Chapter 15Field Forces

The Electric Field

•Gravitational force•Electromagnetic force•Strong nuclear force•Weak nuclear force

How different would life be if therewere more ions around?

We’re used to living in a world dominated by gravity. Let’s compare the force of gravity between two protons to the electrostatic force:

Fg = G mp mp / r2

Fe = k e e / r2

*Note how similar these two look!

Let’s now take the ratio of theelectrostatic force to the gravitationalforce:

Fe k e2 / r2

= Fg G mp

2 / r2

(9 X 109 N m2/C2)(1.6 X 10-19C)2

= (6.7 X 10-11N m2/kg2)(1.67X10-27kg)2

Note: this shouldbe a unit-lessquantity. Right?

Fe 2.3 X 10-28 C2/ C2

= Fg 1.9 X 10-64 kg2/ kg2

= 1.2 X 1036 !!!!!!

Good thing most objects are neutral, eh?

We said that like gravity, the electric force is a

Field Force

The Earth has a gravitational field. Weexperience its effects on a daily basis.In fact, we describe it with the quantity

g = 9.81 m/s2

Does the Earth’s gravity extend to theMoon’s surface?

But is that familiar expression for theEarth’s gravitational field still valid atthe Moon’s surface?

That expression has been derived for conditions at the Earth’s surface. The more general expression for the Earth’s gravitational field is given by...

Ge

e = -Gmr

r2

The minus sign indicates that it is an attractivefield that points toward the Earth’s center...

Ge

e = -Gm

rr2

The familiar g is only one possible value of Ge.

While G and me are constant, if we move away from the Earth (toward the moon, for example),

the magnitude of Ge decreases like 1/r2.

Gravity is a field force -- that is, a force that acts at a distance without requiring physical contact.

Now that we have an expression forthe gravitational field, we can determinethe force on a given mass (like ourselves)from the expression:

F = mG

are you spending so much time on gravity? Didn’t we cover that last semester?

Everything we just did with gravity we can do with electricity, too!

(and I think our intuition about gravity is better…)

An object with a charge (q) produces anelectric field around it given by

E =

k

r 2

q r

So, when we bring a second charge (q) into the neighborhood of an existing charge, the second charge will feel a force due to the electric field of the first charge.

That force is given by….

F = qE

The superposition principle applies to the electric field, too!

So…. E E E Etotal 1 2 3 ...

The proof is rather straightforward…if youbelieve that the electrostatic force obeysthe superposition principle...

Since we have a hard time visualizinga field, it is useful to develop the conceptof field lines. Electric Field Lines indicatethe direction and magnitude of the electricfield at any point in space.

1) Begin on positive charges2) End on negative charges3) Point from positive to negative charge4) Are most dense where the electric field

is the strongest.5) Are the least dense where the electric

field is the weakest.

+ -

Field linesFar apart. Field lines

close together.

Looking at the electric field lines givesus a way to come to understand the 1/r2

nature of the expression for electrostaticforce…

Let’s start in the 2 dimensional world first..

What is the density of lines passing through theblue circle? 8/2rblue

What about the green circle?

8/2rgreen

So in flat-land, the density of lines goesdown by 1/distance away from theconvergence point of the lines.

In the 3-D world we live in, we replacethe circles of flat-land with spheres.

The density of lines passing throughsurrounding shells will decrease like1/distance2, since the surface area ofa sphere is

4r2

Example:q1 q2 q3

q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m

What is the Total Force on q2?

r1 r2

y

x

1) Start by calculating the electric field of charge q1 at the location of charge q2.

r Points in the directionfrom 1 to 2!

E1 2@

k

r

(9 10 Nm / C )( 1 C)

(1 m)

= - 9 10 N/ C

1

12

9 2 2

2

3

qr r

r

(i.e., in the -x direction)

Example (con’t):

q1 q2 q3

q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m

What is the Total Force on q2?

r1 r2

y

x

2) Then examine the electric field of q3 at q2.

rPoints in the directionfrom 3 to 2!

E3 2@

k

r

(9 10 Nm / C )( 1 C)

(2 m)

= + 2.25 10 N/ C

3

22

9 2 2

2

3

qr r

r

(i.e., in the -x direction)

Example (con’t):q1 q2 q3

q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m

What is the Total Force on q2?

r1 r2

y

x

3) Next, use the superposition principle to carefully add together the results.

E E Etot 1 2 3 2

3 3

3

@ @9 10 N/ C 2.25 10 N/ C

= - 11.25 10 N/ C

x x

x (i.e., the electric field at the location of q2 points in the -x direction)

Example (con’t):q1 q2 q3

q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m

What is the Total Force on q2?

r1 r2

y

x

4) Finally, multiply by q2 to get the force...

F ( 2 C)(-11.25 10 N/ C )

= 2.25 10 N tot 2 tot

3

-2

= Eq

x

x (i.e., the electric force at the location of q2 points in the +x direction)