Wouldn't there be a magnetic force?

[agonydrum]

Our professor proposed a scenario during lecture that went like this, There is a wire of infinite length that has a current. At a short distance from the wire is a single charge which is stationary relative to the wire.

What i don't understand is this, since the charge has a velocity relative to the moving electrons which make up the current in the wire, why is there no magnetic force?

 

[Philip Wood]

There will be a magnetic field around the wire. But a stationary charge near the wire won't experience a magnetic force. That's because stationary charges don't experience magnetic forces. In fact the magnetic field strength (strictly, magnetic flux density), $\vec B$ at a point may be defined by the vector equation
$$\vec{F} = q \vec{v} \times \vec{B}$$
in which $\vec F$ is the force on a charge q moving with velocity $\vec v$.

So if the testing charge, q, is at rest in a given frame of reference it can't experience a magnetic force in that frame.

 

[agonydrum]

Thank you for answering I think you may have misunderstood what i was asking, I understand all that, what I am saying is that relative to the moving charges in the wire the charge outside the wire has a velocity. So why wouldn't it undergo a force

 

[Philip Wood]

It's not the relative velocity which matters in this case. Magnetic forces experienced by a 'test' charge are defined according to the velocity of the charge in a given reference frame. If the charge is stationary in that frame it can't, by definition, experience a magnetic force in that frame.

 

[TysonM8]

To answer this question properly, forget about the magnetic force for a moment. If the velocity of a charge outside the wire is stationary relative to the positive metal ions, the positive and negative charge densities in the wire are equal. If however, the charge is moving at some velocity v, let's say for simplicity the same velocity as the electrons in the wire, we can transform into the frame of reference of the moving charge. In this reference frame, the velocity of the electrons in the wire would be zero, with the positive metal ions traveling at -v. As a result of length contraction, the positive charge density becomes larger than the negative charge density, thus producing an electric force on the charge outside the wire. The magnetic force is just the electric force with relativistic effects.

 

[dauto]

Yes the charge is moving relative to the electrons in the wire. But in the reference frame of the electrons in the wire the electrons are not moving and produce no magnetic field. The Magnitude of the electric and magnetic fields and the forces produced by them are not invariants - that is they will be different depending on the reference frame chosen to analyze the problem.

 

[SredniVashtar]

The force between two charges in motion depends on the velocities of both charges (in the reference frame considered).
$$ \vec F_{m_{12}} = \frac{\mu_0}{4 \pi} \frac{q_1 q_2}{{r_{12}}^2} {\vec v}_1 \times ({\vec v}_2 \times {\hat r_{12}}) $$
Hence, unless both charges are moving in the reference frame you are considering, you won't get a magnetic force (actually there is another geometric condition that will make the double cross product zero, but let's try not to be nitpicking).
Please note that in general
$${\vec F}_{m_{12}} ≠ {\vec F}_{m_{21}}$$
Also note that you can isolate the contribute due to one charge and call it "the magnetic field generated by that charge". Then you'll have the force on the other charge is expressed as a Lorentz force.

(Source: Ohanian, "Physics" 2nd edition expanded)

 

[mikeph]

If you want to confuse your brain even more, try working it out in a frame in which the charge and wire electrons have equal and opposite velocities. In this frame we have a nonzero magnetic field due to each charge, and nonzero magnetic forces acting on each as well!

 

[Drakkith]

Per wiki: http://en.wikipedia.org/wiki/Relativistic_electromagnetism#The_origin_of_magnetic_forces

One may think that the picture, presented here, is artificial because electrons, which accelerated in fact, must condense in the lab frame, making the wire charged. Naturally, however, all electrons feel the same accelerating force and, therefore, identically to the Bell's spaceships, the distance between them does not change in the lab frame (i.e. expands in their proper moving frame). Rigid bodies, like trains, don't expand however in their proper frame, and, therefore, really contract, when observed from the stationary frame.

I admit I don't really understand this very well, I merely thought it would help.

 

[agonydrum]

Thank you all for your replies especially TysonM8 that's the exact answer i was looking for