Chapter 17

Electric Potential

The Electric Potential Difference Created by Point Charges

Example: The Total Electric Potential At locations A and B, find the total electric potential.

When two or more charges are present, the potential due to all of the charges is obtained by adding together the individual potentials.

The Electric Potential Difference Created by Point Charges

( )( ) ( )( ) V 240m 60.0

C100.8CmN1099.8m 20.0

C100.8CmN1099.8 82298229

+=×−⋅×

+×+⋅×

=−−

AV

( )( ) ( )( ) V 0m 40.0

C100.8CmN1099.8m 40.0

C100.8CmN1099.8 82298229

=×−⋅×

+×+⋅×

=−−

BV

-9 -9

-9 -9

Equipotential Surfaces and Their Relation to the Electric Field

An equipotential surface is a surface on which the electric potential is the same everywhere.

rkqV =

The net electric force does no work on a charge as it moves on an equipotential surface.

Equipotential surfaces for a point charge (concentric spheres centered on the charge)

Equipotential Surfaces and Their Relation to the Electric Field

The electric field created by any charge or group of charges is everywhere perpendicular to the associated equipotential surfaces and points in the direction of decreasing potential.

In equilibrium, conductors are always equipotential surfaces.

Equipotential Surfaces and Their Relation to the Electric Field

Lines of force and equipotential surfaces for a charge dipole.

Equipotential Surfaces and Their Relation to the Electric Field

sVEΔ

Δ−=

The electric field can be shown to be related to the electric potential. If you move a distance Δs and the electric potential changes by an amount ΔV, the component of the electric field in the direction of Δs is the negative of the ratio of ΔV to Δs, or more compactly,

ΔV/Δs is called the potential gradient.

Equipotential Surfaces and Their Relation to the Electric Field

Example: The Electric Field and Potential Are Related The plates of the capacitor are separated by a distance of 0.032 m, and the potential difference between them is VB-VA=-64V. Between the two equipotential surfaces shown in color, there is a potential difference of -3.0V. Find the spacing between the two colored surfaces.

Equipotential Surfaces and Their Relation to the Electric Field

E = −ΔVΔs

= −−64 V

0.032 m= 2.0×103 V m

Δs = −ΔVE

= −−3.0 V

2.0×103 V m=1.5×10−3m

The gradient equation can be simplified for a parallel plate capacitor with voltage V across it and with plates separated by distance d as: E = V

d

Capacitors and Dielectrics

A parallel plate capacitor consists of two metal plates, one carrying charge +q and the other carrying charge –q. It is common to fill the region between the plates with an electrically insulating substance called a dielectric.

Advantages of dielectrics: •  they break down (spark) less easily than air •  the capacitor plates can be put closer together •  they increase the ability to store charge

Capacitors and Dielectrics

THE RELATION BETWEEN CHARGE AND POTENTIAL DIFFERENCE FOR A CAPACITOR The magnitude of the charge in each place of the capacitor is directly proportional to the magnitude of the potential difference between the plates.

CVq =The capacitance C is the proportionality constant. SI Unit of Capacitance: coulomb/volt = farad (F)

The larger C is, the more charge the capacitor can hold for a given V

Capacitors and Dielectrics

THE DIELECTRIC CONSTANT

If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly.

Dielectric constant EEo=κ

E0

E E0

Some lines of force now terminate or start on induced surface charges E ≤ E0

Since E ≤ E0 --> κ ≥ 1

Capacitors and Dielectrics

Capacitors and Dielectrics

THE CAPACITANCE OF A PARALLEL PLATE CAPACITOR

dVE

E o ==κ

( )AqE oo ε=

VdA

q o !"

#$%

&=κε

dA

C oκε=Parallel plate capacitor

filled with a dielectric

<--> q = CV

C increases for --> increasing κ, increasing A, decreasing d

Capacitors and Dielectrics

Example: A Computer Keyboard One common kind of computer keyboard is based on the idea of capacitance. Each key is mounted on one end of a plunger, the other end being attached to a movable metal plate. The movable plate and the fixed plate form a capacitor. When the key is pressed, the capacitance increases. The change in capacitance is detected, thereby recognizing the key which has been pressed.

The separation between the plates is 5.00 mm, but is reduced to 0.150 mm when a key is pressed. The plate area is 9.50x10-5m2 and the capacitor is filled with a material whose dielectric constant is 3.50. Determine the change in capacitance detected by the computer.

Capacitors and Dielectrics

( ) ( )( )( ) F106.19m100.150

m1050.9mNC1085.850.3 123-

252212−

−−

×=×

×⋅×==

dAC oκε

( ) ( )( )( ) F10589.0m105.00

m1050.9mNC1085.850.3 123-

252212−

−−

×=×

×⋅×==

dAC oκε

F100.19 12−×=ΔC

Example. How large would the plates have to be for a 1 F parallel plate capacitor with plate separation a) d=1 mm in air, b) d=1 µm separated by paper?

a)  d = 1 mm C = κε0A/d = ε0A/d (since κ = 1 for air) --> A = Cd/ε0 = (1)(.001)/(8.85 x 10-12) = 1.13 x 108 m2

If the plates were square, they would be of length and width (A)1/2 = (1.13 x 108)1/2 = 11 km --> very large!! b) d = 1µm = 10-6 m ~ the thickness of paper – use paper: κ = 3.3 --> A = Cd/(κε0) = (1)(10-6)/(3.3 x 8.85 x 10-12) = 3.42 x 104 m2 and (A)1/2 = (3.42 x 104)1/2 = 185 m --> about 2 football fields in length,

improving!

Capacitors and Dielectrics

Conceptual Example: The Effect of a Dielectric When a Capacitor Has a Constant Charge An empty capacitor is connected to a battery and charged up. The capacitor is then disconnected from the battery, and a slab of dielectric material is inserted between the plates. Does the voltage across the plates increase, remain the same, or decrease?

Effect of a Dielectric When a Capacitor Has a Constant Charge -- continued

Before inserting dielectric: q = C0 V0, C0 = ε0A/d

After inserting dielectric: q = C V, C = κε0A/d = κC0

C0 V0 = q = C V = κC0 V --> V = V0/κ --> V < V0 for κ > 1 i.e. the voltage across the plates decreases when the dielectric is inserted if charge is held constant

Capacitors and Dielectrics

ENERGY STORAGE IN A CAPACITOR

( )221Energy Ed

dAo !"

#$%

&=κε

221

VolumeEnergydensity Energy Eoκε==

Ad= Volume

Capacitors store charge at some potential difference, therefore they store EPE, since by definition for a point charge q, EPE = qV . Your book shows that the energy stored in a capacitor can be written several ways:

Using Energy = 1/2 CV2, C = κε0A/d, and V = Ed , the energy of a capacitor can be written as,

We can then calculate the energy density of a capacitor as,

(valid in general for the energy density stored in an electric field)

Energy = 12qV =

12CV 2 =

q2

2C

Example. How much energy is stored in a 10 F capacitor with a potential difference of 3 V across it?

This is the same amount of energy stored in a 4.6 kg mass suspended 1 m above the ground.

Energy = 12CV 2 =

1210( ) 3( )2 = 45J