The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

The Electric Potential, Electric Potential Energy and

Energy Conservation

Electric Potential Energy U is the energy of a charged object in an external electric field (Unit Joule J)

In electricity, it is usually more convenient to use the

Electric Potential V (or Voltage) is electric potential energy per unit charge, measured in joules per coulomb (Unit Volt V)

A charged particle q exerts an electric force (a conservative force) in an electric field. If the charge q is not static then work is done in an electric field by moving the charge q. The work is proportional to the magnitude of charge (More work is needed to move a bigger charge). This is similar to lift/drop an object against/with gravity.

F=q0E F=mg

∆U=Eel=qEd (W=Fs cosθ and ∆U=-W)

(when E-field is homogeneous, E is constant vector everywhere)

∆U=Epot =mgh

Electric Potential Energy Gravitational Potential Energy The electric potential energy increase/decrease in the same

way as like the gravitational energy.

Only differences in potential energy are important

Definition:

The electric potential is electric potential energy per charge (like we derived the electric field).

V = U/q

The change in electric potential energy per charge is known as

potential difference or voltage and is measured in Volt. Only

changes of electric potential are meaningful.

∆V = ∆U/q

= -W/q

1V [Volt] =1 Nm/C

When we dealing with small charges, there is another unit for energy: electron volt = energy of one electron possesses when it

has moved through a potential difference of 1 Volt

1 eV = 1.602 10-19 _{J [Joule]}

Connection between electric field and electric

potential (E=const.)

∆V = ∆U/q

= -W/q

= q

Ed/q

= Ed

E = ∆V/d=∆V/∆s

If E=const. The change in the potential per distance is constant Example: Parallel Plate Capacitor For calculating physical quantities it is the difference in potential which has significance, not the potential itself. Therefore, we may choose any arbitrary point as having zero potential

Energy conservation

When we move a charge in an electric field the total energy – the electric force is conservative – must be conserved.

(Ekin + Eel)initial = (Ekin+Eel)final

(½ mv2_+U)

initial=(½ mv2+U)final

(½ mv2_+qV)

initial=(½ mv2+qV)final

½ mv2

final=(½ mv2)initial + q(Vfinal-Vinitial)

v2

final=v2initial+ 2q(∆V)/m

Rules:

positives charges accelerate in the direction of decreasing electrical potential

negative charges in the opposite direction

In both cases the charge moves to a region with a lower potential energy

The Electric Potential of Point Charges

r

q

q

k

F

=

If we release test charge it will move (accelerate) away At point B: UA-UB=(qEd)A-(qEd)B

= (Fd)A-(Fd)B= kq0q/rA- kq0q/rB

or VA-VB= (UA-UB)/ q0= kq/rA- kq/rB if r ∞ then VB=0

Definition:

] [V

r kq V =

The electric potential energy for a point

charge ] [ 0 0 J r q q k V q U= =

Superposition of two or more point charges

The potential at a given point in space due to several charges can be found by adding the potential from each charge

...

+

+

+

=

r

q

k

r

q

k

r

q

k

V

Another way to draw electric potentials: Contour Maps

The denser the lines the more great is the change in the potential (Equipotential surfaces, Equipotential lines) The electric field points always in the direction of the

decreasing lines

No work is required to move a charge along an equipotential surface, the electric field at every point on an equipotential

The ideal conductor

The electric field is stronger where the conductor is sharply curved. Why?

Let’s assume we have to charged metal spheres both with the same electric potential.

The potential of a metal sphere is given by

R kQ V =

The electric field is given by

2

R kQ E=

If both spheres have the same potential Large sphere: π σ π σ π σ π σ 4 4 4 4 2 2 2 k R R k E R k R R k V = = = = Small sphere: 2 4 ? 2 4 4 ? 2 R k R R k V = π = π

Therefore: ?=σsmall sphere= 2σ higher charge density

π σ π σ 4 2 4 4 4 2 2 2 k R R k E= =

Capacitors

A capacitor is a device that stores energy associated with a configuration of charges. In general, a capacitor consists of two conductors, one with a charge +Q and the other with a charge -Q.

Capacity is the ability to store charge and energy.

The capacitance C is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference

between the conductors:

Q = C V

or C = Q/V

C: Capacitance [F = Farad]

The parallel-plate capacitor

We can calculated according to Gauss law for a parallel plate capacitor the electric field as

d A A Qd Q C A Qd V A Q A Q E 0 0 0 0 0 0 ε ε ε ε ε ε σ = = = = = =

We can calculate the potential as V=Ed

Capacitance

Dielectrics

A way to increase the capacitance is with an insulting material between the plates – a dielectric

Field lines terminate on negative charge and start at positive charge on polarized slab. Less field lines are going through the entire capacitor.

We reduce E-field by a material dependent factor – the dielectric constant κ κ0 E E= Vacuum κ=1; H2O= 80.4 We know κ κ0 0 V d E Ed

V = = = V decrease with increasing κ

0 0 0 C V Q V Q V Q C κ κ κ = = =

= C increase with increasing κ We can expand the equation for the parallel capacitor:

κ

ε

d

A

C

=

=> Capacitance dependence also on the surface area A of the plates and the distance d. That how a keyboard works

A dielectric breakdown appears when the Electric field is too

strong. That means strong enough to tear atoms apart.

Next time you walk across a carpet and get a shock from the doorknob think about the fact that you just have produced an electric field of roughly 3 million V/m.

Dielectric strength (V/m) Substance 3.0 x 106 Air 12 x 106 Neoprene rubber 14 x 106 Pyrex glass 16 x 106 Paper 60 x 106 Teflon 100 x 106 Mica

Electrical Energy Storage

Situation: Parallel plate capacitor with a charge Q on its plates and potential difference of V. Now we transfer small packages of charge from one play to the other. The change in electric potential energy would be ∆U = ∆Q V

Attention! V is not constant. If we would transfer another charge package we have to calculate with another V

U= QVav = ½ QV

With Q = CV U = ½ CV2 V = Q/C U = ½ Q2_/C

For a parallel plate capacitor Q = ε0 E A and V = Ed

Therefore U = ½ Q V = ½ (ε0 E A) (Ed)= ½ ε0 E2 A d

A⋅d is the volume between the plates so we can define a density (i.e. a quantity divided by its volume)

Electric energy density