Electric potential and capacitors

February 10, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 2

Electric Potential Energy !  For a conservative force, the work is path-independent !  Electric potential energy, U, is defined in terms of the work

done by the electric field, We, when the system changes its configuration from some initial configuration to some final configuration •  The change in the electric potential energy is the negative of the

work done by the electric field

!  The work done on a charge q in an electric field is

ΔU =U f −U i = −We

U i is the initial electric potential energy

U f is the final electric potential energy

We = q

!E ⋅d!s

i

f

∫ = q!E ⋅d!s

i

f

∫

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 4

Equipotential surfaces !  A constant electric field has straight, equally spaced, and

parallel field lines, which produces equipotential surfaces in the form of parallel planes

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 5

Single Point Charge !  The electric field lines for point charges are radial !  The equipotential surfaces are concentric spheres in 3D !  The equipotential lines take the form of concentric circles in 2D

Positive charge Negative charge

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 6

Electric Potential Difference ΔV !  The electric potential difference between an initial point i

and final point f is

!  Work and potential are related through

!  The units of electric potential are

!  We express the electric field as

ΔV =Vf −Vi =

U f

q−U i

q= ΔU

q

ΔV = −We

q

1 V = 1 J

1 C

[E]= [F]

[q]= N

C= J/m

C= V

m

Graphical Extraction of the Electric Field !  Calculate the magnitude of the electric field at point P. !  To perform this task, we draw a line through point P

perpendicular to the equipotential line reaching from the next line down to the next line up in potential.

!  Measure the length Δs of this line.

!  The magnitude of the electric field can be approximated as

2/10/14 Chapter 23 7

ES = −

ΔVΔs

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 9

System of Point Charges

!  We calculate the electric potential from a system of n point charges by adding the potential functions from each charge

!  This summation produces an electric potential at all points

in space – a scalar function !  Calculating the electric potential from a group of point

charges is usually much simpler than calculating the electric field •  It’s a scalar

V = Vi

i=1

n

∑ = kqi

rii=1

n

∑

Electric Potential Energy for a System of Particles

!  Now we consider a system of point charges that produce the electric potential themselves.

!  We begin with a system of charges that are infinitely far apart, U = 0, by convention.

!  To bring these charges into proximity with each other, we must do work on the charges, which changes the electric potential energy of the system.

!  We then bring in point charge q1. Because there is no electric field and no corresponding electric force, this action requires no work to be done on the charge.

!  Keeping this charge (q1) stationary, we bring the second point charge (q2) in from infinity to a distance r from q1.

!  This requires work q2V1(r). !  We then bring in all other charges !  The total energy of a system of 3 charges is

2/10/14 Chapter 23 10

U123 =U12 +U13 +U23

U123 =kd

q1q2 + q1q3 + q2q3( )

Physics for Scientists & Engineers 2 11

Review: Capacitance !  The definition of capacitance is

•  Capacitance is a property of the capacitor

!  The capacitance of a parallel plate capacitor depends on the plate area and the plate separation and is given by •  A is the area of each plate •  d is the distance between the plates

qCV

=

0ACdε=

Cylindrical and spherical capacitor !  The capacitance of a

cylindrical capacitor with length L, inner radius r1 and outer radius r2 is

!  The capacitance of a spherical capactitor with inner radius r1 and outer radius r2 is

!  The capacitance of an isolated sphere is

February 10, 2014 Physics for Scientists & Engineers 2, Chapter 24 12

C =

2πε0Lln r2 / r1( )

C = 4πε0

r1r2

r2 − r1

C = 4πε0R

September 17, 2008 Physics for Scientists & Engineers 2, Lecture 14 13

Review !  The electric potential energy stored in a capacitor

is given by

!  The field energy density stored in a parallel plate capacitor is given by

!  The field energy density in general is

212

U CV=

20

12

u Eε=

2

012

Vud

ε ⎛ ⎞= ⎜ ⎟⎝ ⎠

February 10, 2014 Physics for Scientists & Engineers 2, Chapter 24 14

Dielectric

!  Inserting a dielectric between the capacitor plates tends to partially cancel the original electric field

!  The capacitance goes up

E = Eair

κ= qκε0A

= qεA

ΔV = Ed = qd

κε0A C = q

ΔV= κε0A

d C =κCair

Circuit Symbols !  Circuit elements are represented by commonly used

symbols.

February 10, 2014 Chapter 24 15

Capacitors in series and in parallel !  The equivalent capacitance for n capacitors in parallel is

=

!  The equivalent capacitance for n capacitors in series is =

!  For two capacitors in series:

2/10/14 Physics for Scientists & Engineers 2, Chapter 23 16

Ceq = Ci

i=1

n

∑

1Ceq

= 1Cii=1

n

∑

Ceq =C1C2

C1 +C2

February 10, 2014 Physics for Scientists & Engineers 2, Chapter 24 19

Supercapacitor / Ultracapacitor !  Supercapacitors (ultracapacitors) are made using a material

with a very large surface area between the capacitor plates !  Two layers of activated charcoal are given opposite charge

and are separated by an insulating material

!  This produces a capacitor with capacitance millions of times larger than ordinary capacitors

!  However, the potential difference can only be 2 to 3 V