Homework  Assignment I,II , Phy712.   Due Feb 26

(1)  (5 pts)   Consider a current of 10 Amperes in a ring of radius 10 cm.  Find the magnitude
of the  B field at a point that makes an angle of 45 degrees with respect to the axis of symmetry of the ring (e.g. see figure 5.5 in the text) and is a distance of 1 meter from the origin. Express your answer
in terms of Gauss.

|B| =   (Sqrt[(Subscript[B, r])|^2+ |(Subscript[B, θ])|^2]) 

(* use SI units *)
parameters ={a->0.1,μ0->4 π 10^(-7),Ic->10 }
{a->0.1`,μ0->(π/2500000),Ic->10}

(* expression 5.37 in Jackson *)

k=(Sqrt[4 a r Sin[θ]/(a^2+r^2+2 a r Sin[θ])]);
 μ0/(4 π) 4 Ic a/Sqrt[a^2+r^2+2 a r Sin[θ]] (( 2-k^2)EllipticK[k^2]-2 EllipticE[k^2])/k^2
Aϕ[r_,θ_]=%
(1/(4 π r))Ic μ0 Csc[θ] (Sqrt[(a^2)+(r^2)+2 a r Sin[θ]]) (-2 EllipticE[((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))]+EllipticK[((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))] (2-((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))))
(1/(4 π r))Ic μ0 Csc[θ] (Sqrt[(a^2)+(r^2)+2 a r Sin[θ]]) (-2 EllipticE[((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))]+EllipticK[((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))] (2-((4 a r Sin[θ])/((a^2)+(r^2)+2 a r Sin[θ]))))
(* approximate expression 5.39 *)

(*  Aϕ[r_,θ_]=μ0 Ic a^2 r Sin[θ]/4/(a^2+r^2)^(3/2)(1+15 a^2 r^2 Sin[θ]^2/8 /(a^2+r^2)^2) *)

Br[r_,θ_]=1/r/Sin[θ]D[Sin[θ]Aϕ[r,θ],θ];
Bθ[r_,θ_]=-1/r D[r Aϕ[r,θ],r];
Br[1 ,Pi/4] /. parameters
Bθ[1,Pi/4] /. parameters

(Sqrt[%%^2+%^2 ]) 10^4 Gauss
4.459099494359574`*^-8
2.1837634716366335`*^-8
0.0004965117440771395` Gauss

(2)  The permeability of iron is 0.25 H/m. Consider a shell of iron (see Figure 5.13) of inner radius 0.5 meters and outer radius of 0.75 meters. Let (Subscript[B, 0]) in that figure represent the Earth's magnetic field near the surface, which you can take to have the value 0.25 Gauss.  

(a)  What is the maximum value of the B field in the interior of the shielded region. (5 pts)

According to 5.118 in the interior region

Φ(r,θ) = δ1 r (Subscript[P, 1])(Cosθ) = (Subscript[δ, 1])z
Φ[z_]=δ1 z  
z δ1
H[z_]=-D[Φ[z],z]

(* so H field, and thus B field is a constant in the z direction *)
-δ1
δ1=-9 μ/((2μ+1)(μ+2)-2 a^3/b^3 (μ-1)^2)H0
-((9 H0 μ)/(-((2 (a^3) (((-1+μ))^2))/(b^3))+(2+μ) (1+2 μ)))
B[z_]=H[z]/μ0  
((9 H0 μ)/((-((2 (a^3) (((-1+μ))^2))/(b^3))+(2+μ) (1+2 μ)) μ0))
B[z] /. H0->μ0 0.25 Gauss
% /. {μ->0.25/(1.256 10^(-6))}
% /. {a->0.5,b->0.75}
((2.25` Gauss μ)/(-((2 (a^3) (((-1+μ))^2))/(b^3))+(2+μ) (1+2 μ)))
((447850.31847133755` Gauss)/(7.923848964661034`*^10-((7.923669824533644`*^10 (a^3))/(b^3))))
8.031612142248847`*^-6 Gauss

(b) Make a 3D plot for the magnitude of the B field in the interior region. (5 pts)

Hyperlink["http://demonstrations.wolfram.com/MagneticShieldingEffectOfASphericalShell/"]

(c) Repeat part (a) for the case where the shell is, instead of a spherical shell, that of an infinite cylinder. (15 pts)

See John Waite's solution in class