How Should Electric Potential Energy Be Ranked at Point P?

[STEMucator]

Homework Statement

Rank the electric potential energy at point $P$ for the following four cases:

http://gyazo.com/c7d9df3d3d64cda909ddc0d2ab7686bc

Homework Equations

$\Delta U_e = - W_∞$

The Attempt at a Solution

I believe it should be $U_2 > U_1 > U_3 > U_4$, but I am not certain.

 

[ehild]

Why do you think what you believe? How is electric potential defined? And what is the electric potential at distance r from a point charge q?

ehild

 

[STEMucator]

Why do you think what you believe? How is electric potential defined? And what is the electric potential at distance r from a point charge q?

ehild

Ah I see, so $V = k \frac{q}{r}$ in combination with $V = \frac{U_e}{q}$.

This yields $U_2 > U_1 = U_3 = U_4$.

Thank you.

 

[ehild]

Ah I see, so $V = k \frac{q}{r}$ in combination with $V = \frac{U_e}{q}$.

This yields $U_2 > U_1 = U_3 = U_4$.

Thank you.

The solution is correct now.

Yes, the potential is the potential energy of a unit positive charge at a certain point of the electric field. It is defined with the work done by the field:
The potential at a point P is equal to the work done by the electric field on a unit positive charge while it moves from P to the place where the potential is zero.

You know from Gauss Law that the electric field around a charge q is E=kq/r². It is a conservative field. The potential U(r_P) is the work on a unit positive charge when it moves from r_P to infinity: $$U(r_P)=W\big |_{r_P}^{\infty}=\int _{r_P}^{\infty}{\frac{kq}{r^2}dr}=k\frac{q}{r_P}$$.

ehild

 

[HalfLostAndSearching]

Can someone please confirm or correct me if I am wrong?

I would like to clarify that the question is asking for the ranking of electric potential energy at point $P$, not the change in electric potential energy. Therefore, the correct ranking would be: $U_3 > U_4 > U_1 > U_2$. This is because the electric potential energy is directly proportional to the distance between the point and the source of the electric field, with greater distance resulting in lower potential energy. In this case, as the distance from the positive charge increases, the electric potential energy at point $P$ decreases.