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CptXray

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## Homework Statement

There's a thin, straight, infinite wire conducting alternating current:

$$I(t) = I_{0}\exp(-\kappa t^2),$$

where ##\kappa > 0##.

Calculate the force exerted on point charge ##q## that is located in distance ##\rho## from the wire. Consider relativistic effects.

## Homework Equations

Ampere's law:

$$\int\limits_{\partial V} \vec{B} \vec{\mathrm{d}l} = \int \limits_{V} \vec{j} \vec{\mathrm{d} s}$$

$$\nabla \times \vec{E} = - \frac{\partial{\vec{B}}}{\partial{t}}$$

## The Attempt at a Solution

First, the magnetic field of a straight infinite wire:

Using Ampere's law, where ##\partial{V}## is a circle, centered on the wire:

$$\int\limits_{\partial V} \vec{B} \vec{\mathrm{d}l} = \int \limits_{V} \vec{j} \vec{\mathrm{d} s}$$

$$B_{\phi} 2 \pi r = \mu_{0} I$$

$$B_{\phi} = \frac{\mu_{0}I}{2\pi} \frac{1}{r} \hat{\phi},$$

where ##B_{\phi}## is a magnetic field in azimuthal direction and ##r## is the distance from the wire.

Now, I know that the magnetic field is changing with time:

$$- \frac{\partial{B_{\phi}}}{\partial{t}} = 2kt\frac{\mu_{0} I}{2\pi}\frac{1}{r} \hat{\phi}$$

I tried doing double curl ##\nabla \times (\nabla \times \vec{B}) = \nabla (\nabla \cdot \vec{E}) - \Delta E##, but I have a feeling that it gets me nowhere.