Electric Charges,Forces, and

Fields

Electric ChargesElectric charge is a basic property of matterTwo basic charges

Positive and NegativeEach having an absolute value of

1.6 x 10-19 CoulombsExperiments have shown that

Like signed charges repel each otherUnlike signed charges attract each other

For an isolated system, the net charge of the system remains constant

Charge Conservation

Two basics type of materials

ConductorsMaterials, such as metals, that allow the free

movement of charges

InsulatorsMaterials, such as rubber and glass, that don’t

allow the free movement of charges

Coulomb’s LawCoulomb found that the electric force between two

charged objects is

Proportional to the product of the charges on the objects, and

Inversely proportional to the separation of the objects squared

2

21

r

qqkF

k being a proportionality constant, having a value of 8.988 x 109 Nm2/c2

Electric Force

12221

12 r̂r

qqkF

This gives the force on charged object 2 due to charged object 1

The direction of the force is either parallel or antiparallel to this unit vector depending upon the relative signs of the charges

12r̂ is a unit vector pointing from object 1 to object 2

As with all forces, the electric force is a Vector

So we rewrite Coulomb’s Law as

q2q1

Electric ForceThe force acting on each charged object has the same magnitude - but acting in opposite directions

(Newton’s Third Law)2112 FF

More Than Two Charges

q

q1

q2

qqF 1

qqF 2

netF

If q1 were the only other charge, we would know the force on q due to q1 -

qqF 1

If q2 were the only other charge, we would know the force on q due to q2 -

qqF 2

Given charges q, q1, and q2

What is the net force if both charges are present?

The net force is given by the Superposition Principle

21 FFFnet

Superposition of ForcesIf there are more than two charged objects

interacting with each otherThe net force on any one of the charged

objects is The vector sum of the individual Coulomb

forces on that charged object

ji

rr

qkqF ijij

ijj ˆ2

Note on constants

k is in reality defined in terms of a more fundamental constant, known as the permittivity of free space.

2

212

0

0

C 10854.8

4

1

Nmxwith

k

Electric FieldThe Electric Force is like the Gravitational Force

Action at a Distance

The electric force can be thought of as being mediated by an electric field.

What is a Field?A Field is something that can be defined anywhere in space

A field represents some physical quantity (e.g., temperature, wind speed, force)

It can be a scalar field (e.g., Temperature field)

It can be a vector field (e.g., Electric field)

It can be a “tensor” field (e.g., Space-time curvature)

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A Scalar Field

A scalar field is a map of a quantity that has only a magnitude, such as temperature

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A Vector Field

A vector field is a map of a quantity that is a vector, a quantity having both magnitude and direction, such as wind

Electric FieldWe say that when a charged object is put at a

point in space,The charged object sets up an Electric Field throughout the space surrounding the charged object

It is this field that then exerts a force on another charged object

Electric FieldLike the electric force,

the electric field is also a vector

If there is an electric force acting on an object having a charge qo, then the electric field at that point is given by

0q

FE

(with the sign of q0 included)

Electric FieldThe force on a positively charged object is in the same direction as the electric field at that point,

While the force on a negative test charge is in the opposite direction as the electric field at the point

Electric Field

A positive charge sets up an electric field pointing away from the charge

A negative charge sets up an electric field pointing towards the charge

Electric Field

rr

qkE ˆ

2

The electric field of a point charge can then be shown to be given by

ji

rrqk ij

ij

ijj qF ˆ2

Earlier we saw that the force on a charged object is given by

The term in parentheses remains the same if we change the charge on the object at the point in question

The quantity in the parentheses can be thought of as the electric field at the point where the test object is placed

Electric FieldAs with the electric force, if there are several

charged objects, the net electric field at a given point is given by the vector sum of the individual electric fields

i

iEE

Electric Field

rr

dqkE ˆ

2

If we have a continuous charge distribution the summation becomes an integral

Hints1) Look for and exploit symmetries in the

problem.2) Choose variables for integration carefully.3) Check limiting conditions for appropriate

result

Electric FieldRing of Charge

Electric FieldLine of Charge

Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above.

1) What is the direction of the electric field at point A?

a) up b) down c) left d) right e) zero

2) What is the direction of the electric field at point B?

a) up b) down c) left d) right e) zero

Example

Example Two charges, Q1 and Q2, fixed along the x-axis asshown produce an electric field, E, at a point(x,y) = (0,d) which is directed along the negativey-axis.

Which of the following is true?

Q2Q1

(c) E

Q2Q1

(b)

E

Q2Q1 x

y

Ed

(a) Both charges Q1 and Q2 are positive

(b) Both charges Q1 and Q2 are negative

(c) The charges Q1 and Q2 have opposite signs

E

Q2Q1

(a)

Electric Field Lines

Possible to map out the electric field in a region of spaceAn imaginary line that at any given point has its tangent being in the direction of the electric field at that point

The spacing, density, of lines is related to the magnitude of the electric field at that point

Electric Field Lines

At any given point, there can be only one field line

The electric field has a unique direction at any given point

Electric Field LinesBegin on Positive ChargesEnd on Negative Charges

Electric Field Lines

Electric Dipole

An electric dipole is a pair of point charges having equal magnitude but opposite sign that are separated by a distance d.

Two questions concerning dipoles:1) What are the forces and torques acting on a dipole when placed in an external electric field?2) What does the electric field of a dipole look like?

Force on a DipoleGiven a uniform external field

Then since the charges are of equal magnitude, the force on each charge has the same value

However the forces are in opposite directions!

Therefore the net force on the dipole is

Fnet = 0

Potential Energy of a Dipole

Given a dipole in an external field:Dipole will rotate due to torqueElectric field will do workThe work done is the negative of the change in potential energy of the dipole

The potential energy can be shown to be

EdqU

Electric Field of a Dipole