PX157 - C3c - magnetic field of a straight wire

\begin{aligned} {\vec{r}}^{'} & = z^{'} \hat{z} \\ d {\vec{r}}^{'} & = z^{'} d \hat{z} \\ \vec{r} & = R \hat{R} + z \hat{z} \\ d {\vec{r}}^{'} \times & (\vec{r} - {\vec{r}}^{'}) ∥ \hat{ϕ} \end{aligned}

\begin{aligned} \vec{r} - {\vec{r}}^{'} & = R \hat{R} + (z - z^{'}) \hat{z} \\ d {\vec{r}}^{'} \times (\vec{r} - {\vec{r}}^{'}) & = d z^{'} \hat{z} \times R \hat{R} + d z^{'} \hat{z} \times (z - z^{'}) \hat{z} \\ = R d z^{'} \hat{ϕ} \end{aligned}

\begin{aligned} d \vec{B} (\vec{r}) & = \frac{μ_{0}}{4 π} \frac{I R d z^{'}}{| \vec{r} - {\vec{r}}^{'} |^{3}} \hat{ϕ} \\ = \frac{μ_{0}}{4 π} \frac{I R d z^{'}}{| R^{2} + (z - z^{'})^{2} |^{\frac{3}{2}}} \hat{ϕ} \end{aligned}

\begin{aligned} \vec{B} (\vec{r}) & = \frac{μ_{0} I}{4 π} \hat{ϕ} \int_{- \infty}^{\infty} \frac{R}{(R^{2} + (z - z^{'})^{2})^{\frac{3}{2}}} d z^{'} \\ = \frac{μ_{0} I}{4 π R} \hat{ϕ} \frac{- (z - z^{'})}{\sqrt{R^{2} + (z - z^{'})^{2}}} |_{- \infty}^{+ \infty} \end{aligned}

lim_{z^{'} \to \pm \infty} \frac{(z - z^{'})}{\sqrt{R^{2} + (z - z^{'})^{2}}} = lim_{z^{'} \to \pm \infty} \frac{z^{'}}{| z^{'} |} = \pm 1

\vec{B} (\vec{r}) = \frac{μ_{0} I}{2 π R} \hat{ϕ}