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Let an electron be placed at a distance from a $-q$ charge. We learned that electric field lines are into the negative charge. But here direction of force on electron due to negative charge is away from the negative chrage. In this case what will be direction of electric field lines of negative charge on electron be considered? (is it again into the negative charge? But that's impossible, an electron can't have lines of force towards the negative charge!)

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direction of force on electron due to negative charge is away from the negative chrage

Yes, because the direction of force in a negative charge is opposite to the direction of the electric field.

The electric field is defined independently of what charge it acts on, and is even defined when there's no charge present to experience a force. This is very convenient since we don't have to recalculate the field for every possible configuration of charges that it might affect.

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