# Two charges of  -7 C  and  4 C  are at points  (4, 7 ,-8)  and  ( -1 , 3, -8 ), respectively. Assuming that both coordinates are in meters, what is the force between the two points?

May 26, 2016

The force is $- 0.25 \cdot {10}^{12}$ N.

#### Explanation:

The Coulomb force is

$F = \frac{1}{4 \pi {\epsilon}_{0}} \frac{{Q}_{1} {Q}_{2}}{r} ^ 2$. The only unknown quantity is $r$, the distance between the two charges.
We can calculate using the Pitagora's theorem in three dimensions:

$r = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2} + {\left({z}_{1} - {z}_{2}\right)}^{2}} = \sqrt{{\left(4 + 1\right)}^{2} + {\left(7 - 3\right)}^{2} + {\left(- 8 + 8\right)}^{2}} = \sqrt{{5}^{2} + {4}^{2} + {0}^{2}} = \sqrt{{5}^{2} + {4}^{2} + {0}^{2}} = \sqrt{41}$.

The force is

$F = \frac{1}{4 \pi {\epsilon}_{0}} \frac{{Q}_{1} {Q}_{2}}{r} ^ 2 = \frac{1}{4 \cdot 3.14 \cdot 8.854 \cdot {10}^{-} 12} \cdot \frac{\left(- 7\right) \cdot 4}{41} = - \frac{28 \cdot {10}^{12}}{4 \cdot 3.14 \cdot 8.854} =$
$- \frac{28 \cdot {10}^{12}}{111.20624} = - 0.25 \cdot {10}^{12}$ N.
The minus sign means that the two charges are attracting each other.