Electric Potential

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Table of Contents

• Electric Potential
  • Key Definitions
    • $\mathrm{\Delta }U=-W=-\int {\overrightarrow{F}}_{cons}\cdot d\overrightarrow{r}$
  • Potential Energy
    • Electric PE
      • $\mathrm{\Delta }{U}_{elec}=W$
    • Electrostatic PE
      • $\mathrm{\nabla }U=-F$
      • $\mathrm{\Delta }{U}_E=-W_E=-\int {\overrightarrow{F}}_E\cdot d\overrightarrow{r}$
      • $U_E=k_e\frac{qQ}{r}=qV$
      • $U_E(q)=k_{e}q\sum _{i=1}^n\frac{q_i}{r_i}$
  • Conservation of Energy
  • Electric Potential
    • $\mathrm{\nabla }V=-E$
    • $\mathrm{\Delta }V=-{\int }_A^B\overrightarrow{E}\cdot d\overrightarrow{r}$
    • $V=\frac{U_E}{q}=k_e\frac{Q}{r}$
    • $V=k_e\sum _{i=1}^n\frac{Q_i}{r_i}=k_e\int \frac{dq}{r}$
    • Visualization
  • 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
    • $\frac{U_E}{\text{Vol}}=\frac{{ϵ}_0}{2}{E}^2$

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Electric Potential

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Key Definitions

Potential Energy - property of every conservative force given by:

$\mathrm{\Delta }U=-W=-\int {\overrightarrow{F}}_{cons}\cdot d\overrightarrow{r}$

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Potential Energy

Electrostatic forces are conservative

Electric PE

Thus, Electric potential energy, $U_{elec}$ (from ) :

$\mathrm{\Delta }{U}_{elec}=W$

Electrostatic PE

And, electrostatic potential energy, $U_E$ (from electrostatic forces):

$\mathrm{\nabla }U=-F$

$\mathrm{\Delta }{U}_E=-W_E=-\int {\overrightarrow{F}}_E\cdot d\overrightarrow{r}$

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:

$U_E(q)=k_{e}q\sum _{i=1}^n\frac{q_i}{r_i}$

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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”

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Electric Potential

Similar to potential energy for electrostatic forces, $U_E$, static electric fields are also conservative:

Electric Potential, $V$ (from static ):

$\mathrm{\nabla }V=-E$

$\mathrm{\Delta }V=-{\int }_A^B\overrightarrow{E}\cdot d\overrightarrow{r}$

Where usually $A=\mathrm{\infty }$ 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, $k_e=\frac{1}{4\pi {ϵ}_o}\approx 9\times {10}^9\left[\frac{N{m}^2}{C^2}\right]$

Similarity Chart (Gradients):

Visualization

Equipotential Lines - show the “curves” of const. electric potential:

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Common Electric Potentials

Circulars

Ring

Hollow Sphere

Outside ($r>R$)

$V=k\frac{Q}{r}$

Inside ($r<R$)

$V=k\frac{Q}{R}$

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Energy in

In a volume of space, electric energy us proportional to the square of the electric field:

$\frac{U_E}{\text{Vol}}=\frac{{ϵ}_0}{2}{E}^2$

Where ${ϵ}_o\approx 8.854\times {10}^{-12}\left[\frac{C^2}{N{m}^2}\right]$