Return to Question

I've currently trying to learn electromagnetism in depth and I tried to solve the Biot-Savart law, for a magnetic field generated by a current. $$ \vec{B} = \frac{\mu_0}{4 \pi} \int{\frac{I \; \hat{r} \times \mathrm{d} \vec{\ell}}{r^2}} $$ When I looked up for information on how to solve the equation, there are always simplifications, like: $$ B = \frac{\mu_0}{4 \pi} \int{\frac{I \; \mathrm{d} \ell \sin{\theta}}{r^2}} \rightarrow B = \frac{\mu_0 I}{2 \pi R} $$

Edit: But it is possible to solve this integral in a vectorial form and without the need of simplification for the magnetic field in a straight wire carrying a current? Would you be able to arrive to a solution similar to the one of the electric field generated by an electric current in a straight wire $\frac{\lambda}{2 \pi \varepsilon_0 r} \hat{r}$ (which you arrive using Gauss's law)?

I've currently trying to learn electromagnetism in depth and I tried to solve the Biot-Savart law, for a magnetic field generated by a current. $$ \vec{B} = \frac{\mu_0}{4 \pi} \int{\frac{I \; \hat{r} \times \mathrm{d} \vec{\ell}}{r^2}} $$ When I looked up for information on how to solve the equation, there are always simplifications, like: $$ B = \frac{\mu_0}{4 \pi} \int{\frac{I \; \mathrm{d} \ell \sin{\theta}}{r^2}} \rightarrow B = \frac{\mu_0 I}{2 \pi R} $$

Edit: It is possible to solve this integral in a vectorial form and without the need of simplification for the magnetic field in a straight wire carrying a current? Would you be able to arrive to a solution similar to the one of the electric field generated by an electric current in a straight wire $\frac{\lambda}{2 \pi \varepsilon_0 r} \hat{r}$ (which you arrive using Gauss's law)?

I've currently trying to learn electromagnetism in depth and I tried to solve the Biot-Savart law, for a magnetic field generated by a current. $$ \vec{B} = \frac{\mu_0}{4 \pi} \int{\frac{I \; \hat{r} \times \mathrm{d} \vec{\ell}}{r^2}} $$ When I looked up for information on how to solve the equation, there are always simplifications, like: $$ B = \frac{\mu_0}{4 \pi} \int{\frac{I \; \mathrm{d} \ell \sin{\theta}}{r^2}} \rightarrow B = \frac{\mu_0 I}{2 \pi R} $$

Edit: But it is possible to solve this integral in a vectorial form and without the need of simplification? Would you be able to arrive to a solution similar to the one of the electric field generated by a electric current in a wire $\frac{\lambda}{2 \pi \varepsilon_0 r} \hat{r}$ (which you arrive using Gauss's law)?

I've currently trying to learn electromagnetism in depth and I tried to solve the Biot-Savart law, for a magnetic field generated by a current. $$ \vec{B} = \frac{\mu_0}{4 \pi} \int{\frac{I \; \hat{r} \times \mathrm{d} \vec{\ell}}{r^2}} $$ When I looked up for information on how to solve the equation, there are always simplifications, like: $$ B = \frac{\mu_0}{4 \pi} \int{\frac{I \; \mathrm{d} \ell \sin{\theta}}{r^2}} \rightarrow B = \frac{\mu_0 I}{2 \pi R} $$

Edit: But it is possible to solve this integral in a vectorial form and without the need of simplification for the magnetic field in a straight wire carrying a current? Would you be able to arrive to a solution similar to the one of the electric field generated by an electric current in a straight wire $\frac{\lambda}{2 \pi \varepsilon_0 r} \hat{r}$ (which you arrive using Gauss's law)?