# Simplify my integrand to make the function Integrate[] work?

Posted 17 days ago
215 Views
|
2 Replies
|
4 Total Likes
|
 To calculate a physical field, I need to integrate a complicated expression, with Sin and Cos and some 3/2 power. But when I run the Integrate function, it seems running forever. So what could I do to make it possible to get the Integrate result?My code is simple: magneticField = Subscript[\[Mu], 0] * mI * r * (Cross[{0, 0, 1}, {x - r*Cos[\[Theta]], y - r*Sin[\[Theta]], 0}])/EuclideanDistance[{x, y}, {r*Cos[\[Theta]], r*Sin[\[Theta]]}]^3 // ComplexExpand // FullSimplify Integrate[magneticField, {\[Theta], 0, 2*Pi}, GenerateConditions -> False, Assumptions -> (\[Theta] | x | y) \[Element] Reals] 
2 Replies
Sort By:
Posted 17 days ago
 I notice one issue: I think the setup isn't quite correct. Symmetry suggests that the B-field in the x-y plane should only be in the z-direction. Given the Biot-Savart law, I get as my integrand (\[Mu]0 mI)/(4 \[Pi]) Cross[{-a Sin[\[Theta]], a Cos[\[Theta]], 0}, {x - a Cos[\[Theta]], y - a Sin[\[Theta]], 0}]/ Norm[{x - a Cos[\[Theta]], y - a Sin[\[Theta]], 0}]^3 // Simplify {0, 0, (a mI \[Mu]0 (a - x Cos[\[Theta]] - y Sin[\[Theta]]))/( 4 \[Pi] (Abs[x - a Cos[\[Theta]]]^2 + Abs[y - a Sin[\[Theta]]]^2)^( 3/2))} Now, because of the symmetry of our situation, I assume without loss of generality that y=0. Now I do an indefinite integral (because for whatever reason the definite integral still seems to hang after this assumption), and I have temp = Integrate[(a mI \[Mu]0 (a - x Cos[\[Theta]]))/( 4 \[Pi] (a^2 + x^2 - 2 a x Cos[\[Theta]])^(3/2)), \[Theta]]; temp2 = (temp /. \[Theta] -> 2 \[Pi]) - (temp /. \[Theta] -> 0) This returns an answer in terms of EllipticE and EllipticK. As a sanity check, I verify that the field at the origin is as expected, In[5]:= temp2 /. x -> 0 Out[5]= (mI \[Mu]0)/(2 Sqrt[a^2]) and that the field displays the expected behavior as x goes to a, namely that it should be positive, diverge at the coil, and then go negative.