Electric Charges, Forces, and Fields - UTK Department of … · 2008-11-05 · Electric Charges,...

Fall 2008Lecture 1-1Physics 231

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 materialsConductors

Materials, 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 isProportional to the product of the charges on the objects, andInversely proportional to the separation of the objects squared

221

rqq

kF =

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


Electric ForceAs with all forces, the electric force is a Vector

12221

12 r̂rqqkF =

r

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

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

So we rewrite Coulomb’s Law as

q2q1

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


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

2112 FFrr

= (Newton’s Third Law)


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

as shown. When another massive chargedball 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


Example 1Q2

Q1

gd12

Q2

d23

Q3

A charged ball Q1 is fixed to a horizontal surface as shown. When another massive chargedball 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

• 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


Example 1Q2

Q1

gd12

Q2

d23

Q3

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

• 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 Q1so that the total electrical force can be the same!!


More Than Two ChargesGiven charges q, q1, and q2

q

q1

q2

qqF1

r

qqF2

rnetFr

If q1 were the only other charge, we would know the force on qdue to q1 - qqF

1

r

If q2 were the only other charge, we would know the force on qdue to q2 - qqF

2

r

What is the net force if both charges are present?

The net force is given by the Superposition Principle

21 FFFnetrrr

+=


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

rrqkqF ijij

ijj ˆ2

r


Example Two

x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1 θ

qo, q1, and q2 are all point charges where qo = -1µC, q1 = 3µC, and q2 = 4µC

What is the force acting on qo?

We have that 20100 FFFrrr

+=

What are F0x and F0y ?

210

1010

rqqkF = yFF ˆ1010 −=

r

220

2020

rqqkF =

202020 rFF ˆ−=r

2010 FFrr

and calculate to NeedDecompose into its x and ycomponents

20Fr

( ) ( ) yFxFF ˆsinˆcos θθ 202020 −=r

20

02r

xx −=θcos

20

20r

yy −=θsin


Example Two - Continued

10FrNow add the components of and to find and xF0 yF020F

r

X-direction:

θcos200 FF x =

xxx FFF 20100 +=

010 =xF

yyy FFF 20100 x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1

20Fr

10Fr

0Fr

Y-direction: +=

θsin20100 FFF y −−=


Example Two - ContinuedPutting in the numbers . . .

x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1

20Fr

10Fr

0Fr

cm310 =r cm520 =r

8.0cos =θ

N4.1420 =FN3010 =F

We then get for the componentsN52.110 =xF N64.380 −=yF

The magnitude of is0Fr

N32.4020

200 =+= yx FFF

At an angle given by

( ) o40.73)52.11/64.38(tantan 100

1 −=== −−xy FFθ


Note on constantsk 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

41

Nmxwith

k

−=

=

ε

πε


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)


A Scalar Field

7782

8368

5566

8375 80

90 91

757180

72

84

73

82

8892

7788

887364

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


A Vector Field77

82

8368

5566

8375 80

90 91

757180

72

84

73

57

8892

7756

887364

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

0qFEr

r= (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

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

= ∑≠ ji

rrqk ijij

ijj qF ˆ2

rEarlier we saw that the force on a charged object is given byThe 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

rrqkE ˆ2=

r

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


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

iEErr


Electric FieldIf we have a continuous charge distribution the summation becomes an integral

∫= rrdqkE ˆ2

r


Hints

1) Look for and exploit symmetries in the problem.

2) Choose variables for integrationcarefully.

3) Check limiting conditions for appropriate result


Electric FieldRing of Charge


Electric FieldLine of Charge


Example 3

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 4

Q2Q1 x

y

Ed

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?

(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)

Q2Q1

(b)E

Q2Q1

(c) E


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

Electric Field LinesBegin on Positive ChargesEnd 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

( )Edqrrr ×=τ

The torque is then given by τ = qE dsinφ

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


Potential Energy of a DipoleGiven 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

( )EdqUrr

⋅−=


Electric Field of a Dipole

Electric Charges, Forces, and Fields - UTK Department of … · 2008-11-05 · Electric Charges,...

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