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
V_\text{@ P}={\Large \Sigma}_{\tiny {i=1}} ^{\tiny N} \ \frac{k\ q_i}{r_i}
Electrostatics
The Electric Potential
Visualizing the Electric Potential
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)
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
\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:
\Delta V= V_B-V_A
Electrostatics
Video walkthrough this stack
The Electric Potential
Relationship to the Electric Field
Electric Potential
By drmoussaphysics
Electric Potential
Introduction to electric potential
