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Physics 2220. Physics for Scientists and Engineers II. Chapter 23: Electric Fields. Materials can be electrically charged. Two types of charges exist: “ Positive ” and “ Negative ”. Objects that are “charged” either have a net “positive” or a net “negative” charge residing on them.

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physics 2220

Physics 2220

Physics for Scientists and Engineers II

Physics for Scientists and Engineers II , Summer Semester 2009

chapter 23 electric fields
Chapter 23: Electric Fields
  • Materials can be electrically charged.
  • Two types of charges exist: “Positive” and “Negative”.
  • Objects that are “charged” either have a net “positive” or a net “negative” charge residing on them.
  • Two objects with like charges (both positively or both negatively charged) repel each other.
  • Two objects with unlike charges (one positively and the other negatively charged) attract each other.
  • Electrical charge is quantized (occurs in integer multiples of a fundamental charge “e”).

q =  N e (where N is an integer)

electrons have a charge q = - e

protons have a charge q = + e

neutrons have no charge

Physics for Scientists and Engineers II , Summer Semester 2009

material classification according to electrical conductivity
Material Classification According to Electrical Conductivity
  • Electrical conductors: Some electrons (the “free” electrons) can move easily through the material.
  • Electrical insulators: All electrons are bound to atoms and cannot move freely through the material.
  • Semiconductors: Electrical conductivity can be changed over several orders of magnitude by “doping” the material with small quantities of certain atoms, making them more or less like conductors/insulators.

Physics for Scientists and Engineers II , Summer Semester 2009

shifting charges in a conductor by induction

+

+

+

+

+

+

+

+

+

+

-

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-

-

-

-

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Shifting Charges in a Conductor by “Induction”

uncharged metal sphere

Negatively charged rod

Left side of metal sphere

more positively charged

Right side of metal sphere

more negatively charged

Physics for Scientists and Engineers II , Summer Semester 2009

coulomb s law charles coulomb 1736 1806
Coulomb’s Law (Charles Coulomb 1736-1806)

Magnitude of force between two “point charges” q1 and q2 .

Coulomb constant

r = distance between

point charges

Permittivity of free space

Physics for Scientists and Engineers II , Summer Semester 2009

charge
Charge

Unit of charge = Coulomb

Smallest unit of free charge: e = 1.602 18 x 10-19 C

Charge of an electron: qelectron = - e = - 1.602 18 x 10-19 C

Physics for Scientists and Engineers II , Summer Semester 2009

vector form of coulomb s law
Vector Form of Coulomb’s Law

Force is a vector quantity(has magnitude and direction).

unit vector pointing from

charge q1to charge q2

Force exerted by charge q1on charge q2

(force experienced by charge q2 ).

Physics for Scientists and Engineers II , Summer Semester 2009

vector form of coulomb s law1
Vector Form of Coulomb’s Law

Force is a vector quantity(has magnitude and direction).

unit vector pointing from

charge q2to charge q1

Force exerted by charge q2on charge q1

(force experienced by charge q1 ).

Physics for Scientists and Engineers II , Summer Semester 2009

directions of forces and unit vectors

+

+

+

Directions of forces and unit vectors

q2

-

q2

q1

q1

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the resultant forces on charge q 1 in a configuration of 3 charges

+

+

Calculating the Resultant Forces on Charge q1 in a Configuration of 3 charges

a = 1cm

q1

q2

0.5 cm

0.5 cm

q3

-

q3 = - 2.0 mC

q1 = q2 =+2.0 mC

Physics for Scientists and Engineers II , Summer Semester 2009

forces acting on q 1

q1

q2

-

+

+

Forces acting on q1

q3

Total force on q1:

Physics for Scientists and Engineers II , Summer Semester 2009

magnitude of the various forces on q 1
Magnitude of the Various Forces on q1

Note: I am temporarily carrying along extra significant digits in these

intermediate results to avoid rounding errors in the final result.

Physics for Scientists and Engineers II , Summer Semester 2009

adding the vectors using a coordinate system

q1

q2

-

+

+

Adding the Vectors Using a Coordinate System

y

q3

x

Physics for Scientists and Engineers II , Summer Semester 2009

adding the vectors using a coordinate system1
Adding the Vectors Using a Coordinate System

y

x

Physics for Scientists and Engineers II , Summer Semester 2009

doing the algebra
…doing the algebra…

F1 has a magnitude of

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 another example using an even more mathematical approach
Calculating the force on q2 … another example using an even more mathematical approach

Charges

Location of charges

q1 = +3.0 mC

x1=3.0cm ; y1=2.0cm ; z1=5.0cm

q2 = - 4.0 mC

x2=2.0cm ; y2=6.0cm ; z2=2.0cm

In this example, the location of the charges and the distance

between the charges are harder to visualize 

Use a more mathematical approach!

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 another example using an even more mathematical approach1
Calculating the force on q2 … another example using an even more mathematical approach

d12=distance between q1 and q2.

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 mathematical approach

-

+

Calculating the force on q2 … mathematical approach

We need the distance between the charges.

d12 is distance between q1 and q2.

y

q1

q2

x

z

Physics for Scientists and Engineers II , Summer Semester 2009

slide19

Calculating the force on q2 … mathematical approach

Distance between charges q1 and q2 .

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 mathematical approach1

-

+

Calculating the force on q2 … mathematical approach

We need the unit vectors between charges. For example, the unit

vector pointing from q1to q2 is easily obtained by normalizing the

vector pointing from from q1 to q2.

y

q1

q2

x

z

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 mathematical approach2
Calculating the force on q2 … mathematical approach

The needed unit vector:

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 mathematical approach3
Calculating the force on q2 … mathematical approach

You can easily verify that the length of the unit vector is “1”.

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 another example using an even more mathematical approach2
Calculating the force on q2 … another example using an even more mathematical approach

Physics for Scientists and Engineers II , Summer Semester 2009

calculating the force on q 2 another example using an even more mathematical approach3
Calculating the force on q2 … another example using an even more mathematical approach

…and if you want to know just the magnitude of the force on q2:

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field
23.4 The Electric Field

It is convenient to use positive test charges. Then, the direction of

the electric force on the test charge is the same as that of the field

vector. Confusion is avoided.

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field1
23.4 The Electric Field

Q

qo

+ +

+

+ +

test charge

+ +

+ +

Source charge

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field2
23.4 The Electric Field

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of a point charge q
23.4 The Electric Field of a “Point Charge” q

q0

r

q

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of a positive point charge q
23.4 The Electric Field of a Positive “Point Charge” q

(Assuming positive test charge q0)

q0

+

Force on test charge

P

+

Electric field where test charge

used to be (at point P).

The electric field of a positive point charge points away from it.

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of a negative point charge q
23.4 The Electric Field of a Negative “Point Charge” q

(Assuming positive test charge q0)

q0

-

Force on test charge

P

-

Electric field where test charge

used to be (at point P).

The electric field of a negative point charge points towards it.

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of a collection of point charges
23.4 The Electric Field of a Collection of Point Charges

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p
23.4 The Electric Field of Two Point Charges at Point P

y

P

y

q1

q2

x

a

b

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p1
23.4 The Electric Field of Two Point Charges at Point P

y

P

Pythagoras:

r1

r2

y

a

b

x

q1

q2

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p2
23.4 The Electric Field of Two Point Charges at Point P

y

P

x

q1

q2

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p3
23.4 The Electric Field of Two Point Charges at Point P

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p4
23.4 The Electric Field of Two Point Charges at Point P

Special case: q1= q and q2 = -q AND b = a

E from + charge

E from - charge

-

+

q

-q

Physics for Scientists and Engineers II , Summer Semester 2009

23 4 the electric field of two point charges at point p5
23.4 The Electric Field of Two Point Charges at Point P

Special case: q1= q and q2 = q AND b = a

E from other

+ charge

E from + charge

+

+

q

q

Physics for Scientists and Engineers II , Summer Semester 2009

this is called an electric dipole
This is called an electric DIPOLE

Special case: q1= q and q2 = -q AND b = a

E from + charge

E from - charge

-

+

q

-q

For large distances y (far away from the dipole), y >> a:

E falls off proportional to 1/y3

Fall of faster than field of single charge (only prop. to 1/r2).

From a distance the two opposite charges look like they are

almost at the same place and neutralize each other.

Physics for Scientists and Engineers II , Summer Semester 2009