An electron at rest at p = (5,3,7) moves along a path ending at q = (1, 1, 1) under the influence of the electric field (in newtons per coulomb)

f(x, y, z) = 400(x^2 + z^2)^-1 (x, 0, z).

a)find a potential function for f

b) what is the electron’s speed at the point q? (use conservation of energy and the value qe/me = −1.76 × 1011 c/kg, where qe and me are the charge and mass on the electron, respectively

a)

b)

Step-by-step explanation:

The electric field can be written as

a)Find a potential function for F

A potential function for F would be a function f such that

That is, the gradient of f equals F.

If such a function exists then

Integrating the first equation with respect to x

where h(y,z) is a function that does not depend on x. We have then

By taking the partial derivatives with respect y and z we can notice that h(z,y)=0.

Therefore, a potential function for F is

b) What is the electron’s speed at the point Q?

Since F is conservative, the work done to move the particle from P to Q does not depend on the path, but only on the potential function f.

Let W be the work done when moving the particle from P to Q

W = f(Q) - f(P) = f(1,1,1) - f(5,3,7)

joules

According to the law of conservation of energy, the work done to move the electron from P to Q equals the change in kinetic energy of the object.

Since the electron is at rest in P, the kinetic energy at P equals 0 and we have

where  

m = mass of an electron

= speed at point Q

Replacing in the equation