The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r

Force is described as The push or pull on an object with mass that generates it to change its velocity. Force exists as an external agent capable of altering the state of rest or motion of a particular body. It includes a magnitude and a direction.

The straight line path between point (a, b, c) and (l, m, n) exists parametric by the expression

r(t)=(1-t)(a, b, c)+t(l, m, n)

Since we are giving point (4,0,0) and (4,3,4), the parametric equation is providing below

r(t)=(1-t)(4,0,0)+t(4,3,4)

using the dot product system of multiplication, we have

r(t)=(4,3 t, 4 t)

[tex]$\mathrm{t}$[/tex] is between 0,1 .

Next we define the line integral for work done which is express as [tex]$\int_{b}^{a} F \cdot d r$[/tex]

First we define the general expression for the force

[tex]f(x, y, z)=\frac{K(x, y, z)}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}[/tex]

If we substitute our parametric equation we arrive at

[tex]&F(r(t))=\frac{K(4,3 t, 4 t)}{\left(4^{2}+(3 t)^{2}+(4 t)^{2}\right)^{3 / 2}} \\[/tex]

[tex]&F(r(t))=\frac{K(4,3 t, 4 t)}{\left(16+34 t^{2}\right)^{3 / 2}}[/tex]

Also we require to find the expression for d r

r(t)=(4,3 t, 4 t)

d r=(0,3,4) d t

Now we substitute into the integral expression

[tex]$\int_{0}^{1} \frac{k(4,3 t, 4 t)}{\left(16+34 t^{2}\right)^{3 / 2}} \cdot(0,3,4) d t$[/tex]

Using dot product we arrive at

[tex]$\int_{0}^{1} \frac{25 k t}{\left(16+34 t^{2}\right)^{3 / 2}}$[/tex]

Let create a simple substitution so we can simplify the integral,.

let assume[tex]u=16+34 t^{2}[/tex]

[tex]&\frac{d u}{d t}=68 t \\[/tex]

[tex]&d t=\frac{d u}{68 t}[/tex]

And changing setting the new upper and lower limit, we have

[tex]&\frac{25 k}{68} \int_{b}^{a} \frac{1}{u \frac{3}{2}} d u \\[/tex]

a=50

b=16

By simple integral we arrive at

[tex]&-\frac{25 k}{34}\left[\frac{1}{u^{1 / 2}}\right]_{16}^{50} \\[/tex]

[tex]&\frac{25 k}{34}\left[\frac{1}{4}-\frac{1}{50^{1 / 2}}\right][/tex]

Hence the work done is

[tex]$\frac{25 k}{34}\left[\frac{1}{4}-\frac{1}{50^{1 / 2}}\right]$[/tex]

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The complete question is,

The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = x, y, z is F(r) = Kr/|r|3 where K is a constant. Find the work done as the particle moves along a straight line from (4, 0, 0) to (4, 3, 4).