Electric and magnetic fields are two aspects of the same entity

Question

I'm struggling with this doubt. Here it is.... There is a infinitely long current carrying wire. A charged particle is projected with a velocity v parallel to direction of flow of current... then it get deviated due to magnetic force qvB Where q is charge of the particle v is velocity of particle B is field due to wire at the position of particle

But when the same situation is observed in the frame which is moving with same velocity v as that of charged particle there is no magnetic force as velocity of charged particle with respect to that frame is zero....but the particle is deviated from its path... How is this possible?? I read in a book that " what was a pure magnetic field in one frame turns out to be a combination of electric and magnetic field in other frame " I really can't understand this...If it converts into two forces then what is value of each force And I didn't understand why it turns out to be a combination of two forces...it is given in that book that 'electric and magnetic fields are two aspects of Same entity '.... but I didn't clearly why it is so... Please help me with this

Answer 1

If there's no electric field from the wire, then it's electrically neutral. That means that in addition to the charged particles carrying the current, there must also be oppositely charged particles at rest. With respect to your second inertial frame, those particles will be moving and will generate a magnetic field, so the total magnetic field will be nonzero. There's actually no reference frame in which the total magnetic field is zero.

Also, differential Lorentz contraction/expansion of the positive and negative charges mean that the wire will not be electrically neutral in most other reference frames, so there will be an electric field as well.

Formulas for the values of the electric and magnetic fields in any inertial frame as a function of the fields in another inertial frame can be found in any introduction to electromagnetism, or on Wikipedia.

Answer 2

The electric density of current is a 4-vector: $\mathbf j = (\rho,j_x,j_y,j_z)$.

That means, instead of thinking in density of charges and currents as separate entities, they have to be merged together in relativity.

If the direction of the wire is $z$, and the conductor is neutral, we have in the stationary frame:

$\mathbf j = (0,0,0,j_z)$

In order to find $\mathbf j$ at the moving frame it is necessary to apply the Lorentz transformation. If we take c=1 to simplify the formulas:

$\rho' = \gamma(\rho - vj_z) = -\gamma vj_z$
$j_z' = \gamma(j_z - v\rho) = \gamma j_z$

So, the moving frame has not only an electric current flowing in $z$-direction, but also a non zero charge. That zero charge generates an electric field that acts on the charge q.