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Consider the charges \(q, q\) and \(-q\) placed at the vertices of an equilateral triangle, as shown in figure. What is the force on each charge?

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Consider the charges \(q, q\) and \(-q\) placed at the vertices of an equilateral triangle, as shown in figure. What is the force on each charge?
Question: Consider the charges  q, q and -q  placed at the vertices of an equilateral triangle, as shown in figure. What is the force on each charge?

Answer

By observing the figure
Force acting on charge \(q\) at \(A\) due to charge \(q\) at \(B\) is \(F_{12}\) along \(\mathit{BA}\) as both charges are of same nature therefore repulsive in nature.
Force acting on charge \(q\) at \(A\) due to charge \(-q\) at \(C\) is \(F_{13}\) along \(\mathit{AC}\) as both charges are opposite in nature therefore attracts each other.
By the law of parallelogram, the total force \(F_1\) on the charge \(q\) at \(A\) can be found as,
\(F_1=Fȓ_1\)
where as \(ȓ_1\) is a unit vector along \(\mathit{BC}\) which is at an angle of \(120^{\circ}\) from \(x\)- axis .
The force of attraction or repulsion for each pair of charges \((\)either \(\mathit{AB}\) or \(\mathit{BC}\) or \(\mathit{CA})\) has the same magnitude and therefore,
\(F=\frac 1{4\pi \varepsilon _0}\frac{q^2}{l^2}\)
where the magnitude of charge is \(q\) and the distance between each pair is \(l\) .
The total force \(F_2\) on charge \(q\) at \(B\) is therefore \(F_2=Fȓ_2\) , where \(ȓ_2\) is a unit vector along \(\mathit{AC}\) .
And the total force on charge \(-q\) at \(C\) is \(F_3=\sqrt 3F\widehat n\) , where \(\widehat n\) is the unit vector along the direction bisecting the \({\angle}\mathit{BCA}\) .
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