# Biot-Savart law from relativity

Few days ago I came by a derivation of STR from Bio-Savart law. Since then I have been trying to derive Bio-Savart law from STR. The derivation mentioned previously used two parallel current-carrying wires of unit length and used the change of their current density to achieve Lorentz contraction (for the entire derivation, go to http://www.andrijar.com/sr/sr.htm). Using the same idea, I have been able to derive ampere force law(F=2kII'/r) from STR. So,

Is it possible to derive Biot-Savart law from ampere force law? If yes, how?

(N.B. Please don't use Maxwell's equations for I have not yet learnt them)

Consider a positive point charge moving in positive $z$ direction at a constant velocity, at that instant when it's at the origin. The magnetic-field lines generated in plane $z=a$ are closed loops, further we observe circular symmetry in the problem about $z$-axis, henceforth the loops are circles.

A image to develop a idea of how we would be going to proceed: Let's apply Ampere's displacement law to calculate magnetic field. Intuitively we can think, as the particle moves towards the plane $z=a$, the electric flux passing through a surface bounded by any circle centered on the z-asix in that plane will increase.

So, starting with Ampere's Law we have:

$$\oint \vec{B}.d\vec{I} = \mu_o \epsilon_o \frac{d\phi}{dt}$$

We integrate the left hand-side around a circle of radius $b$ on $z$-axis in plane $z=a$, and this is our Amperian Loop. so, $$\oint \vec{B}.d\vec{I} = 2 \pi bB$$

To calculate electric flux $\phi$ enclosed by the circle of radius $b$, we select a spherical surface, of surface area $A$, which is bounded by the circle, is symmetric about z-axis, and has a radius $r$. So it follows from the understanding, we have: $$\phi = \int \vec{E}.d\vec{A} = EA$$, where $$E= \frac{q}{4 \pi \epsilon_o r^{2}}$$

The area $A$ can be obtained using spherical, polar coordinates: $$A= r^{2} \int_0^2\pi d\phi \int_0^\theta \sin x dx = 2 \pi r^2(1-\cos \theta)$$ $$\cos \theta= \frac{z}{\sqrt{z^2+x^2}}$$

Using/combining the above three equations we can have: $$\phi = \frac{q}{2 \epsilon_o}(1-\cos \theta)$$

Differentiating with time gives, $$\frac{d\phi}{dt} = -\frac{q}{2 \epsilon_o} \frac{d \cos \theta}{dt}$$ where, $$\frac{d \cos \theta}{dt} = \frac{d \cos\theta}{dz} \frac{dz}{dt}$$

In the above equation, $z$ is the distance between the particle, which is moving in the positive $z$ direction, and the Amperian Loop, which is fixed in $z=a$ plane. Since $z$ is decreasing with time, $$\frac{dz}{dt} = -v$$ where v is the speed of the particle. Now,

$$\frac{d \cos \theta}{dz}=\frac{y^2}{r^3}$$, where $r=\sqrt{z^2+y^2}$

Combining the first two and last three equations, we arrive at our awaited result as, $$B= \frac{\mu_o}{4\pi} \frac{qv \sin \theta}{r^2}$$, which can be vectorially rewritten as, $$\vec{B} = \frac{\mu_o}{4\pi} \frac{q \vec{v} \times \vec{r}}{r^3}$$

Hence, we are done! :)

• But I asked for the derivation from AMPERE FORCE LAW, not just AMPERE'S LAW. The former states that the force between 2 parallel wires per unit length will be F=2kII'/r where k is the is the magnetic force constant, r is the separation of the wires, and I, I' are the direct currents carried by the wires (refer to link) Mar 14, 2015 at 1:41
• This is a nice derivation so +1 but what does this have to do with SR? Mar 18, 2015 at 14:41
• @FenderLesPaul the OP has already derived Ampere's law from SR and I derived Biot - Savart Law from Ampere's law and hence we can imply deriving Biot - Savart from SR. Mar 18, 2015 at 14:43
• The expression for the electric flux is wrong. The electric field of a moving charge is not the same as the electric field of a stationary charge.
– user4552
Jan 13, 2019 at 20:14