Question 1 · Física II URV Juny 2023

Explain the fundamental physical relationship between work, electrostatic potential, and electrostatic potential energy.

Solución: The relationship is established through the conservative nature of the electrostatic force: • Work and Potential Energy: Because $\vec{F}_e = q\vec{E}$ is conservative, the work done by the electric field on a charge $q$ from $A$ to $B$ equals the negative of the change in electrostatic potential energy: $$W_{A\to B}^{\text{field}} = \int_A^B \vec{F}_e\cdot d\vec{r} = -\Delta U = -(U_B - U_A) = U_A - U_B$$ • Electrostatic Potential: $V$ is defined as the potential energy per unit positive test charge: $$V = \frac{U}{q}$$ Consequently, the potential difference is: $$\Delta V = V_B - V_A = \frac{\Delta U}{q} = -\frac{W_{A\to B}^{\text{field}}}{q} = -\int_A^B \vec{E}\cdot d\vec{r}$$ • Work done by an external agent: At constant kinetic energy, $\vec{F}_{\text{ext}} = -\vec{F}_e$, so: $$W_{A\to B}^{\text{ext}} = -W_{A\to B}^{\text{field}} = \Delta U = q\Delta V = q(V_B - V_A)$$