# Is electric current reference frame dependent?

$$EMF=\frac{dW_{external}}{dq}$$ $$dW_{ext}=\vec{F}_{net}.d\vec{s}$$ $$\vec{F}_{net}=\vec{F}_{inertial}+\vec{F}_{non-inertial}$$ $$dW_{inertial}+dW_{non-inertial}=dW_{external}$$

After reading the mathematical equations above,I think the emf of a battery acting as a source must depend on the reference frame{inertial and non inertial}. It means if emf is a reference dependent quantity then electric current produced in the circuit must also depend on reference frame we choose. Can we conclude that electric current is also a reference dependent quantity?

• EMF of induced electric field for a closed curve in space does depend on the frame. However, in your argument, you seem to be assuming there is some fixed relation between emf and current, valid equally in all frames. Is that true? Are you assuming generalized ohm law is valid in all frames? – Ján Lalinský Aug 1 at 10:33
• @Ján Lalinský I agree with you. – Shreyansh Aug 1 at 13:58
• @Unique well I am not sure that the argument is entirely valid, because it is not very common to describe current in rapidly moving conductors in terms of Ohm's law. The clearer and more general argument which applies even to non-ohmic conductors would be that current is equal to charge passed through the wire over unit time interval, and since time duration is frame-dependent, electric current should be too frame-dependent. – Ján Lalinský Aug 1 at 18:08

Can we conclude that electric current is also a reference dependent quantity?

yes. Electric current density forms part of a four-vector whose timelike component is the electric charge density, in the same way that momentum forms the spacelike part of a four-vector whose timelike component is energy.

In other words, $$j^\mu = (\rho,\vec j)$$ is a four-vector and transforms as such under Lorentz frame transformations. Among other things, this entails that

• the charge density mixes into the current, i.e. a stationary charge produces an electric current when seen from a reference frame where the charge is moving (a fact which is not at all surprising, but)
• if a neutral configuration with nonzero currents is seen from an appropriate reference frame, then a net charge density will appear.

This last bit is pretty surprising, and it is not expected from classical kinematics: a stationary test charge won't feel a force from a neutral set of currents, but a moving test charge will feel a force (which, by continuity, must be proportional to the velocity, at least at low $$v$$). Or, in other words, it sees a magnetic field.

For more details on that connection, I recommend the relativity chapters at the start of Ed Purcell's Electricity and Magnetism.

Yes, the current is RF-dependent quantity. The charge density $$\rho$$ and the current flux $$\vec{j}$$ are RF-dependent quantities - they form a four-vector $$j_{\mu}$$.

In particular, if you take a neutral piece of a wire with a constant current $$I$$, then in a moving RF it will be charged, with the charge density absolute value and its sign RF dependent: $$\rho'\ne 0$$.