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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.

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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 $U_E$.

Hope this helps.

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