- #1
silverwhale
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Homework Statement
We have three point charges [itex] Q_1, Q_2, Q_3 [/itex], forming together an isosceles triangle. Two sides are of equal length [itex] r [/itex].
The point charge [itex]Q_1[/itex] is located at [itex] (0,0)[/itex],
the point charge [itex]Q_2[/itex] is located at [itex] (x_2,0)[/itex]
and The point charge [itex]Q_3[/itex] is located at [itex] (\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2})[/itex].
Calculate the resulting force on [itex]Q_3[/itex] for the cases [itex] Q_2 = Q_1 [/itex] and [itex] Q_2 = - Q_1. [/itex]
Homework Equations
I think two formulas are relevant. Coulombs law and the superposition principle.
The Attempt at a Solution
From the Superposition principle and Coulombs law we have:
[tex] \vec{F}_3 = \frac{1}{4 \pi \epsilon_0} Q_3 Q_1 \frac{\vec{x}_3 - \vec{x}_1}{|\vec{x}_3 - \vec{x}_1|^3} + \frac{1}{4 \pi \epsilon_0} Q_3 Q_2 \frac{\vec{x}_3 - \vec{x}_2}{|\vec{x}_3 - \vec{x}_2|^3},[/tex]
where [itex]\vec{F}_3[/itex] is the resulting force on [itex]Q_3[/itex].
Next, I calculated [itex] \vec{x}_3 - \vec{x}_1 = (\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2}) [/itex]
and [itex] \vec{x}_3 - \vec{x}_2 = (-(\frac{x_2}{2}),\sqrt{r^2 - ({\frac{x_2}{2}})^2}). [/itex]
Then I calculated the norm of both to get [tex] |\vec{x}_3 - \vec{x}_1| = |\vec{x}_3 - \vec{x}_2| = r. [/tex]
So I get for [itex] \vec{F}_3 [/itex],
[tex] \vec{F}_3 = \frac{1}{4 \pi \epsilon_0} \frac{Q_3 Q_1}{r^3} (\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2}) + \frac{1}{4 \pi \epsilon_0} \frac{Q_3 Q_2}{r^3} (-\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2})[/tex]
Now for [itex] Q_2 = Q_1 [/itex]:
[tex] \vec{F}_3 = \frac{1}{4 \pi \epsilon_0} \frac{Q_3 Q_1}{r^3}( (\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2}) + (-\frac{x_2}{2},\sqrt{r^2 - ({\frac{x_2}{2}})^2}))[/tex]
thus,
[tex] \vec{F}_3 = \frac{1}{4 \pi \epsilon_0} \frac{Q_3 Q_1}{r^3}( 0,2 \sqrt{r^2 - ({\frac{x_2}{2}})^2}).[/tex]
And for the case [itex] Q_2 = - Q_1 [/itex]:
[tex] \vec{F}_3 = \frac{1}{4 \pi \epsilon_0} \frac{Q_3 Q_1}{r^3} (x_2,0).[/tex]
Is this correct? What do you think?
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