Solve by creating a oython code The off$-$axis $B_2$ from a circular current loop based on Biot$-$Savart Law as derived in Note$\_05$ vector potential is$,$

$|)/(b|=\sqrt[\phantom{2}]{y^2+z^2+R^2-2yRsin{}^{\text{'}}}$

You may want to check the validity of the expression $|)/(b|=\sqrt[\phantom{2}]{y^2+z^2+R^2-2yRsin{}^{\text{'}}}$

In view of the Helmholtz coil arrangement, choose $|)/(b|=\sqrt[\phantom{2}]{y^2+(\frac{R^2}{4})+R^2-2yRsin{}^{\text{'}}}.$

In the $\frac{e}{m}$ experiment, we need a uniform $B_z$ on the $xy-$plane between the two coils. The cyclotron equation relates the radius $y$ of the glowing ring and the calculated $B-$field. The anti$-$derivative of the above integrant does not exist, but can be summed numerically.

$($a$)$ Calculate $f(\frac{y}{R})=[\frac{(1\mathrm{.}25{)}^{\frac{3}{2}}}{2}]xn{t}_{{}^{\text{'}}}=0^2\frac{[1-\frac{y}{R}sin{}^{\text{'}}]d{}^{\text{'}}}{[(\frac{y}{R}{)}^2+(1\mathrm{.}25)-2\frac{y}{R}sin{}^{\text{'}}{]}^{\frac{3}{2}}}$ for $\frac{y}{R}=0\mathrm{.}0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5.$

Use either $($i$)$ There is a scipy.integrate sub$-$package in Python that can calculate $f(\frac{y}{R})$ numerically. Or $($ii$)$ Alternatively, use EXCEL to create as many rows and columns as necessary. Divide $2$ into $35$ parts $)=(5,15,25,$cdots$355.$ Have Excel calculate the numerator, the denominator of the integrant and then SUM the cells in that column. Repeat the above procedure for each value of $\frac{y}{R}.$

$($b$)$ Plot $f(\frac{y}{R})$ verus $\frac{y}{R}$ over the interval $(-0\mathrm{.}5,+0\mathrm{.}5)$ and comment on the uniformity of $B_z$ on the $xy-$plane. The function is even, i$.$e$.f(\frac{-y}{R})=f(\frac{y}{R}).$ Replace $f(\frac{-y}{R})$ with $f(\frac{+y}{R})$ in the plot.