A uniform electric field of magnitude 260 V/m is directed in the positive x direction. A +14.0 µC charge moves from the origin to the point (x, y) = (20.0 cm, 50.0 cm).
(a) What is the change in the potential energy of the charge field system?
To calculate the change in potential energy of the charge-field system, you need to find the initial and final potential energy and then calculate the difference between them.
The potential energy of a charged particle in an electric field is given by the equation:
PE = q * V
where q is the charge of the particle and V is the electric potential.
In this case, the charge (q) is +14.0 µC, which is equivalent to 14.0 x 10^-6 C. The electric potential (V) can be calculated using the equation:
V = E * d
where E is the magnitude of the electric field and d is the distance.
Given that the electric field magnitude (E) is 260 V/m, and the charge travels from the origin (0,0) to the point (x, y) = (20.0 cm, 50.0 cm), we need to find the distance (d) between the two points.
The distance (d) can be calculated using the Pythagorean theorem:
d = sqrt((x^2) + (y^2))
Substituting the values, we have:
d = sqrt((0.2 m)^2 + (0.5 m)^2)
Now we can calculate the initial and final potential energies:
Initial Potential Energy (PEi) = q * Vi
Final Potential Energy (PEf) = q * Vf
To find Vi, we use the equation:
Vi = E * di
where di is the initial distance (from the origin to (x, y) = (0, 0)). Since the charge starts at the origin, di is 0.
Now, we calculate Vf using the equation:
Vf = E * df
where df is the final distance (from the origin to (x, y) = (20.0 cm, 50.0 cm)).
Substituting the values into the equations, we have:
Vi = 260 V/m * 0 = 0 V
df = sqrt((0.2 m)^2 + (0.5 m)^2)
Finally, we can calculate the change in potential energy:
Change in Potential Energy (ΔPE) = PEf - PEi = q * Vf - q * Vi
Substituting the values, we get:
ΔPE = (14.0 x 10^-6 C) * Vf - 0 V
ΔPE = (14.0 x 10^-6 C) * (260 V/m) * df
Now, you can calculate the change in the potential energy of the charge-field system by substituting the value of df into the equation.