PX157 - C3b - the biot-savart law applied to a wire

d Q = n A d r^{'} q

d Q {\vec{v}}_{d} = n q A d r^{'} {\vec{v}}_{d} = n q A v_{d} d {\vec{r}}^{'} = I d {\vec{r}}^{'}

d \vec{B} = \frac{μ_{0}}{4 π} d Q \frac{{\vec{v}}_{d} \times (\vec{r} - {\vec{r}}^{'})}{| \vec{r} - {\vec{r}}^{'} |^{3}} = \frac{μ_{0}}{4 π} I \frac{d {\vec{r}}^{'} \times (\vec{r} - {\vec{r}}^{'})}{| \vec{r} - {\vec{r}}^{'} |^{3}}

∴ \vec{B} = \frac{μ_{0}}{4 π} \int I \frac{d {\vec{r}}^{'} \times (\vec{r} - {\vec{r}}^{'})}{| \vec{r} - {\vec{r}}^{'} |^{3}}