1/10/2006 184 Lecture 3 1

PHY 184PHY 184

Spring 2007Lecture 3

Title: The Coulomb Force

1/10/2006 184 Lecture 3 2

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1/10/2006 184 Lecture 3 3

Outline1 – Review2 – Electrostatic charging3 – Coulomb’s Law

1/10/2006 184 Lecture 3 4

ReviewReview

There are two types of charge: negative and positive. Most objects are electrically neutral; they have equal numbers

of negative and positive charges (net charge is 0). An object becomes charged by adding or removing electrons. An electron carries negative charge of magnitude e =

1.602×10-19 C. Law of Charges

• Like charges repel and opposite charges attract. Law of charge conservation

• The total charge of an isolated system is strictly conserved. Conductors are materials where some of the electrons can

move freely. Insulators are materials where none of the charges can move

freely.

1/10/2006 184 Lecture 3 5

Electrostatic ChargingElectrostatic Charging

There are two ways to charge an object• Conduction• Induction

Charging by conduction• We can charge an object by connecting a source of

charge directly to the object and then disconnecting the source of charge

• The object will remain charged– Conservation of charge

1/10/2006 184 Lecture 3 6

Charging by ConductionCharging by Conduction

We brought charge onto the electrode by contact.

+ + + + + + + + +

Electroscope

1/10/2006 184 Lecture 3 7

InductionInduction

InductionThe presence of the positively charged rod leads to a redistribution of charge (a kind of polarization).

It pulls electrons up to the electrode.

+ + + + + + + + + - - - - -- - - -

1/10/2006 184 Lecture 3 8

Charging by InductionCharging by Induction

We can also charge an object without physically connecting to it• First we charge a rod positively• Then we ground the object to be charged

• Connecting the object to the Earth provides an effectively infinite sink for charge

• We bring the charged rod close to the object but do not touch it

• We remove the ground connection and move the rod away

• The object will be charged by induction

1/10/2006 184 Lecture 3 9

InductionThe presence of the positively charged rod leads to a redistribution of charge.

Grounding pushes positive charge to Earth (or rather pulls electrons from Earth!) leaving the electroscope negative.

+ + + + + + + + +

Charging by inductionCharging by induction

- - - - -- - - -

ground

1/10/2006 184 Lecture 3 10

Coulomb’s Law

1/10/2006 184 Lecture 3 11

Electric Force - Coulomb’s LawElectric Force - Coulomb’s Law

Consider two electric charges: q1 and q2

The electric force F between these two charges separated by a distance r is given by Coulomb’s Law

The constant k is called Coulomb’s constant and is given by

221

r

qkqF

229 /CNm1099.8 k

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Coulomb’s Law (2)Coulomb’s Law (2)

The coulomb constant is also written as

0 is the “electric permittivity of vacuum”• A fundamental constant of nature

2

212

00 Nm

C 1085.8 where

4

1

k

221

04

1

rqq

F

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Example:What is the force between two charges of 1 C separated by 1 meter?

Answer: 8.99 x 109 N,i.e., huge!

1/10/2006 184 Lecture 3 14

Electric ForceElectric Force

The electric force is given by

The electric force, unlike the gravitational force, can be positive or negative• If the charges have opposite

signs, the force is negative• Attractive

• If the charges have the same sign, the force is positive• Repulsive

221

r

qqkF

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Electric Force VectorElectric Force Vector

Electric force in vector form

rr

qqkF ˆ

221

2

x

y

q1

q2

r2

r1

r

rrr

rr

r

rrr

12

12

ˆ

rr

qqkF ˆ

221

1

1/10/2006 184 Lecture 3 16

Superposition PrincipleSuperposition Principle

The net force acting on any charge is the vector sum of the forces due to the remaining charges in the distribution.

znF

zF

zF

zF

ynF

yF

yF

yF

xnF

xF

xF

xF

nnet

,1,13,121

,1,13,121

,1,13,121

,13,12,1,1

FFFF

1/10/2006 184 Lecture 3 17

Example - The Helium NucleusExample - The Helium Nucleus

Part 1: The nucleus of a helium atom has two protons and two neutrons. What is the magnitude of the electric force between the two protons in the helium nucleus?

Answer: 58 N

Part 2: What if the distance is doubled; how will the force change?

Answer: 14.5 N

Inverse square law: If the distance is doubled then the force is reduced by a factor of 4.

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Consider two charges located on the x axis

The charges are described by• q1 = 0.15 C x1 = 0.0 m

• q2 = 0.35 C x2 = 0.40 m

Where do we need to put a third charge for that charge to be at an equilibrium point?• At the equilibrium point, the forces from the two

charges will cancel.

x1 x2

Example - Equilibrium PositionExample - Equilibrium Position

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Example - Equilibrium Position (2)Example - Equilibrium Position (2)

The equilibrium point must be along the x-axis.

Three regions along the x-axis where we might place our third charge

x1 x2

x3 < x1

x1 < x3 < x2

x3 > x2

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Example - Equilibrium Position (3)Example - Equilibrium Position (3)

x3<x1

• Here the forces from q1 and q2 will always point in the same direction (to the left for a positive test charge)

• No equilibrium

x2<x3

• Here the forces from q1 and q2 will always point in the same direction (to the right for a positive test charge)

• No equilibrium

x1 x2

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Example - Equilibrium Position (4)Example - Equilibrium Position (4)

x1 x2

0)()( 2

32

322

13

31

xx

qqk

xxqq

k

x1 < x3 < x2

Here the forces from q1 and q2 can balance.

Answer: x3 = 0.16 m

q3

Check the signs!!

1/10/2006 184 Lecture 3 22

Example - Charged PendulumsExample - Charged Pendulums

Consider two identical charged balls hanging from the ceiling by strings of equal length 1.5 m (in equilibrium). Each ball has a charge of 25 C. The balls hang at an angle = 25 with respect to the vertical. What is the mass of the balls?

Step 1: Three forces act on

each ball: Coulomb force, gravity and the tension of the string. mgTF

d

kqTF

y

x

cos

sin

:lefton Ball

2

2

x

y

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Example - Charged Pendulums (2)Example - Charged Pendulums (2)Step 2: The balls are in equilibrium positions. That means the sum of all forces acting on the ball is zero!

tan

/

cos

sin

2

2

22

dkq

mg

mgdkq

TT

Answer: m = 0.76 kg

A similar analysis applies to the ball on the right.

d=2 l sin

1/10/2006 184 Lecture 3 24

Electric Force and Gravitational Force

Electric Force and Gravitational Force

Coulomb’s Law that describes the electric force and Newton’s gravitational law have a similar functional form

Both forces vary as the inverse square of the distance between the objects.

Gravitation is always attractive. k and G give the strength of the force.

Felectric kq1q2

r2Felectric k

q1q2

r2 Fgravity Gm1m2

r2Fgravity G

m1m2

r2

1/10/2006 184 Lecture 3 25

Example - Forces between Electrons

Example - Forces between Electrons

What is relative strength of the electric force compared with the force of gravity for two electrons?

422

2

102.4 n)calculatio the(do Gm

keFF

gravity

electric

2

2

2

2)(

r

mGF

r

ekF

egravity

electric

• Gravity is irrelevant for atomic and subatomic processes – the electric force is much much stronger.

• But sometimes gravity is most important; e.g, the motion of the planets. Why?

1/10/2006 184 Lecture 3 26

Example - Four ChargesExample - Four Charges

Consider four charges placed at the corners of a square with sides of length 1.25 m as shown on the right. What is the magnitude of the electric force on q4 resulting from the electric force from the remaining three charges?

Set up an xy-coordinate system with its origin at q2.

1/10/2006 184 Lecture 3 27

Example - Four Charges (2)Example - Four Charges (2)

Answer:

F (on q4) = 0.0916 N

… and the direction?