# Does an observer measure magnetic field when moving inside a static E field?

Imagine a static electric field shaped as a sinusoid created by an array of electrodes on an infinite line. Does an observer flying parallel inside this e-field measure an h-field ?

• Some sort of diagram might help. In general yes, if you have a static E-field in one frame of reference then in a different moving frame of reference there will be both an E-field and B- (or H-) field. Nov 17, 2015 at 16:52

A bit of special relativity is required to understand this. Electric and magnetic fields form a 4-vector with their potentials:

$$A^{\mu} = (\phi, \vec{A})$$

Where $\phi$ is the scalar electric potential and $\vec{A}$ is the magnetic potential. In the case you described, this would be:

$$A^{\mu} = (\phi, 0)$$

Where $\phi$ is constant in time and $\vec{A} = 0$. The observer, assuming they are moving in an inertial frame at velocity $v$ in the x-direction, has undergone a Lorentz boost. In the observers frame, they measure $A'^{\mu}$:

$$A'^{\mu} = \Lambda^{\mu}_{\nu} A^{\nu}$$

Using Einstein notation. Just think of this as a linear transformation acting on the 4-vector $A^{\nu}$ (the original 4-potential). Therefore the elements of the new 4-vector are, using $\Lambda$ from the link provided, produced upon using the matrix:

$$\lambda^{\mu}_{\nu} = \begin{bmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Which results in a new potential:

$$A'^{\mu} = (\gamma \phi, -\beta \gamma \phi, 0, 0)$$

As you can see the 4-potential to the observer moving at constant velocity $v$ has picked up a magnetic potential component in the direction of motion. The magnetic potential looks to him to be:

$$\vec{A} = (-\beta\gamma\phi, 0, 0)$$

Which has a non-zero curl, and therefore produces a magnetic field. Notice that both terms have a factor of $\gamma$, but only the second term has the factor of $\beta$. For non-relativistic speeds $\gamma \approx 1$ and $\beta = v/c \approx 0$, which reduces the potentials back to their places.