Electric Potential
| |
Table of Contents
- Electric Potential
#UCLA #Y1Q3 #Physics1B
Electric Potential
Key Definitions
Potential Energy - property of every conservative force given by:
\(\Delta U=-W=-\int \vec F_{cons} \cdot d\vec r\)
Potential Energy
Electrostatic forces are conservative
Electric PE
Thus, Electric potential energy, $U_{elec}$ (from ) :
\(\Delta U_{elec}=W\)
Electrostatic PE
And, electrostatic potential energy, $U_E$ (from electrostatic forces):
\(\nabla U=-F\)
\(\Delta U_E=-W_E=-\int \vec F_E \cdot d\vec r\)
Similarly, of two static charges is derived from :
\(U_E=k_e\frac{qQ}{r}=qV\)
Between multiple charges, we can use the Superposition Principle:
\(U_E(q)=k_eq\sum_{i=1}^n\frac{q_i}{r_i}\)
Conservation of Energy
Energy is Conserved - as a charge accelerates in the direction of its Coulomb force, its electric potential energy decreases
Additionally, systems tend towards lower “energy configurations”
Electric Potential
Similar to potential energy for electrostatic forces, $U_E$, static electric fields are also conservative:
Electric Potential, $V$ (from static ):
\(\nabla V=-E\)
\(\Delta V=-\int_A^B \vec E\cdot d\vec r\)
Where usually $A=\infty$ if finding potential at a point (reference point; possibly 0 depending on reference)
\(V=\frac{U_E}{q}=k_e\frac{Q}{r}\)
\(V=k_e\sum_{i=1}^n\frac{Q_i}{r_i}=k_e\int\frac{dq}{r}\)
Where Coulomb’s force constant, $k_e=\frac{1}{4\pi\epsilon_o}\approx9\times10^9\space\left[\frac{Nm^2}{C^2}\right]$
Similarity Chart (Gradients): 
Visualization
Equipotential Lines - show the “curves” of const. electric potential: 
Common Electric Potentials
Circulars
Ring
Hollow Sphere
Outside ($r > R$)
\(V=k\frac Q r\)
Inside ($r<R$)
\(V=k\frac Q R\)
Energy in
In a volume of space, electric energy us proportional to the square of the electric field:
\(\frac{U_E}{\text{Vol}}=\frac{\epsilon_0}{2}E^2\)
Where $\epsilon_o\approx8.854\times10^{-12}\space \left[\frac{C^2}{Nm^2}\right]$