# Electric Current in Special Relativity

• jimmd
In summary, we discussed the transformation of charge density and current density in special relativity, and how this applies to an electric circuit at rest in the lab frame observed from a different frame with a velocity v. We also explored the transformation of electrical resistance and how relativity unifies electrostatics and magnetism.

#### jimmd

Consider an electric circuit at rest in the lab inertial reference frame. The circuit carries a steady electric current I. What is the current in the circuit as observed from a different inertial reference frame with a velocity of v relative to the lab frame?

jimmd said:
Consider an electric circuit at rest in the lab inertial reference frame. The circuit carries a steady electric current I. What is the current in the circuit as observed from a different inertial reference frame with a velocity of v relative to the lab frame?

Special relativity considers the transformation of charge density and current density. Have a look at
W.G.V. Rosser, Classical electromagnetism via relativity Butterworth London 1968 pp165-173
For the OX(O'X') components of the current density J(X) in I and J'(X) you find there
J'(X)=g(V)(J(x)-Vro')
where g(V) stands for the gamma factor and ro' for rhe charge density in I'.
The other two components have the same magnitude in all inertial referfence frames in relative motion.
The transformation for the current depends on the orientation of the conductor through which the current flows.

If you write your current and charge density in a 4-vector:

$$j^a = [c\rho, j_x,j_y,j_z]$$

then this object transforms as a 4-vector under boosts and rotations (i.e. in exactly the same way as the displacement 4-vector). The form of transformation is given in, for example, this Wikipedia article section.

electrical resistence

jimmd said:
Consider an electric circuit at rest in the lab inertial reference frame. The circuit carries a steady electric current I. What is the current in the circuit as observed from a different inertial reference frame with a velocity of v relative to the lab frame?

what about the electrical resistance? how does it transform. has it the same magnitude in all inertial reference frames

masudr said:
If you write your current and charge density in a 4-vector:

$$j^a = [c\rho, j_x,j_y,j_z]$$

then this object transforms as a 4-vector under boosts and rotations (i.e. in exactly the same way as the displacement 4-vector). The form of transformation is given in, for example, this Wikipedia article section.

Yep. If we apply this to the particular problem, we see that we need to know the charge density in the lab frame.

If we assume that the current carrying wire has no charge in the lab frame, and that the direction of the wire and current is the same as the direction of the velocity of the moving observer, we see that in the moving frame the wire acquires a charge, and that both the current and current density are increased by a factor of gamma = 1/sqrt(1-v^2/c^2).

Note that if we have a complete circuit, some parts of the wire will acquire a positive charge and others will acquire a negative charge, but the total charge on the wire will remain constant at 0.

Note that this means that moving charge parallel to a wire experiences an electrostatic force in its own rest frame. In the laboratory rest frame, the wire is neutral, so the charge experiences no electric force, but rather a magnetic force.

This is why it is often said that relativity unifies electrostatics and magnetism.

## 1. What is electric current in special relativity?

Electric current in special relativity refers to the flow of charged particles, such as electrons, in a conductive material. It is described by the laws of special relativity, which take into account the effects of motion and the constancy of the speed of light.

## 2. How does special relativity affect electric current?

Special relativity states that the laws of physics, including those governing electric current, are the same for all observers in uniform motion. This means that the behavior of electric current may appear different to observers in different frames of reference, but the underlying laws governing it remain the same.

## 3. Can electric current travel at the speed of light in special relativity?

No, according to special relativity, nothing can travel at the speed of light. As electric current is made up of charged particles, it must obey the laws of special relativity and cannot exceed the speed of light.

## 4. How does the concept of time dilation in special relativity affect electric current?

Time dilation, which is the slowing down of time for objects in motion, can affect electric current in that the perception of time for moving particles may be different for observers in different frames of reference. This can lead to differences in measured current values, but the underlying laws remain the same.

## 5. Are there any practical applications of electric current in special relativity?

Yes, special relativity plays a crucial role in the development and understanding of technologies such as particle accelerators, which use electric current to accelerate charged particles to high speeds. It also helps in the design and operation of electronic devices, such as GPS systems, which rely on precise timing and synchronization between moving objects.