Law of Electrical ForceCharles-Augustin

Coulomb(1785)" The repulsive force between two small spheres

charged with the same sort of electricity is in the inverse ratio of the squares of the distances between the centers of the spheres"

1 22

q qFr

q2q1

r

What We CallCoulomb's

Law

MKS Units:• r in meters• q in Coulombs• in Newtons• is a unit vector

pointing from 1 to 2• This force has same spatial dependence as the gravitational

force, BUT there is NO mention of mass here!! • The strength of the force between two objects is determined by

the charge of the two objects, and the separation between them.

q2q1

The force from 1 acting on 2

12212

21

012 ˆ

41 r

rqqF

F

We call this group of constants “k” as in:

1

4 = 9 · 109 N-m2/C2

0

221

rqqkF

12r

12r̂

Coulomb Law Qualitativeq2

r

q1

• What happens if q1 increases?

F (magnitude) decreases

F (magnitude) increases

• What happens if q1 changes sign ( + - )?

1

The direction of is reversed

• What happens if r increases?

F

• 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.

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

Q2

Q1

gd12

Q2

d23

Q3

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

• To be in equilibrium, the total force on Q2 must be zero.• The only other known (from 111) 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

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.

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

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

• 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.

• When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at distance d23 (< d12) directly above Q3.1b: 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

Gravitational vs. Electrical Force

FelecFgrav

= q1q2m1m2

140G

r

F Fq1

m1

q2

m2

Felec = 1

40 q1q2r2

Fgrav = G m1m2r 2

®

* smallest charge seen in nature!

For an electron:* |q| = 1.6 ´ 10-19 Cm = 9.1 ´ 10-31 kg ® F

Felec

grav ´ +417 10 42.

Notation For Vectors and ScalarsVector quantities are written like this : F, E, x, rTo completely specify a vector, the magnitude (length) and direction must be known.

r rˆˆ

Where Fx, Fy, and Fz (the x, y, and z components of F ) are scalars.

A unit vector is denoted by the caret “^”. It indicates only a directionand has no units.

zFyFxFF zyx ˆˆˆ ++

The magnitude of is ; this is a scalar quantity

The vector can be broken down into x, y, and z components:

221

rqqkFF

F

F

rrqqkF ˆ2

21

For example, the following equation shows specified in terms of , q1, q2, and r :

F

r̂

ˆ rrr

Vectors: an Example

Now, plug in the numbers.

F12 = 67.52 N cm 2.828cm 8 r

To do this, use Coulomb’s Law: 221

12 rqqkF

221

22121 )()( where yyxxrrr +

q1 and q2 are point charges, q1 = +2mC and q2 = +3 mC. q1 is located at and q2 is located at

Find F12 (the magnitude of the force of q1 on q2).

)2cm,1cm(1 r

)4cm,3cm(2 r

y (cm)

x (cm)1 2 3 4

321 q1

q2r

Vectors: an Example continued

Now, find Fx and Fy, the x and y components of the force of q1 on q2.

y (cm)

x (cm)1 2 3 4

321 q1

q2rq

qcos12FFx rxx 12cos

q

qsin12FFy ryy 12sin

q

Symbolically Now plug in the numbers

Fx = 47.74 N

Fy = 47.74 N7071.22sin q

7071.22cos q

What happens when youconsider more than two charges?

• What is the force on q when both q1 and q2 are present??– The answer: just as in mechanics, we have the

Law of Superposition:• The TOTAL FORCE on the object is just

the VECTOR SUM of the individual forces.

2

F®

F1®

F2®

q

+q1

+q2

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

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

F®

= F1®

+ F2®

•Two balls, one with charge Q1 = +Q and the other with charge Q2 = +2Q, are held fixed at a separation d = 3R as shown.

+2Q

+2Q

Q2Q1 3R

+Q

R

Q2Q1

+QQ3

2R

(a) The force on Q3 can be zero if Q3 is positive.(b) The force on Q3 can be zero if Q3 is negative.(c) The force on Q3 can never be zero, no matter what

the (non-zero!) charge Q3 is.

• Another ball with (non-zero) charge Q3 is introduced in between Q1 and Q2 at a distance = R from Q1.

• Which of the following statements is true?

• Two balls, one with charge Q1 = +Q and the other with charge Q2 = +2Q, are held fixed at a separation d = 3R as shown.

• Another ball with (non-zero) charge Q3 is introduced in between Q1 and Q2 at a distance = R from Q1.– Which of the following statements is true?

The magnitude of the force on Q3 due to Q2 is proportional to (2Q Q3 /(2R)2) The magnitude of the force on Q3 due to Q1 is proportional to (Q Q3 /R2)

These forces can never cancel, because the force Q2 exerts on Q3 will always be 1/2 of the force Q1 exerts on Q3!!

(a)

(c)(b)

The force on Q3 can be zero if Q3 is positive.The force on Q3 can be zero if Q3 is negative.The force on Q3 can never be zero, no matter what the (non-zero) charge Q3 is.

Q2Q1 3R

+Q

R

Q2Q1

+QQ3

2R

Another Example

What is the force acting on qo ( ) ?

qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram.

0F

What are F0x and F0y ?

Decompose into its x and y components

20F

yFxFF ˆsinˆcos 202020 qq

20

02cosrxx

q20

20sinryy

q

x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1 q

We have

210

1010 r

qqkF 220

2020 r

qqkF

yFF ˆ1010

20ˆ2020 rFF

2010 FF

and Find

0 10 20F F F +

Another Example continued

qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram.

qcos200 FF x 0 10 20 sinyF F F q

2010 FF

Now add the components of andto find and

yx FF 00

xxx FFF 20100 + yyy FFF 20100 +

x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1

10F 20F

0F

Another Example continued

qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram.

N52.110 xF N64.380 yF

N32.4020

200 + yx FFF

The magnitude of is0F

x (cm)

y (cm)

1 2 3 4 5

4321

qo

q2q1

10F 20F

0F

Let’s put in the numbers . . .

cm310 r cm520 r

8.0cos q

N4.1420 FN3010 F

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)

Electric field, introductionOne problem with the above simple description of forces is that it doesn’t describe the finite propagation speed of electrical effects.In order to explain this, we must introduce the concept of the electric field.

7782

8368

5566

8375 80

90 91

757180

72

84

73

82

8892

7788

887364

A Scalar Field

These isolated temperatures sample the scalar field(you only learn the temperature at the point you choose,

but T is defined everywhere (x, y)

7782

8368

5566

8375 80

9091

7571

80

72

84

73

57

8892

77

5688

7364

A Vector FieldIt may be more interesting to know which way the wind is blowing...

That would require a vector field(you learn both wind speed and direction)

It may be more interesting to know which way the wind is blowing…

Summary• Charges come in two varieties

– negative and positive– in a conductor, negative charge means extra mobile

electrons, and positive charge means a deficit of mobile electrons

• Coulomb Force– bi-linear in charges– inversely proportional to square of separation– central force

12212

2112 ˆ

41 r

rqqF

o

• Law of Superposition F®

= F1®

+ F2®

• Fields

Appendix A: Electric Force Example• Suppose your friend can push their arms apart with a

force of 100 lbs. How much charge can they hold outstretched?

+Q -Q

2mF= 100 lbs = 450 N

9 2 2

F 450Q r 2mk 9 10 /

NNm C

´

= 4.47•10-4 C

-4 1519

14.47 10 2.8 101.6 10

eC eC ´

31

15 9.1 10 kg2.8 10 ee

´ 152.54 10 kg

That’s smaller than one cell in your body!

2

2

kQFr

Appendix B: Outline of physics 212

• Coulomb’s Law gives force acting on charge Q1 due to another charge, Q2.

– superposition of forces from many charges

– Electric field is a function defined throughout space

– here, Electric field is a shortcut to Force

– later, Electric field takes on a life of it’s own!

12212

2112 ˆ

41 r

rQQF

o

totaltotal EQF

1

n

nn

n

ototal r

rQQF 121

1 ˆ4

1

12r̂

Q1

Q2

Q313r̂