Doubt with Ampere and Biot-Savart

[atomqwerty]

Hello,

I think I'm terribly wrong by supossing Ampère-Maxwell and Biot-Savart are referred to the same concept of magnetic field B. For example, for calculating B near an infinite line, I used both, as I understood them, obtaining different expressions (see image). What is that that I don't get?

Thanks

 

[Muphrid]

You didn't do the Biot-Savart integral right. Let the wire go up the z-axis, and the integral is

$$B(\rho) = \int_{-\infty}^\infty \frac{\mu_0}{4\pi} \frac{I e_z \; dz' \times (\rho e_\rho - z' e_z)}{(\rho^2 + {z'}^2)^{3/2}}$$

You treated the denominator like it doesn't depend on $z'$, but it does. I write it $z'$ to emphasize that it is the integration variable (not the position we want to find the magnetic field at). Consult a table of integrals to easily find the antiderivative.

In general, for some current density $j$, the Biot-Savart Law is

$$B(r) = \int_{V'} \mu_0 j(r') \; dV' \times \frac{r - r'}{4 \pi |r - r'|^3}$$

Actual line currents (not densities) just reduce this integral from a volume to a line. Compare with the electric field from some charge density:

$$E(r) = \int_{V'} \frac{\rho(r')}{\epsilon_0} \; dV' \frac{r - r'}{4 \pi |r - r'|^3}$$

for vectors $r, r'$. You can see these are both really the "same" law. The function $(r-r')/4\pi|r-r'|^3$ has special significance in 3D space. Wiki "Green's functions" if you're interested in learning about it.

 

[atomqwerty]

Thank you very much, that was really helpful, I appreciate it!

Carlos