# Attraction due to magnetic field produced by moving charges

When two current carrying conductors are placed next to each other and current flows in same direction, say $$i$$, then the current carrying conductors attracts each other with a force $$Bil$$.

Now suppose two electron streams/ beams and flowing next to each other parallelly , will beams merge/attract or repel each other?

They should merge/attract each other due magnetic force

They should repel due to electrostatic force

And also it's given that we are in a frame such that electron beam is in motion relative to us all the time

So which event is likely to occur and why?

This question is an extension of this post

Do moving charged particles have both magnetic and electric fields?

• I'm so naughty: consider 2 wires with electric current flowing in the same direction... Jan 24, 2020 at 10:18

The short answer is, that the total force between the streams is the same in all inertial frames. In the moving frame where the electrons are at rest, the force is repellant and therefore, the force will be repellant in all inertial frames.

To see this, consider a dense stream of electrons with a line charge density $$\lambda$$ [charge per length] with a parallel speed $$v$$ such that the current is $$I=\lambda v$$. The magnetic field $$B$$ and the electric field $$E$$ a distance $$r$$ away from this infinite line current is

$$B = \frac{\mu_0 I}{2\pi r}$$ $$E = \frac{\lambda}{2\pi\epsilon_0 r}$$

As you state, the attractive magnetic force $$f_B$$ per length between the streams is

$$f_B = B I = \frac{\mu_0 (\lambda v)^2}{2\pi r}$$

At the same time, the repellant electric force $$f_E$$ per length between the streams is

$$f_E = E\lambda = \frac{\lambda^2}{2\pi\epsilon_0 r}$$

The total force per length is therefore (attractive positive, repellant negative)

$$f_\mathrm{total}= f_B-f_E = -\frac{\lambda^2}{2\pi \epsilon_0 r^2}\Big(1-\frac{v^2}{c^2}\Big)$$

where $$c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}$$ is the speed of light. In a relativistic calculation, one should replace $$\lambda\rightarrow \frac{\lambda_0}{\sqrt{1-\frac{v^2}{c^2}}}$$, due to the Lorentz contraction where $$\lambda_0$$ is the proper line charge density measured in the coordinate system where the electrons are at rest. The total force per length $$l$$ between the streams is therefore the same in all inertial frames,

$$f_\mathrm{total} = -\frac{\lambda_0^2}{2\pi \epsilon_0 r^2} =- (f_E)_0/l$$

where $$(f_E)_0/l$$ is the electric force per length between the streams measured in the coordinate system where the electrons are at rest.