The Electric Field January 13, 2006

Calendar  Quiz Today  Continuation with Coulomb’s Law and the concept of the Electric Field  MLK Holiday on Monday  Next week we continue with same topics – see the schedule on the website  Quiz on Friday

Last Time  We found two kinds of charge.  Like charges repel, unlike charges attract.  Discussed induction … a bit  Fundamental unit of charge is the COULOMB.  Coulomb’s Law  Assignment: read text about induction.

One example of induction

Polarize

Ground

Remove Ground

Positive !

Coulomb’s Law The Unit of Charge is called THE COULOMB Smallest Charge: e ( a positive number) 1.6 x Coul. electron charge = -e Proton charge = +e

What would be the magnitude of the electrostatic force between two C charges separated by the following distances, if such point charges existed (they do not) and this configuration could be set up? (a) 1.25 m __________N (b) 1.25 km __________N

Three point charges are located at the corners of an equilateral triangle as shown in Figure P23.7. Calculate the resultant electric force on the 7.00-μC charge.

Two small beads having positive charges 3q and q are fixed at the opposite ends of a horizontal, insulating rod, extending from the origin to the point x = d. As shown in Figure P23.10, a third small charged bead is free to slide on the rod. At what position is the third bead in equilibrium? Can it be in stable equilibrium?

This is WAR You are fighting the enemy on the planet Mongo. The evil emperor Ming’s forces are behind a strange green haze. You aim your blaster and fire … but …… Ming the merciless this guy is MEAN !

Nothing Happens! The Green thing is a Force Field! The Force may not be with you ….

Side View The FORCE FIELD Force Position o |Force| Big!

Properties of a FORCE FIELD  It is a property of the position in space.  There is a cause but that cause may not be known.  The force on an object is usually proportional to some property of an object which is placed into the field.

EXAMPLE: The Gravitational Field That We Live In. m M mgmg MgMg

Mysterious Force F

Electric Field  If a charge Q is in an electric field E then it will experience a force F.  The Electric Field is defined as the force per unit charge at the point.  Electric fields are caused by charges and consequently we can use Coulombs law to calculate it.  For multiple charges, add the fields as VECTORS.

Two Charges

Doing it Q r q A Charge The spot where we want to know the Electric Field F

General-

Force  Field

Two Charges What is the Electric Field at Point P?

The two S’s S uperpositio n S ymmetry

What is the electric field at the center of the square array?

Kinds of continuously distributed charges  Line of charge  or sometimes = the charge per unit length. dq=ds (ds= differential of length along the line)  Area  = charge per unit area dq=dA dA = dxdy (rectangular coordinates) dA= 2rdr for elemental ring of charge  Volume =charge per unit volume dq=dV dV=dxdydz or 4r 2 dr or some other expressions we will look at later.

Continuous Charge Distribution

ymmetry

Let’s Do it Real Time Concept – Charge per unit length  dq=  ds

The math Why?

A Harder Problem A line of charge  =charge/length setup dx L r   x dEdE dE y

(standard integral)

Completing the Math 1/r dependence

Dare we project this??  Point Charge goes as 1/r 2  Infinite line of charge goes as 1/r 1  Could it be possible that the field of an infinite plane of charge could go as 1/r 0 ? A constant??

The Geometry Define surface charge density  =charge/unit-area dq=  dA dA=2  rdr (z 2 +r 2 ) 1/2 dq=  x dA = 2  rdr

(z 2 +r 2 ) 1/2 

Final Result

Look at the “Field Lines”

What did we learn in this chapter?? FIELD  We introduced the concept of the Electric FIELD. We may not know what causes the field. (The evil Emperor Ming) If we know where all the charges are we can CALCULATE E. E is a VECTOR. The equation for E is the same as for the force on a charge from Coulomb’s Law but divided by the “q of the test charge”.

What else did we learn in this chapter?  We introduced continuous distributions of charge rather than individual discrete charges.  Instead of adding the individual charges we must INTEGRATE the (dq)s.  There are three kinds of continuously distributed charges.

Kinds of continuously distributed charges  Line of charge  or sometimes = the charge per unit length. dq=ds (ds= differential of length along the line)  Area  = charge per unit area dq=dA dA = dxdy (rectangular coordinates) dA= 2rdr for elemental ring of charge  Volume =charge per unit volume dq=dV dV=dxdydz or 4r 2 dr or some other expressions we will look at later.

The Sphere dq r thk=dr dq=  dV=  x surface area x thickness =  x 4  r 2 x dr

Summary (Note: I left off the unit vectors in the last equation set, but be aware that they should be there.)

To be remembered …  If the ELECTRIC FIELD at a point is E, then  E = F /q (This is the definition!)  Using some advanced mathematics we can derive from this equation, the fact that: