Electric Potential Energy of the System of Charges Potential energy of a test charge q 0 in the presence of other charges Potential energy of the system.

Electric Potential Energy of the System of Charges

Potential energy of a test charge q0

in the presence of other charges0

04i

ii

q qU

r

Potential energy of the system of charges(energy required to assembly them together)

0

1

4i j

iji j

q qU

r

Potential energy difference can be equivalently described as a work done by external force required to move charges into the certain geometry (closer or farther apart). External force now is opposite to the electrostatic force

( )a b b a extW U U F d l

Electric Potential Energy of System

• The potential energy of a system of two point charges

• If more than two charges are present, sum the energies of every pair of two charges that are present to get the total potential energy€

U = ke

q1q2

r12

ji ij

jietotal r

qqkU

,

23

32

13

31

12

21

rqq

rqq

rqq

kU etotal

Electric potential is electric potential energy per unit charge

Finding potential (a scalar) is often much easier than the field (which is a vector). Afterwards, we can find field from a potential

0

UV

q Units of potential are Volts [V]

1 Volt=1Joule/Coulomb

If an electric charge is moved by the electric field, the work done by the field

0 0

( )a ba b

W UV V

q q

Potential difference if often called voltage

Two equivalent interpretations of voltage:

1.Vab is the potential of a with respect to b, equals the work done by the electric force when a UNIT charge moves from a to b.

2. Vab is the potential of a with respect to b, equals the work that must be done to move a UNIT charge slowly from b to a against the electric force.

Potential due to the point charges

0

1

4

dqV

r Potential due to a continuous

distribution of charge

Finding Electric Potential through Electric Field

0

ba b

a ba

WV V E d l

q

Some Useful Electric Potentials

• For a uniform electric field

• For a point charge

• For a series of point charges

€

V = −r E • d

r l = −

r E • d

r l = −

r E •

r l ∫∫

rq

kV e

i

ie r

qkV

Potential of a point charge

Moving along the E-field lines means moving in the direction of decreasing V.

As a charge is moved by the field, it loses it potential energy, whereas if the charge is moved by the external forces against the E-field, it acquires potential energy

• Negative charges are a potential minimum

• Positive charges are a potential maximum

Positive Electric Charge Facts

• For a positive source charge– Electric field points away from a positive source charge

– Electric potential is a maximum

– A positive object charge gains potential energy as it moves toward the source

– A negative object charge loses potential energy as it moves toward the source

Negative Electric Charge Facts

• For a negative source charge– Electric field points toward a negative source charge

– Electric potential is a minimum

– A positive object charge loses potential energy as it moves toward the source

– A negative object charge gains potential energy as it moves toward the source

Unit: 1 Volt= 1 Joule/Coulomb (V=J/C)

Field: N/C=V/m

1 eV= 1.6 x 10-19 JJust as the electric field is the electric force per unit charge, the electrostatic potential is

the potential energy per unit charge.

Electron Volts

Electron volts – units of energy

abU eV

1 eV – energy a positron (charge +e) receives when it goes through the potential difference Vab =1 V

ExamplesA small particle has a charge -5.0 C and mass 2*10-4 kg. It moves from point A, where the electric potential is a =200 V and its speed is V0=5 m/s, to point B, where electric potential is b =800 V. What is the speed at point B? Is it moving faster or slower at B than at A?

A B

E

F

2 20

2 2a bmV mV

q q

~ 7.4 /bV m s

In Bohr’s model of a hydrogen atom, an electron is considered moving around a stationary proton in a circle of radius r. Find electron’s speed; obtain expression for electron’s energy; find total energy. 2 2

2e ee V

F k mrr

T K U 2

UK

115.3 10r m

13.6T eV

Calculating Potential from E field

• To calculate potential function from E field

€

V = −r E • d

r s

i

f

∫= − (Ex

ˆ i + Eyˆ j + E z

ˆ k )i

f

∫ • dxˆ i + dyˆ j + dz ˆ k ( )

− = Exdx + Eydy + E zdzi

f

∫

Generally, in electrostatics it is easier to calculate a potential (scalar) and then find electric field (vector). In certain situation, Gauss’s law and symmetry consideration allow for direct field calculations. Moreover, if applicable, use energy approach rather than calculating forces directly (dynamic approach)

When calculating potential due to charge distribution, we calculate potential explicitly if the exact distribution is known. If we know the electric field as a function of position, we integrate the field.

b

a

E d l

Example: Solid conducting sphere

Outside: Potential of the point charge

0

1

4

qV

r

Inside: E=0, V=const

Potential of Charged Isolated Conductor

• The excess charge on an isolated conductor will distribute itself so all points of the conductor are the same potential (inside and surface).

• The surface charge density (and E) is high where the radius of curvature is small and the surface is convex

• At sharp points or edges (and thus external E) may reach high values.

• The potential in a cavity in a conductor is the same as the potential throughout the conductor and its surface

At the sharp tip (r tends to zero), large electric field is present even for small charges.

Corona – glow of air due to gas discharge near the sharp tip. Voltage breakdown of the air

6max 3 10 /V V m max maxV RE

Lightning rod – has blunt end to allow larger charge built-up – higher probability of a lightning strike

Example: Potential between oppositely charged parallel plates

From our previous examples

0( )

( )

U y q Ey

V y Ey

abVE

d

Easy way to calculate surface charge density

0 abV

d

Remember! Zero potential doesn’t mean the conducting object has no charge! We can assign zero potential to any place, only difference in potential makes physical sense

Calculating E field from Potential

• Remembering E is perpendicular to equipotential surfaces

ˆˆ ˆ

x y z

E V

V V VE i j k

x y z

V V VE E E

x y z

Example: Charged wireWe already know E-field around the wireonly has a radial component

0

1;

2rEr

0

ln2

bb

aa

rE dr

r

Vb = 0 – not a good choice as it follows

Why so?

aV

We would want to set Vb = 0 at some distance r0 from the wire

0

0

ln2

rV

r

r - some distance from the wire

Example: Sphere, uniformly charged inside through volume

3' r

q QR

Q - total charge

Q

V - volume density of charge

( )r R r

R rR

E dr

eR

k Q

R

2

23

2e

rk Q r

R R

This is given that at infinity

rE03

R

R

2

3|

2re

R Rk Q r

R