OpenStax University Physics/E&M/Electric Potential

$ℰ\mathrm{\&}ℳ$ (Vol. 2):    5:Electric Charges and Fields    6:Gauss's Law    7:Electric Potential    8:Capacitance    9:Current and Resistance    10:Direct-Current Circuits    11:Magnetic Forces and Fields    12:Sources of Magnetic Fields    13:Electromagnetic Induction    14:Inductance    15:Alternating-Current Circuits    16:Electromagnetic Waves

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Chapter 7

Electric Potential

Electric potential $\mathrm{\Delta }{V}_{AB}=V_A-V_B=-{\int }_A^B\overrightarrow{E}\cdot d\overrightarrow{\ell }$. Change in potential energy $=q\mathrm{\Delta }V=\mathrm{\Delta }U$

▭  Electron (proton) mass = 9.11×10⁻³¹kg (1.67× 10⁻²⁷kg). Electron volt: 1 eV = 1.602×10⁻¹⁹J
▭  Near an isolated point charge $V(r)=k\frac{q}{r}$ where $k=\frac{1}{4\pi {\epsilon }_0}$ =8.99×10⁹ N·m/C² is the Coulomb constant.
▭ Work done to assemble N particles $W_{\mathrm{12...}N}=\sum _{i=1}^N\sum _{j=1}^{i-1}\frac{q_i{q}_j}{r_{ij}}=\frac{k}{2}\sum _{i=1}^N\sum _{j=1}^N\frac{q_i{q}_j}{r_{ij}}\text{ for }i\ne j$
▭ Electric potential due to N charges. $V_P=k\sum _1^N\frac{q_i}{r_i}$. For continuous charge $V_P=k\int \frac{dq}{r}$. For a dipole, $V=k\frac{\overrightarrow{p}\cdot \overrightarrow{\hat{r}}}{r^2}$.
▭ Electric field as gradient of potential $\overrightarrow{E}=-\frac{\partial V}{\partial x}\hat{i}-\frac{\partial V}{\partial y}\hat{j}-\frac{\partial V}{\partial z}\hat{k}=-\overrightarrow{\mathrm{\nabla }}V$ ▭ Del operator^note: Cartesian $\overrightarrow{\mathrm{\nabla }}=\hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}\text{; }$Cylindrical $\overrightarrow{\mathrm{\nabla }}=\hat{r}\frac{\partial }{\partial r}+\hat{\varphi }\frac{\partial }{\partial \varphi }+\hat{z}\frac{\partial }{\partial z}\text{; }$Spherical $\overrightarrow{\mathrm{\nabla }}=\hat{r}\frac{\partial }{\partial r}+\hat{\theta }\frac{\partial }{\partial \theta }+\hat{\varphi }\frac{\partial }{\partial \varphi }\text{.}$

For quiz at QB/d_cp2.7

$\mathrm{\Delta }{V}_{AB}=V_A-V_B=-{\int }_A^B\overrightarrow{E}\cdot d\overrightarrow{\ell }$ = electric potential

$\overrightarrow{E}=-\frac{\partial V}{\partial x}\hat{i}-\frac{\partial V}{\partial y}\hat{j}-\frac{\partial V}{\partial z}\hat{k}=-\overrightarrow{\mathrm{\nabla }}V$

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

$Power=\frac{\mathrm{\Delta }U}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }t}V=IV=e\frac{\mathrm{\Delta }N}{\mathrm{\Delta }t}$

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

$K=\frac{1}{2}m{v}^2$=kinetic energy. 1 eV = 1.602×10⁻¹⁹J

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

Many charges: $V_P=k\sum _1^N\frac{q_i}{r_i}\to k\int \frac{dq}{r}$.