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https://texercises.com/exercise/two-point-charges-1/
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Exercise:
A po charge + Q is placed at distance d to the left of a po charge -n Q. Determine the pos along the straight line connecting the two charges where the total electric field vanishes.

Solution:
The only possible location for the solution is to the left of +Q since the two partial field vectors vecE_ and vecE_ have to have opposite directions and the same magnitude. With x being the distance from +Q the two partial fields have magnitudes E_+ k_CfracQx^ E_- k_Cfracn Qd+x^ Since they have to be equal E_E_ it follows that k_CfracQx^ k_Cfracn Qd+x^ fracx^ fracnd+x^ d+x^ n x^ d^+ d x + x^ n x^ nm x^- d x - d^ This is a quadratic with solutions x_pm fracdpmsqrtd^+nf d^den fracdpmsqrtd^ +nmdenfracdpm dsqrtnden fracd pmsqrtnden ifthenelsedenX dpmsqrtnfracdpmsqrtnnm The positive solution x_+ifthenelsedenX d+sqrtnd+sqrtn/nm is the one we are looking for. The negative solution x_-ifthenelsedenX d-sqrtnd-sqrtn/nm is between the two charges. At this po the two partial field vectors have the same magnitude but they po in the same direction.
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Exercise:
A po charge + Q is placed at distance d to the left of a po charge -n Q. Determine the pos along the straight line connecting the two charges where the total electric field vanishes.

Solution:
The only possible location for the solution is to the left of +Q since the two partial field vectors vecE_ and vecE_ have to have opposite directions and the same magnitude. With x being the distance from +Q the two partial fields have magnitudes E_+ k_CfracQx^ E_- k_Cfracn Qd+x^ Since they have to be equal E_E_ it follows that k_CfracQx^ k_Cfracn Qd+x^ fracx^ fracnd+x^ d+x^ n x^ d^+ d x + x^ n x^ nm x^- d x - d^ This is a quadratic with solutions x_pm fracdpmsqrtd^+nf d^den fracdpmsqrtd^ +nmdenfracdpm dsqrtnden fracd pmsqrtnden ifthenelsedenX dpmsqrtnfracdpmsqrtnnm The positive solution x_+ifthenelsedenX d+sqrtnd+sqrtn/nm is the one we are looking for. The negative solution x_-ifthenelsedenX d-sqrtnd-sqrtn/nm is between the two charges. At this po the two partial field vectors have the same magnitude but they po in the same direction.
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Electrostatics
Tags
electric field, electrostatics, point charge
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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