Question

A charge of `2 C` is at the origin. How much energy would be applied to or released from a `1 C` charge if it is moved from `( -2 , 1 )` to `( -4 , 2)`?

Answer 1

`W_(12) = 8.99xx10^1 2/(sqrt(20) - sqrt(5)) = 8.04xx10^9 J`

Explanation:

Given two charges:
`q_1 = 2C, " located at " P_0(0,0)`
`q_2 = 1C, " located at " P_1(-2,1)`
`q_2 " is moved to "P_1(-2,1) => P_2(-4,2)`
Find the energy required to move from `P_1(-2,1) => P_2(-4,2)`

Solution:
First let's list the key definition and principles we need to solve this problem:
1) `color(grey) "Electric Force between two charged particles"`:
`F_e = K(q_1q_2)/(Deltar)^2; q_(i=1,2) = "charge"; K = 8.99xx10^9 Nm^2/C^2`
`Deltar = "distance between charges 1 and 2"`

2) `"Work done to move " q_2: P_1 => P_2`
`color(red)(W_(12) = K(q_1q_2)/(Deltar_(12))`
`Deltar_(12) = |r_(01) - r_(02)| = r` change in distance of separation
3) `color(red)"Distance formula"` :
`r_(12) = sqrt((x_0-x_2) + (y_0 -y_2)) - sqrt((x_0-x_1) + (y_0 -y_1 )) =`
`r_(12) = sqrt(20) - sqrt(5)`
`W_(12) = 8.99xx10^1 2/(sqrt(20) - sqrt(5)) = 8.04xx10^9 J`

That is a lot of energy, the reason is that a Coulomb is a huge charge. Typically you would have smaller charges and shorter distances. BTW I made an assumption that all distances are meters your question is silent on that...

The relevant principles are 20 Work formula and distance formula
Technically what we need is:
`W_12 = int_(r_(1))^(r_(2)) F_e(r) *dr = Kq_1q_2int_(r_(1))^(r_(2)) 1/r^2 dr`
but by inspection all you need the linear separation
`Deltar_(02) - Deltar_(01) = sqrt(20) - sqrt(5)`