Electric Charges, Forces, and Fields

Fall 2008Lecture 1-1Physics 231

Electric Charges,Forces, and

Fields


Electric ChargesElectric charge is a basic property of matterTwo basic charges

Positive and Negative

Each having an absolute value of 1.6 x 10-19 Coulombs

Experiments have shown thatLike 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


Example 1A charged ball Q1 is fixed to a horizontal surface

as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1.

When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at a distance d23 (< d12) directly above Q3.

For 1a and 1b which is the correct answer

1a: A) The charge of Q3 has the same sign of the charge of Q1

B) The charge of Q3 has the opposite sign as the charge of Q1

C) Cannot determine the relative signs of the charges of Q3 & Q1

1b: A) The magnitude of charge Q3 < the magnitude of charge Q1

B) The magnitude of charge Q3 > the magnitude of charge Q1

C) Cannot determine relative magnitudes of charges of Q3 & Q1

Q2

Q1

gd12

Q2

d23

Q3


• To be in equilibrium, the total force on Q2 must be zero.• The only other known force acting on Q2 is its weight.• Therefore, in both cases, the electrical force on Q2 must be directed

upward to cancel its weight.• Therefore, the sign of Q3 must be the SAME as the sign of Q1

A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1.


1a: A) The charge of Q3 has the same sign of the charge of Q1

B) The charge of Q3 has the opposite sign as the charge of Q1

C) Cannot determine the relative signs of the charges of Q3 & Q1

Q2

Q1

gd12

Q2

d23

Q3

Example 1


• The electrical force on Q2 must be the same in both cases … it just cancels the weight of Q2

• Since d23 < d12 , the charge of Q3 must be SMALLER than the charge of Q1 so that the total electrical force can be the same!!

A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1.


1b: A) The magnitude of charge Q3 < the magnitude of charge Q1

B) The magnitude of charge Q3 > the magnitude of charge Q1

C) Cannot determine relative magnitudes of charges of Q3 & Q1

Q2

Q1

gd12

Q2

d23

Q3

Example 1


More Than Two Charges

q

q1

q2

qqF1

qqF2

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


What are the directions of the forces acting on q1?


Superposition of ForcesIf there are more than two charged objects

interacting with each other

The 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


Example Twoqo, q1, and q2 are all point charges where qo = -1C, q1 = 3C, and q2 = 4C

What are F0x and F0y ?

yFxFF ˆsinˆcos 202020

20

02

r

xx cos

20

20

r

yy sin

x (cm)

y (cm)

1 2 3 4 5

4

3

2

1

qo

q2q1

210

1010

r

qqkF yFF ˆ1010

220

2020

r

qqkF

202020 rFF ˆ

2010 FF

and calculate to Need

What is the force acting on qo?

We have that 20100 FFF

Decompose into its x and y components

20F


Example Two - Continued

10F

Now add the components of and to find and xF0 yF020F

cos200 FF x

xxx FFF 20100

010 xF

X-direction:

yyy FFF 20100

sin20100 FFF y

Y-direction:x (cm)

y (cm)

1 2 3 4 5

4

3

2

1

qo

q2q1

20F

10F

0F


Example Two - Continued

N52.110 xF N64.380 yF

N32.4020

200 yx FFFThe magnitude of is0F

Putting in the numbers . . .

cm310 r cm520 r

8.0cos

N4.1420 FN3010 F

We then get for the components

At an angle given by

40.73)52.11/64.38(tantan 100

1 xy FF

x (cm)

y (cm)

1 2 3 4 5

4

3

2

1

qo

q2q1

20F

10F

0F


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 Field

The 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)


7782

8368

5566

8375 80

9091

7571

80

72

84

73

82

8892

77

8888

7364

A Scalar Field

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


77

82

8368

5566

8375 80

9091

7571

80

72

84

73

57

8892

77

5688

7364

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 Field

We 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 Field

The 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 Field

As 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


Hints

1) 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 3


Example 4Two 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



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

The electric field has a unique direction at any given point


Begin on Positive Charges

End on Negative Charges




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


Torque on a Dipole

The individual forces acting on the dipole may not necessarily be acting along the same line.

If this is the case, then there will be a torque acting on the dipole, causing the dipole to rotate.


Torque on a Dipole

Edq

The torque is then given by = qE dsin

d is a vector pointing from the negative charge to the positive charge


Potential Energy of a Dipole

Given a dipole in an external field:

Dipole will rotate due to torque

Electric field will do work

The 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

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Electric Charges, Forces, and Fields