Worked example — Three charges at the vertices of a triangle

Problem

Three charges $q_1 = 2\cdot 10^{-4}\ \text{C}$, $q_2 = -3\cdot 10^{-4}\ \text{C}$ and $q_3 = 5\cdot 10^{-4}\ \text{C}$ are placed at the vertices of an equilateral triangle of side $\ell = 0{,}1\ \text{m}$. Find the resultant force acting on $q_3$.

Solution

Use $k = 8{,}99\cdot 10^9\ \text{N·m}^2/\text{C}^2$. Two Coulomb forces act on charge $q_3$: $\vv{F}_{13}$ (from $q_1$) and $\vv{F}_{23}$ (from $q_2$). Both have magnitude given by Coulomb’s law with $r = \ell$.

Magnitude of $\vv{F}_{13}$ (repulsive: $q_1,q_3$ both positive):

$F_{13} = k\,\frac{q_1 q_3}{\ell^2} = 8{,}99\cdot 10^9 \cdot \frac{(2\cdot 10^{-4})(5\cdot 10^{-4})}{(0{,}1)^2} = 8{,}99\cdot 10^9 \cdot \frac{10^{-7}}{10^{-2}} = 8{,}99\cdot 10^4\ \text{N} \approx 89{,}9\ \text{kN}$

Magnitude of $\vv{F}_{23}$ (attractive: $q_2<0$, $q_3>0$):

$F_{23} = k\,\frac{|q_2|\,q_3}{\ell^2} = 8{,}99\cdot 10^9 \cdot \frac{(3\cdot 10^{-4})(5\cdot 10^{-4})}{(0{,}1)^2} = 8{,}99\cdot 10^9 \cdot \frac{1{,}5\cdot 10^{-7}}{10^{-2}} \approx 134{,}9\ \text{kN}$

Directions. Place $q_3$ at the top vertex (C). $\vv{F}_{13}$ is repulsive, so it points along the line $A\to C$ extended, i.e. in direction $(\cos 60^\circ,\ \sin 60^\circ) = (0{,}5;\ 0{,}866)$. $\vv{F}_{23}$ is attractive, so it points from $C$ towards $B$, direction $(0{,}5;\ -0{,}866)$.

Components (in kN):

$F_x = F_{13}\cdot 0{,}5 + F_{23}\cdot 0{,}5 = 89{,}9\cdot 0{,}5 + 134{,}9\cdot 0{,}5 = 44{,}95 + 67{,}45 = 112{,}4\ \text{kN}$

$F_y = F_{13}\cdot 0{,}866 - F_{23}\cdot 0{,}866 = (89{,}9 - 134{,}9)\cdot 0{,}866 = -45\cdot 0{,}866 = -38{,}96\ \text{kN}$

Resultant magnitude:

$F = \sqrt{F_x^2 + F_y^2} = \sqrt{112{,}4^2 + 38{,}96^2} = \sqrt{12634 + 1518} \approx 118{,}9\ \text{kN}$

Direction (angle below the horizontal):

$\theta = \arctan\!\frac{|F_y|}{F_x} = \arctan\frac{38{,}96}{112{,}4} \approx 19{,}1^\circ$

$\ev{F \approx 118{,}9\ \text{kN},\quad \theta \approx 19{,}1^\circ \text{ below the horizontal}}$

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Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Principio di sovrapposizione