# Difference between potential and potential energy mathematically

I search google, quora, and reddit: What is the difference between potential and potential energy?

• Potential is the ability to do work. Potential Energy is the amount of energy it acquires.

• Potential is the work done. Potential Energy is the energy stored when the work is done.

• Potential Energy means only the stored energy due to position and potential means stored energy in any Field.

Please explain the difference between potential and potential energy mathematically; What is the mathematical formula for potential and potential energy? I know two expressions for potential energy but not the derivations; $$U = mgh$$ and, $$U = -G \frac{m_1 M_2}{r}$$.

I am learning gravitation in my school; my teacher derived gravitational potential from gravitational potential energy, but I missed the class and class notes are not available. Also this question-answer is not explained and is limited to electromagnetic potential only. I am sorry but please help me. Please explain potential and potential energy with the example of gravitation.

For the gravitational case, the definition of potential ($$V$$) is: $$V = \frac{U_g}{m}$$ Mathematically they represent different physical quantities. Potential energy is a property of a system of two masses, while potential is only a property of the source mass.
For a spherical source mass $$M$$, the potential at a radial distance $$r> a$$ where $$a$$ is it's radius, is given by: $$V = \frac{-\frac{GmM}{r}}{m}= -\frac{GM}{r}$$
This result does not apply for $$r < a$$ as $$U_g \ne \frac{-GmM}{r}$$ for $$r < a$$, but the first definition still holds.
For the electrostatic analog, replace mass with charge and $$U_g$$ with $$U_E$$.