Electric Potential and
Capacitance
What’s a volt anyway?
Presentation 2001 Dr. Phil Dauber
as modified by R. McDermott
Why Potential?
Electric potential can be visualized as
“height”.
 This allows us to make comparisons to
gravity.
 Potential is independent of the object in the
electric field.

What is Potential?

By agreement from fields, we always define electrical
quantities in terms of positive charge.

Positive charges will always move away from other
positive charges and toward negative charges, much as a
ball always rolls downhill.

By analogy, we consider a region of positive charge to be
at a high potential (hill), and a region of negative charge to
be at a low potential (valley).

Field lines, then, show the most direct “downhill” path.
Equipotential Lines

Maps dealing with hills and valleys are called contour or
topographical maps, and consist of closed lines that connect
points that have the same height or altitude.

We draw similar “maps” for dealing with charges using
Equipotential lines, which connect points that have the same
electrical height (potential).

An equipotential line will always be perpendicular to the
electric field, since the field always points “downhill”.

Stationary charges will not move on their own along an
equipotential line, because that will not lead them
“downhill”.
Hills and Valleys, Oh My!

As a ball rolls downhill, the gravitational field does work on
the ball due to the unbalanced force (weight) acting on the
ball, causing it to gain speed (and kinetic energy). We say
that potential energy stored in the field between the ball and
Earth is converted into kinetic energy.

Similarly, as a positive charge moves from high potential to
low the electric field does work on the charge due to the
unbalanced force acting on it, causing it to gain speed (and
kinetic energy). We say that potential energy stored by the
field and charge is converted into kinetic energy.
Hills and Valleys, Oh My!
Electric and Gravitational Fields
Gravity:
 Property: mass
Electricity:
 Property: charge

1 Sign: positive

2 Signs: pos, neg

Dependency: 1/r2

Dependency: 1/r2
 G = 6.67x10-11 n-m2/kg2
 k = 8.99x109 n-m2/C2
Electric and Gravitational Fields
Gravity:
Electricity:


F= (GmEarth/rEarth2) m2
F = gfieldm2

PE = mgh

PE = qEx (x = d)

PE is in Joules

PE is in Joules


F = (kq1/r2) q2
F = Efieldq2
Electric and Gravitational Fields
Gravity:
 We want to treat all
Electricity:
 We want to treat all objects
objects the same way, so.. the same way, so..
 “Liftage”, L = mgh/m = gh  “Potential”, V = Eqd/q = Ed
L= (PE)/mass
 Units = Joules/kg

V = (PE)/charge
 Units = Joules/Coul.

Electric and Gravitational Fields
Gravity:
g = F/m
 Units = N/kg
 g is the grav. field
strength

Electricity:
E = F/q
 Units = N/c
 E is the electric field
strength

Electric Potential and Electric
Field
Can describe charge distribution in terms
field or potential. Consider uniform field:
 F = Eq
 W = qVba
 W = Fd = qE d
 Thus Vba = Ed
or E = Vba/d
 Alternate units for E: volts per coulomb

Electrical Potential
Potential is potential energy per unit charge
 Analogous to field which is force per unit
charge
 Symbol of potential is V;
Va = PEa/q
 Only potential differences are measurable;
zero point of potential is arbitrary
 Vab = Va – Vb = - Wba/q
 Wba is work done to move q from b to a

Units:
Unit of electric potential is the volt
 Abbreviation V
 1 V = 1 Joule/Coulomb = 1 J/C
 Thus electrical work = qV
If q is in coulombs, the work is in joules; if
q is in elementary charges, the work is in
electron-volts (eV).
 Potential difference is called voltage

Electrical Potential Energy
The change in potential energy equals the
negative of the work done by the field.
 Electrical PE is transferred to the charge as
kinetic energy.
 A positive charge has its greatest PE near
another positive charge or positive plate.
 Only differences in potential energy are
measurable

V = 0 Arbitrary
Usually ground is zero point of potential
 Sometimes potential is zero at infinity
 + terminal of 12V battery is
said to be at 12V higher
potential than – terminal

Electric Potential and Potential
Energy
PE = PEb – PEa = qVba
If object with charge q moves through a potential
difference Vba its potential energy changes by qVba


Example: What is the gain of electrical PE when 1
C of charge moves between the terminals of a 12
volt battery?
Water Analogy:
Voltage is like water pressure (depth or height)
Example: Electron in Computer
Monitor



An electron is accelerated from rest through a
potential difference of 5000 volts
Find its change in potential energy
Find its speed after acceleration
Vba = 5000 v = Vb - Va
PE = 8x10-16 J
V = 4.2x107 m/s
a
b
Questions on Preceding Example

Does the energy depend on the particle’s
mass?

Does the final speed depend on the mass?
Electric Potential Due to Point
Charges
V = k Q/r derived from Calculus
 Here V = 0 at r = infinity; V represents
potential difference between r and infinity

Example: Work to force two
point charges together

What work is needed to bring a 2mC charge
to a point 10cm from a 5 mc charge?
Work
required = change in potential energy
W = qVba = q{kQ/rb – kQ/ra}
The Electron Volt

The energy acquired by a particle carrying a
charge equal to that of the electron when
accelerated through one volt.
W = qV, if q is in Coulombs, then W is in Joules.
If q is in elementary charges, W is in electron volts
(eV).
1 eV = 1.6 x 10-19 joules

It’s about ENERGY!

1 KeV = 1000 eV; 1 MeV = 106 eV
1 Gev = 109 eV



Potential at an Arbitrary Point
Near Several Point Charges
Add the potentials due to each point charge
 Use the right sign for the charge
 Relax; potential isn’t a vector
 What is true about the mid-plane between
two equal point charges of opposite sign?

Capacitors
Store charge
 Two conducting plates
 NOT touching
 May have insulating
material between

Q = CV
Capacitance
Symbol C
 Unit: coulombs per volt = farad
 1 pF = 1 picofarad = 10-12 farad
 1nf = 1 nanofarad = 10-9 f
 1mf = 1 microfarad = 1-6 f

C is Constant for a Given
Capacitor
Does not depend on Q or V
 Proportional to area
 Inversely proportional to distance between
plates

C

= e0 A/d
If dielectic like oil or paper between plates
use e = Ke0; K is called dielectric constant
Find the Capacitance


A capacitor can hold 5 mC of charge at a
potential difference of 100 volts. What is its
capacitance
C = Q/V = 5 x 10-6 C/ 100V =
5 x 10-8 f = 50000 pf
Capacitors
Photos courtesy Illinois Capacitor, Inc
Applications
In automotive ignitions
 In strobe lights
 In electronic flash
 In power supplies
 In nearly all electronics

A Capacitor Stores Electric
Energy



A battery produces electric energy bit by bit
A capacitor is NOT a type of battery
A battery can be used to charge a capacitor
Energy in a Capacitor Holding
Charge Q at Voltage V




U = QV/2 = CV2/2 = Q2/2C
Derivation: the work needed to charge a capacitor
by bringing charge onto a plate when some is
already there (use W =QV)
Initially V = 0
Average voltage during the charging process is
V/2