Changes

This gives us the equation<math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{r'^2}\int_0^a rdr \frac{d}{k\sin{\theta}'dr} J_1(kr\sin{\theta}')</math><br><br>

This gives us the equation<math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{r'^2}\int_0^a rdr \frac{d}{k\sin{\theta}'dr} J_1(kr\sin{\theta}')</math><br><br>

We invoke the principle that the integral of a derivative is the function evaluated at the end points to give us the equation<br><br><math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}\left[aJ_1(ka\sin{\theta}')-0J_1(0k\sin{\theta}')\right]=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')</math><br><br> and <math>A(r')=\frac{z'\dot{A_0}}{2\pi c}e^{-ikct'} 2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')=\frac{z'\dot{A_0}a}{c}\frac{e^{ikr'-ikct'}}{k\sin{\theta}'r'^2}J_1(ka\sin{\theta}')</math><br><br>

We invoke the principle that the integral of a derivative is the function evaluated at the end points to give us the equation<br><br><math>\mathcal{J}(r')=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}\left[aJ_1(ka\sin{\theta}')-0J_1(0k\sin{\theta}')\right]=2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')</math><br><br> and <math>A(r')=\frac{z'\dot{A_0}}{2\pi c}e^{-ikct'} 2\pi\frac{e^{ikr'}}{k\sin{\theta}'r'^2}aJ_1(ka\sin{\theta}')=\frac{z'\dot{A_0}a}{c}\frac{e^{ikr'-ikct'}}{k\sin{\theta}'r'^2}J_1(ka\sin{\theta}')</math><br><br>

To find the angle to the first diffraction minimum, we must find the first zero of this amplitude function.  This will occur when <math>J_1(ka\sin{\theta}')=0 \quad</math><br><br>

To find the angle to the diffraction minimum, we must find the zeroes of this amplitude function.  This will occur when <math>J_1(ka\sin{\theta}')=0 \quad</math><br><br>

[[Image:Bessels_J0.svg|thumb|300px|right|Plot of Bessel function of the first kind, J_&alpha;(x), for integer orders &alpha;=0,1,2.]]

[[Image:Bessels_J0.svg|thumb|300px|right|Plot of Bessel function of the first kind, J_&alpha;(x), for integer orders &alpha;=0,1,2.]]