# Drift speed and current in two different inertial frames

We have a long, cylindrical wire carrying a constant current $I$ in an inertial frame. At a distance of $R$ from the center of the wire, the magnitude of magnetic field is $μI/2 \pi R$. What is the magnitude of magnetic field in another inertial frame moving at a certain speed opposite to the direction of electric current?

Since the observed drift speed of the electrons is now greater, does this mean that the current as observed from this frame is greater?

If so, what about the magnetic field due to this current? Does this change too?

In general how do currents vary between different frames of reference?

• Hi user31150, and welcome to Physics Stack Exchange! What prompted you to ask this question? Perhaps if you explain your motivation some more, it would help us approach the question better. – David Z Oct 16 '13 at 20:57
• Hello, see assume that electrons are moving to the right, and you( you are +ve ly chaged) started moving to the left, now you see that electrons are speeding up, but since protons in the wire were initially a rest w.r.t you now they seem to appear moving in the opposite direction, I think due to length contraction both the charges will balance in such a way that electric current remains same, and so does magnetic. Field, I am not sure though.. – Sarthak Sharma Mar 1 '17 at 12:08

The magnetic field arises because of relativity. Imagine a frame of reference, $A$ in which a charge Q is at rest. If another charge q is brought close, it will experience an electrostatic force. On another inertial frame of reference $B$ with velocity $v$ with respect to $A$, both the charges are moving. The static charges on $A$ appear as charges and currents. The electrostatic field of $A$ becomes as an electrostatic field of different magnitude and a magnetic field.

magnetism due to a current can be considered to be a relativistic effect.

Permanent magnetism of a bar magnet is not a relativistic effect because of the electron's spin and not orbital motion.

• Well, you could argue that you get a spin on an electron due to relativity. YOu certainly need relativity to get the relationship between the electron spin and the magnitude of it's magnetic moment correct. – Jerry Schirmer Oct 22 '14 at 21:05

Current can be represented as

$I = n e v_d A$ Here $n$ is no of electron per unit volume, $e$is charge of electron, $A$ is area of cross section from where they are flowing and $v_d$ is drift speed.

In a general case, we define $v_d = eE\tau /m_e$, here $E$ is field electrons move in, $m_e$ is mass of electron and $\tau$ is relaxation time

But when you see from a frame moving with velocity $v$, the electrons seem to move with net velocity $v_d + v$. In this case the net current would be given as :

$I = n e (v_d + v) A$

Since the current changes, so does the magnetic field.

The above equation can also be written as:

$I = n e A v + n e v_d A$ This is like $y = mx + c$

Therefore current varies linearly with the speed of observer.

• You forgot to add the current of ions which make up the wire itself, so your calculation went completely wrong. For small velocities, $I$ changes with the second power of $(v/c)$, and not with the first (assuming that wire is electrically neutral in its rest frame). It is most easy to calculate these things in the 4-vector formalism. – firtree Aug 20 '14 at 14:23
• I thought current was with respect to the material it travels across (such as a wire)? Using your formulas, doesn't that mean if I just ran across a neutral ordinary wire I would observe a current flowing? – Joshua Lin Dec 25 '14 at 1:30
• @Joshua Lin: only if the electrons were at rest when you were at rest what you say would have happened, but it doesnt. – Rijul Gupta Dec 29 '14 at 18:32
• So that means if I have two wires and run currents through them, if I move with the flow, that I would observe the force between the two wires to be stronger than if I were a stationary observer? And also that the magnetic field would get stronger the faster I get? Wow that's hectic – Joshua Lin Dec 29 '14 at 22:32
• That would happen only if all electrons moved together, but they don't. It's much more complicated than what you are thinking on particle scale. – Rijul Gupta Dec 30 '14 at 17:44