Electrical Potential Energy & Electrical Potential Work, Potential Energy and the Electric Force ...

Electrical Potential Energy & Electrical Potential

Work, Potential Energy and the Electric Force

Electrical Potential

Equipotential Surfaces

Dielectrics, Capacitors, and Energy



Electrical Potential Equipotential Surfaces Dielectrics, Capacitors, and Energy

Work, Potential Energy, and the Electric Force

Consider a positive charge q, moving west through a displacement S at a constant velocity in a region occupied by a uniform eastward electric field E.

S

E

+ q


The charge experiences an eastward electric force. For it to move at constant velocity, an equal westward force is required.

S

E

+ q

F

F


Work done by the westward force:

Work done by the electrical force:

Net work = kinetic energy = zero

S

E

+ q

F

F

qESqESWwest 0cos

qESqESWelec 180cos


The electric force is conservative. Its work does not change the energy of the object.

The westward force is nonconservative. Its work changes the total energy:

S

E

+ q

F

F

qESWE west


The increase, qES, in the energy of the object is

electrical potential energy (EPE).

Like any other form of energy, it has SI units of joules (J).

Every conservative force is associated with a form of potential energy.

S

E

+ q

F

F


The work done by the forces is the same, even if we go from the starting to the ending point by a meandering path. Work done by a conservative force is path-independent.

S

E

+ q

F

F


The increase in electrical potential energy (EPE) is also unaffected by our choice of path. EPE depends only on E, q, and the endpoint locations.

S

E

+ q

F

F


Call the endpoints of our travel A (beginning) and B (end).

Travel from A to B with a “test charge” q0.

Divide through by q0 :

ABABAB EPEEPEEPEW

0000 q

EPE

q

EPE

q

EPE

q

W ABABAB


Define a new quantity, electrical potential:

and

SI units of electrical potential:

0000 q

EPE

q

EPE

q

EPE

q

W ABABAB

0q

WV

ABABAB VVVq

W

0

(V) volt coulomb

joule




Equipotential Surfaces Dielectrics, Capacitors, and Energy


Since we’ve talked about the electrical potential of points A and B, let’s look at what that means. Consider a positive point charge, Q, and a point A that is a distance r from Q:

ABABAB VVVq

W

0

r

Y

X

x = r

+ Qpoint A


What is the electrical potential at point A?

To answer that question, we need to know how much work is required to bring a test charge from a point infinitely distant from Q, to point A.

r

Y

X

x = r

+ Qpoint A


The force required to move the test charge at a constant velocity is not constant with the distance x from Q:

In fact, the force isn’t even linear with x … so we can’t calculate and use an average force.

r

Y

X

x = r

+ Qpoint A

20

x

QqkF


Calculating the potential at point A is a calculus problem. The work required to move the test charge an infinitesimal distance dx is dW:

r

Y

X

x = r

+ Qpoint A

dxx

QqkFdxdW

20


Integrate to calculate W:

r

kQ

q

WV

r

Qkq

rQkqdx

xQkqW

dxx

QqkFdxdW

r

0

0020

20

111


The potential of a point is relative to zero potential, which is located infinitely far away.

When more than one charge is present, the potential is the algebraic sum of the potentials due to each of the charges, individually.

“Electric potential” is not the same thing as “electrical potential energy.”


Gravitational Electrical

Another way to express the magnitude of the electric field: as a “potential gradient:”

ghm

W

mghW

distanceforce

ESVq

W

SqEW

distanceforce

er)(volts/met S

VE


Work, Potential Energy and the Electric Force Electrical Potential




Equipotential surface: a collection of points that all have the same electrical potential.


The electric field vector is everywhere perpendicular to equipotential surfaces. Why?


The electric force does no work on charges moving on an equipotential surface. Why?


Work, Potential Energy and the Electric Force Electrical Potential Equipotential Surfaces



Consider a parallel-plate capacitor, with a charge q and a plate area A, the plates separated by a distance d. The internal electric field:

A constant electric field E, and a distance d between the plates, gives the potential difference between the plates:

A

qE

0

A

qdEdV

0


Solve for the ratio of the charge to the voltage, and define a new quantity, capacitance:

SI units: coulomb/volt = C2/J = farad (F)

A

qdEdV

0

d

A

V

qC 0


Dielectric material: fancy term for an insulating material.

Separation of surface charge causes a reduction of the internal electric field by a factor of , the dielectric constant:

E

E0


Fill the capacitor with a dielectric:

Its capacitance is increased by a factor of , the dielectric constant.

d

A

V

qC 0

A

qE

0

A

qdEdV

0


Springs Capacitors

Work is required to compress a spring

Work is required to charge a capacitor

Work per unit length increases linearly with compression

Work per unit charge increases with the amount of charge

Energy: Energy:

2

2

1kxE 2

2

1CVE

Electrical Potential Energy & Electrical Potential Work, Potential Energy and the Electric Force ...

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