r is described as

$r = \sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}$ Eq.(1-3)

We can describe Eq.(1-2), using Eq.(1-1)

$\overset{⃑}{B} = \overset{⃑}{\nabla} \times \frac{μ_{0}}{4 π} ∭_{V}^{} \frac{\overset{⃑}{ｉ}}{r} d V'$

$= \frac{μ_{0}}{4 π} ∭_{V}^{} \{(\frac{\partial}{\partial y} \frac{i_{z}}{r} - \frac{\partial}{\partial z} \frac{i_{y}}{r}) \overset{⃑}{e_{x}} + (\frac{\partial}{\partial z} \frac{i_{x}}{r} - \frac{\partial}{\partial x} \frac{i_{z}}{r}) \overset{⃑}{e_{y}} + (\frac{\partial}{\partial x} \frac{i_{y}}{r} - \frac{\partial}{\partial y} \frac{i_{x}}{r}) \overset{⃑}{e_{z}}\} d V'$

$= ∭_{V}^{} [\{i_{z} \frac{\partial}{\partial y} (\frac{1}{r}) - i_{y} \frac{\partial}{\partial z} (\frac{1}{r})\} \overset{⃑}{e_{x}} + \{i_{x} \frac{\partial}{\partial z} (\frac{1}{r}) - i_{z} \frac{\partial}{\partial x} (\frac{1}{r})\} \overset{⃑}{e_{y}} + \{i_{y} \frac{\partial}{\partial x} (\frac{1}{r}) - i_{x} \frac{\partial}{\partial y} (\frac{1}{r})\} \overset{⃑}{e_{z}}] d V'$

$= - \frac{μ_{0}}{4 π} ∭_{V}^{} \overset{⃑}{i} \times \overset{⃑}{\nabla} \frac{１}{r}$ Eq.(1-4)

Using Eq.(1-3) above equation can described as

$\overset{⃑}{\nabla} \frac{１}{r} = \frac{\partial}{\partial x} \frac{1}{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}} \overset{⃑}{e_{x}} + \frac{\partial}{\partial y} \frac{1}{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}} \overset{⃑}{e_{y}} + \frac{\partial}{\partial z} \frac{1}{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}} \overset{⃑}{e_{z}}$

$= \frac{- (x - x')}{{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}}^{3}} \overset{⃑}{e_{x}} + \frac{- (y - y')}{{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}}^{3}} \overset{⃑}{e_{y}} + \frac{- (z - z')}{{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}}^{3}} \overset{⃑}{e_{z}}$

$= - \overset{⃑}{r} \frac{1}{r^{3}}$ Eq.(1-5)

So finally Eq.(1-4) become expressed as below.

$\overset{⃑}{B} = \frac{μ_{0}}{4 π} ∭_{V}^{} \overset{⃑}{ｉ} \times \overset{⃑}{r} \frac{1}{r^{3}} d V'$ 式(1-6)

Publication Date：16May2015 　　　　updated：27Jan2018
Author：Nobuo Kojima
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