# OpenStax University Physics/E&M/Electric Potential

## Chapter 7

#### Electric Potential

Electric potential ${\displaystyle \Delta V_{AB}=V_{A}-V_{B}=-\int _{A}^{B}{\vec {E}}\cdot d{\vec {\ell }}}$. Change in potential energy ${\displaystyle =q\Delta V=\Delta U}$

▭  Electron (proton) mass = 9.11×10−31kg (1.67× 10−27kg). Electron volt: 1 eV = 1.602×10−19J
▭  Near an isolated point charge ${\displaystyle V(r)=k{\tfrac {q}{r}}}$ where ${\displaystyle k={\tfrac {1}{4\pi \varepsilon _{0}}}}$ =8.99×109 N·m/C2 is the Coulomb constant.
▭ Work done to assemble N particles ${\displaystyle W_{12...N}=\sum _{i=1}^{N}\sum _{j=1}^{i-1}{\tfrac {q_{i}q_{j}}{r_{ij}}}={\tfrac {k}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\tfrac {q_{i}q_{j}}{r_{ij}}}{\text{ for }}i\neq j}$
▭ Electric potential due to N charges. ${\displaystyle V_{P}=k\sum _{1}^{N}{\frac {q_{i}}{r_{i}}}}$. For continuous charge ${\displaystyle V_{P}=k\int {\frac {dq}{r}}}$. For a dipole, ${\displaystyle V=k{\tfrac {{\vec {p}}\cdot {\vec {\hat {r}}}}{r^{2}}}}$.
▭ Electric field as gradient of potential ${\displaystyle {\vec {E}}=-{\tfrac {\partial V}{\partial x}}{\hat {i}}-{\tfrac {\partial V}{\partial y}}{\hat {j}}-{\tfrac {\partial V}{\partial z}}{\hat {k}}=-{\vec {\nabla }}V}$ ▭ Del operatornote: Cartesian ${\displaystyle {\vec {\nabla }}={\hat {i}}{\tfrac {\partial }{\partial x}}+{\hat {j}}{\tfrac {\partial }{\partial y}}+{\hat {k}}{\tfrac {\partial }{\partial z}}{\text{; }}}$Cylindrical ${\displaystyle {\vec {\nabla }}={\hat {r}}{\tfrac {\partial }{\partial r}}+{\hat {\phi }}{\tfrac {\partial }{\partial \phi }}+{\hat {z}}{\tfrac {\partial }{\partial z}}{\text{; }}}$Spherical ${\displaystyle {\vec {\nabla }}={\hat {r}}{\tfrac {\partial }{\partial r}}+{\hat {\theta }}{\tfrac {\partial }{\partial \theta }}+{\hat {\phi }}{\tfrac {\partial }{\partial \phi }}{\text{.}}}$

#### For quiz at QB/d_cp2.7

${\displaystyle \Delta V_{AB}=V_{A}-V_{B}=-\int _{A}^{B}{\vec {E}}\cdot d{\vec {\ell }}}$ = electric potential

${\displaystyle {\vec {E}}=-{\tfrac {\partial V}{\partial x}}{\hat {i}}-{\tfrac {\partial V}{\partial y}}{\hat {j}}-{\tfrac {\partial V}{\partial z}}{\hat {k}}=-{\vec {\nabla }}V}$

${\displaystyle q\Delta V}$ = change in potential energy (or simply ${\displaystyle U=qV}$)

${\displaystyle Power={\tfrac {\Delta U}{\Delta t}}={\tfrac {\Delta q}{\Delta t}}V=IV=e{\tfrac {\Delta N}{\Delta t}}}$

Electron (proton) mass = 9.11×10−31kg (1.67× 10−27kg). Elementary charge = e = 1.602×10−19C.

${\displaystyle K={\tfrac {1}{2}}mv^{2}}$=kinetic energy. 1 eV = 1.602×10−19J

${\displaystyle V(r)=k{\tfrac {q}{r}}}$ near isolated point charge

Many charges: ${\displaystyle V_{P}=k\sum _{1}^{N}{\frac {q_{i}}{r_{i}}}\to k\int {\frac {dq}{r}}}$.