# A charge of -1 C is at the origin. How much energy would be applied to or released from a  -1 C charge if it is moved from  ( -1, 3 )  to (-9 ,7 ) ?

Jan 28, 2018

The energy needed is $= 2.06 \cdot {10}^{9} J$

#### Explanation:

The potential energy is

$U = k \frac{{q}_{1} {q}_{2}}{r}$

The charge ${q}_{1} = - 1 C$

The charge ${q}_{2} = - 1 C$

The Coulomb's constant is $k = 9 \cdot {10}^{9} N {m}^{2} {C}^{-} 2$

The distance

${r}_{1} = \sqrt{{\left(- 1\right)}^{2} + {\left(3\right)}^{2}} = \sqrt{10} m$

The distance

${r}_{2} = \sqrt{{\left(- 9\right)}^{2} + {\left(7\right)}^{2}} = \sqrt{130}$

Therefore,

${U}_{1} = k \frac{{q}_{1} {q}_{2}}{r} _ 1$

${U}_{2} = k \frac{{q}_{1} {q}_{2}}{r} _ 2$

$\Delta U = {U}_{2} - {U}_{1} = k \frac{{q}_{1} {q}_{2}}{r} _ 2 - k \frac{{q}_{1} {q}_{2}}{r} _ 1$

$= k \left({q}_{1} {q}_{2}\right) \left(\frac{1}{r} _ 1 - \frac{1}{r} _ 2\right)$

$= 9 \cdot {10}^{9} \cdot \left(\left(- 1\right) \cdot \left(- 1\right)\right) \left(\frac{1}{\sqrt{10}} - \frac{1}{\sqrt{130}}\right)$

$= 2.06 \cdot {10}^{9} J$

The energy needed is $= 2.06 \cdot {10}^{9} J$