February 16, 2010

Potential Difference and Electric Potential

Conservative forces

The electrostatic force is conservative.What characterizes a conservative force?

Path independence of work.Work along a closed path is zero.Work is related to potential energy change.

Electrical Potential Energy

Like all other forms of potential energy, the change in electrical potential energy is equal to the negative of the work done by the conservative electrical force.

Ue = -We

Field Refresher

+++++++++++++++

---------------

HighPotential(voltage)

LowPotential(voltage)

Electric Field

Equi-potential Surfaces

+

A

+

B

+C

Work Done by Electrical FieldThe work done by the electrical force in moving a

charge q from point A to point B is given by:

This results in a potential energy change of

uniform field

non-uniform field

B B

A A

W F r qE r

W d q d

F r E r

uniform field

non-uniform field

B

A

U qE r

U q d

E r

Electric Potential

The electric potential is defined to be the potential energy per unit charge. Thus...

or U U

V Vq q

q E drV E dr

q

Sample ProblemA proton is accelerated through a potential difference

of -20,000 V. What is the potential and kinetic energy change of the proton? How much work did the electric field do on the proton?

Sample ProblemHow fast is a 2 MeV alpha particle moving? What potential

difference is needed to stop this alpha particle?

January 20, 2009

Potential Differences in a Uniform Electric Field

How electrostatic concepts are related

Field and Force are closely relatedBoth are vectorsF = qE

Potential and Potential Energy are closely relatedBoth are scalarsU = qV

Electric Potential in Uniform Field

B B

A A

U q d qE dr qE r E r =

B B

A A

V d E dr E r E r =

Sample Problem

An electric field is given by E = 250 i V/m. A 12.0 C charge moves from the origin to (0.20 m, 0.50 m). What is the change in potential energy and electric potential for the charge?

Sample Problem

An electron is released from rest in an electric field of magnitude by 5900 V/m. Through what potential difference will it have passed after moving 1.00 cm? How fast will it be moving at this time?

January 21, 2009

Electric Potential and Potential Energy due to Point Charges

Point Charge Concept Map(turn to back page of packet)

Force

Field Potential

Potential Energy

Potential Energy Graphs(turn to back page of packet)

r

U

gravityElectric+ and -

Electric(like charges)

Thursday, January 22, 2009

Obtaining the Electric Field due to the Electric Potential

Equi-potential or Iso-potential surfaces

Definition:

Characteristics of equi-potential surfaces:

Equipotential Surfaces: + charge

+

Sample Problem

Find the potential at a distance of 1.00 cm from a proton. Repeat for an electron.

Sample Problem

Two point charges (5.0 nC and -3.0 nC) are separated by 35 cm. What is the potential energy of the pair? What is the electric potential at a point midway between the charges?

Obtaining the Field from Potential

V E r dV Edr

r

dVE

dr

V V V

x y z

E i j k

The field is strongest where the potential is changing most rapidly.

Sample Problem The electric potential in a region is given by the function

V = -9.0 x V/m – 3.0 x2 V/m2.

What is the magnitude and direction of the electric field at x = 2.0 m?

Sample Problem Over a region of space, the electric potential is given by

V = -3.0 x + 6.0 x y – 2.0 x2y.

Derive the vector representing the electric field at (1, 2) m.

January 26, 2009

Potential Due To Continuous Charge Distribution I

Potential is a Scalar Sum

V = Vi

The potential at a point in space is due to the sum of the potentials due to individual charges.

V = dVWhen these charges are infinitesimally small

and in a continuous distribution, an integral is needed.

dqV k

r

Sample Problem: Determine the electric potential at a point P located on the perpendicular axis of a uniformly charged ring of radius R and total charge Q.

Tuesday, January 26, 2009

Electric Potential Due to Charged Conductors

Sample Problem: Determine the electric potential along the perpendicular central axis of a uniformly charged disk of radius R and surface charge density .

Sample Problem: Find the potential at a point inside a charged non-conducting solid sphere of radius R as a function of its distance from the center of the sphere. Assume charge Q is distributed uniformly.

Conductors

Charged conductors store all excess charge on the exterior of the conductor, whether the conductor is hollow or solid.

Charge will rearrange itself on a neutral conductor immersed in an external electric field in order to nullify the field.

Wednesday, January 27, 2009

Defining and Calculating Capacitance

Sample ProblemTwo charged conductors are connected by a long conducting wire, and a charge of 10 nC is placed on the combination. Sphere A has a diameter of 10 mm, and sphere B has a diameter of 5 mm. How much charge is on each sphere? What is the electric potential of each sphere?

What is a Capacitor?

A capacitor is a device designed to store electrical energy.

It consists of two conducting “plates” in close proximity.

When “charged”, there is a voltage across the plates, and they bear equal and opposite charges.

dielectric

Parallel Plate Capacitor

E

+Q

-Q

V1V2

V3

V4

V5

Cylindrical Capacitor

-Q

+Q

Capacitance

C = Q / VC: capacitance in FaradsQ: charge (on positive plate) in CoulombsV: potential difference between plates in Volts

The capacitance of a capacitor depends upon its geometry and whether or not there is a dielectric material inserted between the plates.

The larger the capacitance, the more charge the capacitor can hold at a given voltage.

Thursday, January 29, 2009

Deriving Capacitance

Equivalent Capacitance

Energy in Capacitors

Sample Problem

How much charge is on each plate of a 4.00 F capacitor when connected to a 12-V battery?

Parallel Plate Capacitor

A parallel plate capacitor can be built so as to have a given capacitance.

C = o A / d : dielectric constant of the filling materialC: Capacitance (Farads) o: electrical permittivity of free space

8.85 x 10-12 F/m

A: area of one plate (m2)d: distance between places (m)

Deriving Capacitance -- Steps

1. Draw the capacitor; identify symmetry

2. Draw Gaussian surface

3. Write Gauss’ Law

4. Solve Gauss’ Law for E

5. Develop function for V from E

6. Develop function for C from V

Derive C for Parallel Plate Capacitor

Derive C for Charged Sphere

Derive C for a Spherical Capacitor

Derive C for a Cylindrical Capacitor

Capacitors in Circuits

+Q -Q +Q -Q

Equivalent Capacitance - Series

C1 C2 C3

1/Ceq = 1/C1+ 1/C2 + 1/C3

Series circuit

Capacitors in SeriesCapacitors in series all have the same

charge.If the capacitor combination has charge Q,

then each capacitor in the series also has charge Q.

This is because capacitors must obtain charge from adjacent capacitors.

The voltage difference across the combination varies inversely with the capacitance according to V = Q/C

Equivalent Capacitance - parallel

Ceq = C1+ C2 + C3

Parallel circuit

C2

C1

C3

Capacitors in Parallel

Capacitors in parallel all have the same potential difference across their plates.

If the capacitor combination has charge Q, then the charges on all capacitors in the combination will sum to equal Q.

The larger the capacitance, the larger the fraction of the total charge on that capacitor according to Q = CV.

Sample problem: Determine equivalent capacitance of the configuration shown

C C

C

C CC

Sample problem: Determine equivalent capacitance between a and b. If the potential difference between a and b is 10V, what charge is stored on C3? (C1=5F, C2=10F, C3=2F)

a

b

C1 C1

C2 C2

C2 C2

C3

Announcements

Chapter 26, problems 23, 25, 27

Energy in Capacitors

Capacitors “store” electrical energy by creating an electric field between their plates. This is a type of potential energy.

C: capacitance V: voltageUE: electric potential energy

221 1

( )2 2E

qU C V

C

Sample Problem

A 3.00 mF capacitor is connected to a 12.0 V battery. How much energy is stored in the capacitor?

Dielectrics in Capacitors

Dielectrics are “fillings” that are inserted between the plates of a capacitor and increase its capacitance, or ability to separate charge.

C = Co

Dielectrics are polar non-conductors that work by decreasing the field strength between the plates. The non-conducing molecules orient themselves such that their electric dipoles subtract from the field produced by the charged plates.

Since the field is reduced, the voltage between the plates is reduced.

Sample Problem

Find the capacitance of a parallel plate capacitor that uses Bakelite as a dielectric, if each of the plates has an area of 5.0 cm2 and the plate separation is 2.00 mm.