www.IrPDF.com 80
` 4 Ampere, Faraday, and Maxwell
By superposition the field for two filaments is simply the vectorial summation of the field of the individual filaments displaced to x = d and x = −d Bx =
−µI y µI y + 2 2 2π ((x + d) + y ) 2π ((x − d)2 + y 2 )
By =
−µI (x + d) µI (x − d) − 2 2 2π ((x + d) + y ) 2π ((x − d)2 + y 2 )
(4.29)
This field is plotted in Fig. 4.2c using the following commands in Mathematica In[1]:= In[2]:= In[3]:= In[4]:= In[5]:= In[6]:= In[7]:= In[8]:= In[9]:=
Bx = -y/((x+1)^2+y^2) + y/((x-1)^2+y^2) By = (x+1)/((x+1)^2+y^2) - (x-1)/((x-1)^2+y^2) B = {Bx,By}; <<Graphics'PlotField' bfield = PlotVectorField[B,{x,-5,5},{y,-4,4}]; cur1 = Point[{-1,0}]; cur2 = Point[{1,0}]; cur = {cur1,cur2}; Show[bfield,Graphics[{PointSize[.02],cur}]] }
Magnetic vector potential Earlier we showed that we can also define a vector A such that B = ∇ × A. Since ∇ × B = µJ ∇ × ∇ × A = µJ
(4.30)
We can apply the vector identity for ∇ × ∇ × A ∇ × ∇ × A = ∇ (∇ · A) − ∇ 2 A
(4.31)
Given that the divergence of A is arbitrary, let us choose the most convenient value. In magnetostatics that is ∇ · A = 0. Then we have ∇ 2 A = −µJ
(4.32)
We have met this equation before!
Equations for potential The vector Laplacian can be written as three scalar Laplacian equations (using rectangular coordinates). For instance, the x-component is given by ∇ 2 A x = −µJx
(4.33)