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I know that a static electric field can store energy (which is proportional to $E^2$). I also know that if you have both electric and magnetic fields, linear and angular momenta will be stored in the fields (both of which can be acquired using the Poynting vector). But, my question is can a static electric field (with no magnetic field present) store linear momentum?

Thanks

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No, both an electric and magnetic field are necessary.

As the momentum density is proportional to $\vec{E} \times \vec{B}$, if the magnetic field is zero, so is the momentum density.

When viewed from a moving frame, there will now be an electric and magnetic field, and there will be non-zero momentum. So in this sense, when you see a static electric field, you are in the fields "rest frame" as there is energy in the fields but not momentum.

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In my humble opinion, there is linear momentum stored in electric field. Think about a charged particle, initially at rest, placed at that field. It will increase its velocity. Then, its linear momentum is different from zero. Where did this moment came from? I fail to see another cause but static electric field.

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