Electric Potential

Electrostatics

The Electrostatic Potential

Electrostatics

The Electrostatic Potential

Learning Outcomes

Learn how to:

Calculate the electrostatic potential at a given point in space due to

a configuration of point charges.
a "continuous" distribution of electric charges.

Relate the electrostatic potential to

the electrostatic potential energy
the electrostatic field

Electrostatics

The Electric Charge

Charge Distribution

Electrostatics

Electric Charge

Charge Distributions

\lambda = dq / dl

Linear Charge Density

\sigma=dq / dA

Surface Charge Density

\rho=dq / dV

Volume Charge Density

\lambda_\text{\tiny uniform} = \frac{Q_\text{\tiny total}}{l_\text{\tiny total}}

\sigma\text{\tiny uniform} = \frac{Q_\text{\tiny total}}{A_\text{\tiny total}}

\rho\text{\tiny uniform} = \frac{Q_\text{\tiny total}}{V_\text{\tiny total}}

Electrostatics

Electric Charge

Point Charge

Charge distribution A

Charge distribution B

What do we mean by point charges?

\text{size$_\text{charge distribution}$}

\text{scale of analysis}

\lt\lt\lt

Electrostatics

The Electric Charge

... and the rest of the cast

Electrostatics

The influence & interaction of electric charges

The Cast

q

\vec{E}

V

\Phi

\vec{F}

U

potential

potential energy

field

force

charge

flux

influence

interaction

Electric ....

Electrostatics

The influence & interaction of electric charges

The Cast - relationship map

Electric ....

\vec{E}

V

\Phi

influence

interaction

\ \vec{E}=-\vec{\nabla} V\

\ \Delta V = -\int \vec{E}\cdot d\vec{s}\

\ \Phi = \int \vec{E}\cdot d\vec{A}\

\vec{F}

U

\ \vec{F}=-\vec{\nabla} U\

\ \Delta U = -\int \vec{F}\cdot d\vec{s}\

\ \vec{F} = q_0\ \vec{E}\

\ \vec{E} = \vec{F}/q_0\ \

\ U = q_0\ V\

\ V = U/q_0\ \

q

Electrostatics

The influence & interaction of electric charges

The Cast - relationship map

Electric ....

\vec{E}

V

\Phi

influence

interaction

\ \vec{E}=-\vec{\nabla} V\

\ \Delta V = -\int \vec{E}\cdot d\vec{s}\

\ \Phi = \int \vec{E}\cdot d\vec{A}\

\vec{F}

U

\ \vec{F}=-\vec{\nabla} U\

\ \Delta U = -\int \vec{F}\cdot d\vec{s}\

\ \vec{F} = q_0\ \vec{E}\

\ \vec{E} = \vec{F}/q_0\ \

\ U = q_0\ V\quad

\ V = U/q_0\ \

q

\Phi_g

influence

\ \Phi_g = \int \vec{g}\cdot d\vec{A}\

\vec{g}

gy

\ \vec{g}=-\vec{\nabla} (gy)\

\ \Delta V = \int \vec{g}\cdot d\vec{s}\

\vec{F}_g

mgy

interaction

\ \vec{F}=-\vec{\nabla} (mgy)\\ =-mg\hat j

\ \Delta U = -\int \vec{F}\cdot d\vec{s}\

m

\ \vec{F_g} = m\ \vec{g}\

\ \vec{g} = \vec{F_g}/m\ \

\ U_g = m\ V_g\

\ V_g = U_g/m\ \

gravitational....

analogus to

Electrostatics

The Electrostatic Potential

conceptual overview

Electrostatics

The Electric Potential

Conceptual overview

Electric Charges create an influence in their vicinity that we call the Electrostatic Potential.

Picture it

as a fictious glow surrounding each charge that fades with the

d i s t a n c e

from the charge.

The "glow" at any point in space is the combined result of all the "glow" from the distribution of charges.

Positive charges create positive "glow", and

negative charges create negative "glow".

In this simulator, you can change the source of the potential and the parameters of the view. You can also change the camera angle by clicking and dragging on the image.

Electrostatics

The Electric Potential

Conceptual overview - the basics

The Electrostatic Potential due to a point charge is proportional to the charge.

The Electrostatic Potential due to a point charge is ...

inversely proportional to the distance from that charge.

The Electrostatic Potential at some location due to a point electric charge depends on the material(s) in the region(s) separating the charge and that location.

1

2

3

Electrostatics

The Electric Potential

Point Charge

The Electrostatic Potential due to a point charge is

V(r)=\frac{kq}{r}

given by ...

charge creating potential.

distance from the charge q to the point of interest P

material-dependent proportionality constant

k:

q:

r:

Electrostatics

The Electrostatic Potential

Point Charge

V(r)=\frac{kq}{r}

\text{the Electrostatic Potential due to a point-charge $q$ at some distance $r$}

Visualizingthe Electric Potential from a point-charge

Electrostatics

The Electrostatic Potential

due to a point charge

Electrostatics

The Electric Potential

Point-charge

V(r)=\tfrac{1}{4\pi\epsilon}\frac{q}{r}

\text{the Electric Potential due to a point-charge $q$ at some distance $r$}

\text{permittivity}

\text{SI unit: Volt (V)}

\text{Volt}= \text{Joule / Coulomb}

\text{Scalar Quantity } (\pm)

\kappa : \text{relative permittivity (Dielectric Constant)}

\text{Permittivity of free space: } \epsilon_0=8.854\times10^{-12}\ C^2/Nm^2

\text{Absolute Permittivity (material dependent): } \epsilon = \kappa \epsilon_0

\text{In free space: } k=1/4\pi\epsilon_0 = 8.99\times10^{9}\ Nm^2/C^2

Electrostatics

The Electric Potential

Point-charge

V(r)=\tfrac{1}{4\pi\epsilon_0}\frac{q}{r}

\text{**Calculate the electric potential produced by}\\\text{ a point-charge of $3.0\mu C$ at a distance of 2.0mm}

=\tfrac{(8.99\times10^9 Nm^2/C^2) \ \times (3\times10^{-6} \ C)}{(2 \times10^{-3} m)}

=1.3\times10^7 V

Electrostatics

The Electric Potential

Point-charge

V=\tfrac{1}{4\pi\epsilon_0}\frac{q}{r}

Suppose that a point charge of $q=8.0\text{nC}$ is located at $\vec{r}_q =\ \langle 0.50\text{m}, 1.5\text{m}\rangle $ according to some 2D Cartesian coordinate system.
Calculate the electric potential produced by the charge $q$ at the shown locations $P_1$ and $P_2$. (hover on each point to see the calculation.)

x

y

0.50 m

=\tfrac{(8.99\times10^9 Nm^2/C^2) \ \times (8\times10^{-9} \ C)}{\sqrt{(6 \times0.5 m)^2+(2 \times0.5 m)^2})}

=23 V

=\tfrac{1}{4\pi\epsilon_0}\frac{q}{|\vec{r}_{P_1}-\vec{r_q}|}

P_1

P_2

=\tfrac{(8.99\times10^9 Nm^2/C^2) \ \times (8\times10^{-9} \ C)}{\sqrt{(2 \times0.5 m)^2+(-2 \times0.5 m)^2})}

=50 V

=\tfrac{1}{4\pi\epsilon_0}\frac{q}{|\vec{r}_{P_2}-\vec{r_q}|}

{|\vec{r}_{P_1}-\vec{r_q}|}

6\times 0.5\text{m}

2\times 0.5\text{m}

Electrostatics

The Electric Potential

due to a configuration of point charges

Electrostatics

The Electric Potential

Multiple point charges

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

\text{The Net Electric Potential due to multiple point-charges }

q_1

q_2

q_N

\text{algebriac sum}

\text{the contribution due to $q_i$}

Electrostatics

The Electric Potential

Multiple point charges

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

Electrostatics

The Electric Potential

Multiple point charges

(a)

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

=\frac{k\ (+q)}{2a} + \frac{k\ (-q)}{a}

=\frac{-kq}{2a}

(b)

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

=\frac{k\ (+q)}{a} + \frac{k\ (+q)}{a}

=\frac{2kq}{a}

e.g. For each of the shown charge configurations, find an expression for the net Electric Potential at point P

(c)

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

=\frac{k\ (+q)}{a/\sqrt{2}} + \frac{k\ (+q)}{a/\sqrt{2}}+\frac{k\ (-q)}{a/\sqrt{2}} + \frac{k\ (-q)}{a/\sqrt{2}}

=0

Electrostatics

The Electric Potential

Representation and Visualization

Electrostatics

The Electric Potential

Visualizing the Electric Potential

Visualizing the Electric Potential from a point-charge -- phet

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

Electrostatics

The Electric Potential

Visualizing the Electric Potential

Visualizing the Electric Potential from a point-charge

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

Electrostatics

The Electric Potential

Visualizingthe Electric Potential from a point-charge (1D)

V

x

y

Visualizing the Electric Potential

V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}

Electrostatics

The Electric Potential

Equipotential Surfaces

Electrostatics

The Electric Potential

Equipotential Surfaces

The "landscape" analogy

Electrostatics

The Electric Potential

Equipotential Surfaces

toggle between

3D view and Equipotential view

Vary the charges and their locations

Electrostatics

The Electric Potential

Equipotential Surfaces

In this simulation, the shown curves represent the intersection of the equipotential surfaces with the shown plan

use the pencil button on the cross-hairs tool.

Electrostatics

The Electric Potential

Equipotential Surfaces

In this simulation, the shown curves represent the intersection of the equipotential surfaces with the chosen plan

show different slices to investigate the whole 3D space

Electrostatics

The Electric Potential

The Potential Difference (aka Voltage)

Electrostatics

The Electric Potential

The Electric Potential Difference (aka Voltage)

\Delta V

\Delta V

ANALOG VOLTMETER

The Instruments measure the electric potential difference between any two points in space.

Electrostatics

The Electric Potential

The Electric Potential Difference (aka Voltage)

\Delta V

ANALOG VOLTMETER

The Electric Potential is defined up to an arbitrary scalar shift

Therefore, it is the Electric Potential Difference that really matters

The Electric Potential Difference between any two points in space, A and B, is given by: