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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)?

Qmechanic
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Álvaro Rodrigo
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    What is the example you want to solve it for? – Brendan Darrer Nov 18 '21 at 23:13
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    For deriving the magnetic field expression in a vectorial form, in a similar way of the electric field $E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \hat{r}$. – Álvaro Rodrigo Nov 18 '21 at 23:16
  • Not sure what you mean? I think you need to apply it to a certain situation, such as B-field on circular coil, B-field on a solenoid or due to a straight current carrying wire etc. – Brendan Darrer Nov 18 '21 at 23:27
  • Ah sorry, let me clarify the post. – Álvaro Rodrigo Nov 18 '21 at 23:30
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    Related : (1) A generalization of the Biot-Savart law. (2) Magnetic field due to a single moving charge. See equation (02) in the first link. – Frobenius Nov 18 '21 at 23:57
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    I'm still a bit confused by the question. As far as I'm concerned, the Biot-Savart law is the solution - if you provide the current density $\mathbf J$ everywhere in space, then you just integrate it as per Biot-Savart to obtain the magnetic field (assuming that the conditions for Biot-Savart to be applicable are satisfied). Of course, for all but the simplest current configurations the integral will not have a nice closed form, but it's unclear what else you are looking for beyond the explicit formula for $\mathbf B$ which you've provided. – J. Murray Nov 19 '21 at 01:36
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    $\uparrow$ The form of Biot-Savart I refer to is $\mathbf B(\mathbf r) = \frac{\mu_0}{4\pi} \int \mathrm d^3 r' \mathbf J(\mathbf r') \times (\mathbf r-\mathbf r')/ |\mathbf r-\mathbf r'|^3$ – J. Murray Nov 19 '21 at 01:38
  • https://physics.stackexchange.com/questions/545027/is-biot-savart-law-valid-to-derive-the-magnetic-field-for-a-point-charge-moving/679040#679040 another derivation starting from the volume form of the magnetic vector potential using the dirac definition of a point charge instead of a dl element – jensen paull Dec 19 '21 at 13:17

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What about:

$$d \vec{B} = \frac{\mu_0 I d \vec{\ell} \times \vec{r}}{4 \pi r^3}$$

or

$$d \vec{B} = \frac{\mu_0 I d \vec{\ell} \times \hat{r}}{4 \pi r^2}$$

$B$ = magnetic flux density; $I$ = current in wire; $\ell$ = length of wire; $r$ = distance of a point in the field to a segment of the wire; $\mu_0$ = magnetic permeability of free space.

Brendan Darrer
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