Electric potential, Systems of charges

04/19/23 Lecture IV 1

Physics 114

Electric potential,Systems of charges


Concepts

• Primary concepts:– Electric potential– Electric energy– capacitance

• Secondary concepts:– Equipotentials– Electonvolt


Potential electric energy

Just like gravity electric force can do work work does not depend on the path it depends only on the initial and final position there is a potential energy associated with electric field.

High PE High PE

Low PE

Low PE


Electric potential

• PE/q is a property of the field itself – called electric potential V

qqPE )(


Electric potential

q

PEV

• V – electric potential is the potential energy of a positive test charge in electric field, divided by the magnitude of this charge q.

• Electric potential is a scalar (so much nicer!).• Electric potential is measured in Volts (V=J/C).

• Potential difference between two points V=Vb-Va is often called voltage.


Charges in electric fieldsE=constForce on charge q:F=qEWork done by the field to move this chargeW=Fd=dqE

W=PEa-PEb=qVa-qVb=-qV

• d E=V

• E= V/d, points from high potential to low• Sometimes electric field is measured in

V/m =N/C

+E F

d

a

b


Non-uniform electric field

rdEV

EdxqFdxW

xEqxFxEb

a

b

a

)()()(

+


Determine E from V • Think ski slopes• If V depends on one coordinate x• E is directed along x from high V to low

• If V depends on x,y,zdx

dVxE )(

z

VE

y

VE

x

VE zyx

;;


Electric field and potential in conductors

E=0 in good conductors in the static situation.

E is perpendicular to the surface of conductor.

Metal hollow boxes are used to shield electric fields.

When charges are not moving (!!) conductor is entirely at the same potential.

+

externalE

-- - - -

+ + + +

constV

EEE ernalexernal

0int


Electronvolt

• Energy that one electron gains when being accelerated over 1V potential difference is called electronvolt eV:

• 1eV=1.6x10-19C 1V= 1.6x10-19J• Yet another unit to measure

energy,• Commonly used in atomic

and particle physics.eVEnergy 5000


EquipontentialsEquipotentials • are surfaces at the same

potential;• are always perpendicular to

field lines;• Never cross;• Their density represents the

strength of the electric field• Potential is higher at points

closer to positive charge


Potential of a point chargePotential V of electric field

created by a point charge Q at a radius r is

Q>0 V>0Q<0 V<0Do not forget the signs!Potential goes to 0 at infinity.Equipotentials of a point charge

are concentric spheres.

020

0

)(r

Qk

r

drkQEdrrV

r

+

0)( V


Superposition of fields

Principle of superposition:

Net potential created by a system of charges is a scalar (!) sum of potentials created by individual charges:

1 2

-+

+21 VVV

....321 VVVV

Potential is a scalar no direction to worry about, But signs are important.

01 V

02 V


Work to move a chargeHow much work has to be

done by an external force to move a charge

q=+1.5 C

from point a to point b?

Work-energy principle

Q1=10C-+

+

aaa VVV 21

ab PEPEPEKEW

+

bbb VVV 21

)( 21aa

aa VVqqVPE

)( 21bb

bb VVqqVPE Q2=-20C

20cm

30cm

25cm

15cm


Capacitance• Two parallel plates are called a

capacitor. When capacitor is connected to a battery plates will charge up.

CVQ • C – coefficient, called

capacitance, property of the capacitor.

• Capacitance is measured in Farad (F=C/V)

Note: net charge =0


Electric field in a capacitor

d

AC

Vd

AQ

d

V

A

Q

0

0

0

E=const• E= V/d • points from high potential to low• When V is fixed (same battery), E depends

only on the d.• Potential

– High next to + plate

– Low – next to - plate

A

QE

00


Capacitance• Capacitance depends on the

geometry of a capacitor

d

AC 0

0 = 8.85. 10-12 C2/N m2-

• permittivity of free space

• A – area of plates (m2)

• Same sign charges want to “spread out” – to hold more charge need large area

• d – distance between plates (m)

• Opposite sign charges “hold” each other, attraction is stronger for shorter d

A

d


Dielectrics • Put non-conductive

material (dielectric) between plates

• Can hold more charge capacitance increases

d

AKC 0

• K(>1) – dielectric constant


Charging up a capacitor

• Find the work needed to charge a capacitor C to voltage V

• Take small charge dq and move it across the capacitor, which is at voltage V at this moment

• dW=Vdq

C

Qqdq

CVdqW

QQ

2

1 2

00


Energy storage

• Work to charge a capacitor= potential energy stored in the capacitor

222

22 CVQV

C

QU

• To use the right formula, watch what is kept constant• V=const – if C connected to a

battery• Q=const - if C disconnected


Inserting dielectric• Capacitor is connected to a

battery, supplying voltage V. How will the energy stored in the capacitor change if we insert a dielectric (K=2)?

• CKC=2C – capacitance increases

UKCVCV

U 222

22

2/22

22

UKC

Q

C

QU

V stays const – same battery

Q changes


Inserting dielectric• Capacitor is charged to charge

Q and disconnected from a battery. How will the energy stored in the capacitor change if we insert a dielectric (K=2)?

• CKC=2C – capacitance increases

UKCVCV

U 222

22

2/22

22

UKC

Q

C

QU

V can change

Q stays const – charge conservation


Inserting dielectric

KUU KUU /

Connected battery – energy increasesDielectric will be “pushed out”

Disconnected battery – energy decreasesDielectric will be “sucked in”


• Between two very large oppositely charged parallel plates at which of the three locations A, B and C electric potential is the greatest?– A A– B B– C C– D Equal at all three locations.

Test problem


E near metal sphere

• Find the largest charge Q that a conductive sphere radius r=1cm can hold.

• Air breakdown E=3x106V/m

2'

1)(

r

Qk

rkQ

dr

dVrE

CErk

Q 033.01033.0)10(109

1031 7229

62

Larger spheres can hold more charge


Test problem• What is wrong with this picture?

– A Equipotentials must be parallel to field lines

– B Field lines cannot go to infinity

– C Some field lines point away from the negative charge

– D Equipotentials cannot be closed

Documents

Electric potential, Systems of charges