OpenStax University Physics/E&M/Electric Potential

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

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

For quiz at QB/d_cp2.7

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

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

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

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

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

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

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

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

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