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| | & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ | | & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g + 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\ |
| | & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\ | | & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\ |
| − | \left( \frac{q}{k} \right)^2
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| − | & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right)
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| − | \end{alignat}
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| − | </math>
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| − |
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| − | ====Check of Total Magnitude #2: Doesn't work====
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| − | We define:
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| − | ::<math>
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| − | \begin{alignat}{2}
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| − | d^{\prime} & = \sqrt{x^2 + d^2 + z^2} = \| \mathbf{v}_1 \| \\
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| − | ( v_{2y} ) & = ( d \cos \theta_g - z \sin \theta_g ) \\
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| − | ( v_{2y} )^2 & = ( d \cos \theta_g - z \sin \theta_g )^2 \\
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| − | & = d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g
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| − | \end{alignat}
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| − | </math>
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| − |
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| − | And calculate:
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| − | :<math>
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| − | \begin{alignat}{2}
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| − | q^2 & = [ (q_x)^2 + (q_y)^2 + (q_z)^2 ] \\
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| − | \left ( \frac{q}{k} \right )^2 d^{\prime 2}
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| − | & = \begin{alignat}{2} [ & \left( x \cos \phi_g -\sin \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right)^2 \\ & + \left( x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) - d^{\prime} \right)^2 \\ & + \left( d \sin \theta_g + z \cos \theta_g \right)^2 ] \end{alignat} \\
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| − |
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| − | & = \begin{alignat}{2} [
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| − | & \left( x \cos \phi_g -\sin \phi_g ( v_{2y} ) \right)^2 \\
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| − | & + \left( x \sin \phi_g + \cos \phi_g ( v_{2y} ) - d^{\prime} \right)^2 \\
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| − | & + \left( d \sin \theta_g + z \cos \theta_g \right)^2 ] \end{alignat} \\
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| − |
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| − | & = \begin{alignat}{2} [
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| − | & x^2 \cos^2 \phi_g - 2 x \cos \phi_g \sin \phi_g ( v_{2y} ) + \sin^2 \phi_g ( v_{2y} )^2 \\
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| − | & + x^2 \sin^2 \phi_g + x \sin \phi_g \cos \phi_g ( v_{2y} ) - d^{\prime} x \sin \phi_g \\
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| − | & + x \sin \phi_g \cos \phi_g ( v_{2y} ) + \cos^2 \phi_g ( v_{2y} )^2 - d^{\prime} \cos \phi_g ( v_{2y} ) \\
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| − | & - d^{\prime} x \sin \phi_g - d^{\prime} \cos \phi_g ( v_{2y} ) + d^{\prime 2} \\
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| − | & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\
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| − | \end{alignat}
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| − | </math>
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| − | Grouping and rearranging:
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| − | :<math>
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| − | \begin{alignat}{2}
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| − | \left ( \frac{q}{k} \right )^2 d^{\prime 2}
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| − | & = \begin{alignat}{2} [
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| − | & x^2 + ( v_{2y} )^2 \\
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| − | & - 2 d^{\prime} x \sin \phi_g \\
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| − | & - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
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| − | & + d^{\prime 2} \\
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| − | & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\
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| − |
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| − | & = \begin{alignat}{2} [
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| − | & d^{\prime 2} + x^2 + ( d^2 \cos^2 \theta_g - 2dz \cos \theta_g \sin\theta_g + z^2 \sin^2 \theta_g ) \\
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| − | & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) \\
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| − | & + d^2 \sin^2 \theta_g + 2 d z \sin \theta_g \cos \theta_g + z^2 \cos^2 \theta_g ] \end{alignat} \\
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| − |
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| − | & = \begin{alignat}{2} [
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| − | & d^{\prime 2} + x^2 + d^2 + z^2 \\
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| − | & - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} ) ] \end{alignat} \\
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| − |
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| − | & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( v_{2y} )\\
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| − |
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| − | & = 2 d^{\prime 2} - 2 d^{\prime} x \sin \phi_g - 2 d^{\prime} \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \\
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| − | & = 2 d^{\prime} \left( d^{\prime} - x \sin \phi_g - \cos \phi_g ( d \cos \theta_g - z \sin \theta_g ) \right) \\
| |
| | \left( \frac{q}{k} \right)^2 | | \left( \frac{q}{k} \right)^2 |
| | & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right) | | & = 2 \left( 1 - \frac{x \sin \phi_g + \cos \phi_g ( d \cos \theta_g - z \sin \theta_g )}{d^{\prime} } \right) |
| | \end{alignat} | | \end{alignat} |
| | </math> | | </math> |
![{\displaystyle {\begin{alignedat}{2}\left({\frac {q}{k}}\right)^{2}d^{\prime 2}&={\begin{alignedat}{2}[&\left(x\cos \phi _{g}-\sin \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})\right)^{2}\\&+\left(x\sin \phi _{g}+\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})-d^{\prime }\right)^{2}\\&+\left(d\sin \theta _{g}+z\cos \theta _{g}\right)^{2}]\end{alignedat}}\\&={\begin{alignedat}{2}[&x^{2}\cos ^{2}\phi _{g}-x\cos \phi _{g}\sin \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+\sin ^{2}\phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})^{2}\\&+x^{2}\sin ^{2}\phi _{g}+x\sin \phi _{g}\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})-d^{\prime }x\sin \phi _{g}\\&+\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})x\sin \phi _{g}+\cos ^{2}\phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})^{2}-d^{\prime }\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})\\&-d^{\prime }x\sin \phi _{g}-d^{\prime }\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+d^{\prime 2}\\&+d^{2}\sin ^{2}\theta _{g}+2d\sin \theta _{g}z\cos \theta _{g}+z^{2}\cos ^{2}\theta _{g}]\end{alignedat}}\\&={\begin{alignedat}{2}[&x^{2}\cos ^{2}\phi _{g}-x\cos \phi _{g}\sin \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+\sin ^{2}\phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})^{2}\\&+x^{2}\sin ^{2}\phi _{g}+2x\sin \phi _{g}\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})-2d^{\prime }x\sin \phi _{g}\\&+\cos ^{2}\phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})^{2}-2d^{\prime }\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+d^{\prime 2}\\&+d^{2}\sin ^{2}\theta _{g}+2d\sin \theta _{g}z\cos \theta _{g}+z^{2}\cos ^{2}\theta _{g}]\end{alignedat}}\\&={\begin{alignedat}{2}[&x^{2}-x\sin \phi _{g}\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+(d\cos \theta _{g}-z\sin \theta _{g})^{2}\\&+2x\sin \phi _{g}\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})-2d^{\prime }x\sin \phi _{g}\\&-2d^{\prime }\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})+d^{\prime 2}\\&+d^{2}\sin ^{2}\theta _{g}+2dz\sin \theta _{g}\cos \theta _{g}+z^{2}\cos ^{2}\theta _{g}]\end{alignedat}}\\&={\begin{alignedat}{2}[&x^{2}+d^{2}\cos ^{2}\theta _{g}-2dz\cos \theta _{g}\sin \theta _{g}+z^{2}\sin ^{2}\theta _{g}\\&+(-x\sin \phi _{g}\cos \phi _{g}+2x\sin \phi _{g}\cos \phi _{g}-2d^{\prime }\cos \phi _{g})(d\cos \theta _{g}-z\sin \theta _{g})\\&-2d^{\prime }x\sin \phi _{g}\\&+d^{\prime 2}\\&+d^{2}\sin ^{2}\theta _{g}+2dz\sin \theta _{g}\cos \theta _{g}+z^{2}\cos ^{2}\theta _{g}]\end{alignedat}}\\&={\begin{alignedat}{2}[&d^{\prime 2}+x^{2}+d^{2}+z^{2}-2dz\cos \theta _{g}\sin \theta _{g}\\&+(x\sin \phi _{g}\cos \phi _{g}-2d^{\prime }\cos \phi _{g})(d\cos \theta _{g}-z\sin \theta _{g})\\&+2dz\sin \theta _{g}\cos \theta _{g}-2d^{\prime }x\sin \phi _{g}]\end{alignedat}}\\&=2d^{\prime 2}-2d^{\prime }x\sin \phi _{g}+(x\sin \phi _{g}\cos \phi _{g}-2d^{\prime }\cos \phi _{g})(d\cos \theta _{g}-z\sin \theta _{g})\\&=2d^{\prime 2}-2d^{\prime }x\sin \phi _{g}+(x\sin \phi _{g}-2d^{\prime })\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})\\&=?\\&=?\\&=2d^{\prime 2}-2d^{\prime }x\sin \phi _{g}+2d^{\prime }\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})\\&=2d^{\prime }\left(d^{\prime }-x\sin \phi _{g}+\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})\right)\\\left({\frac {q}{k}}\right)^{2}&=2\left(1-{\frac {x\sin \phi _{g}+\cos \phi _{g}(d\cos \theta _{g}-z\sin \theta _{g})}{d^{\prime }}}\right)\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d68c5043400ca1ea8111f1b9af10b554eaa6be9)
