How to Calculate Work Required for Moving a Charge in 2D Electric Field

[linda300]

Hey,

I've been working on this problem, it starts by asking for the potential, then from that the electric field, and finally it asks to calculate the work required to move a charge from infinite to the point it was originally.

It's a 2D problem so the electric field was E = (E_x,E_y) and the location of the particle is at (a, b).

W = ∫F . dl

and F = qE = q (E_x,E_y),

My question is, how do i integrate both x,y in the vector separately from infinity to a or b?

Do I first do the only the Ex component and set y=b, then to work calculated from that add the work calculated using the Ey component and setting x=a ?

Or use parametrization ψ(x) = 1/t(a,b) 0<t<1 so infinity initially and (a,b) at t=1

then do ∫F(ψ) . -1/t^2(a,b) between 0 and 1,

I'm just unsure if your allowed to have a parametrization like that, which involve infinity since should be continuously differentiable. But in this case I want it to be infinity initially

 

[darkys]

and then (a,b) at t=1.Thanks in advance for any help Yes, you can use a parametrization like that, as long as the path from infinity to (a, b) is continuous and differentiable. The integral would be of the form W = ∫F(ψ(t))*(-1/t^2(a,b))dtwhere F(ψ(t)) is the force field at the point (a, b). Integrating with respect to t should give you the total work required.