Introductory Physics Volume Two

2.1 Electric Potential 39
Example
Suppose that an electron moved from a region where the electric potential
is 150 volts to a region where the electric potential is 100 volts.
There is only the electric field acting on the electron. What would be
the change in the kinetic energy of the electron? We know that the
potential energy and electric potential are related: ∆V = ∆U/q.
so that
∆K + ∆U = 0
∆K = −∆U = −q∆V = −(−e)(100V − 150V) = −8.0 × 10 −18 J
In this example we see that negatively charged particles slow down
when they go from a region of high electric potential to a region of low
electric potential. A positively charged particles would speed up.
Example
Now let’s try an example where the field is not uniform.
⃗E = α [ y 2 î + 2xyĵ ]
If we move from the origin to the point (a, b) what is the change in the
electric potential. First we need to pick a path from the starting point
to the ending point. A straight line will do. Let ⃗r(t) = x(t)î + y(t)ĵ =
atî + btĵ, we see that as the parameter t is varied from 0 to 1, that
the vector ⃗r(t) points along the path from the origin to the final point
(a, b): that is ⃗r(t) is the trajectory of a particle following our path. This
function is called a parameterization of the path. Now we can compute
the line integral ∫ E ⃗ · dr ⃗ since we can now write
⃗dr = d⃗r
dt dt = [ dx
dt î + dy
dt ĵ
]
dt = [aî + bĵ] dt