To find the change in the electric potential energy associated with the proton, we need to calculate the work done by the electric field in moving the proton from the origin to x = 2.5 m. The work done is equal to the change in electric potential energy.
The formula to calculate the work done by the electric field is given by:
Work = Electric Field Strength x Displacement x Cosine of the angle between the electric field direction and the displacement direction.
In this case, the electric field is acting in the positive x-direction, and the displacement is also in the positive x-direction. Thus, the angle between the electric field direction and the displacement direction is 0 degrees.
Plugging in the given values:
Electric Field Strength (E) = 850 N/C
Displacement (d) = 2.5 m
Angle (θ) = 0 degrees
We can calculate the work done using the formula:
Work = E * d * cos(θ)
Since the angle is 0 degrees, the cosine of 0 is 1. Thus, the work done simplifies to:
Work = E * d
Plugging in the values:
Work = 850 N/C * 2.5 m
Work = 2125 J (joules)
Therefore, the change in electric potential energy associated with the proton is 2125 J.