Point charge moving in an electric field

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A point charge accelerating through space generates an electric field and radiates energy. But how is this influenced if the point charge is moving along the field lines of an external electric field? Specifically how is calculation of radiated energy effected? Would you add the external field to the radiative component of the electric field (the 1/r dependent part) when calculating the Poynting vector, or is wrong?
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As long as the externally applied electric field is constant, it has no additional effect on the radiated energy.
 
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As long as the externally applied electric field is constant, it has no additional effect on the radiated energy.
What if, say the charged particle was moving along a fixed axis, and the applied electric field, was also on that axis, BUT dependent upon position.
 

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