)]()([ 12 rUrUW veconservati

y

yxUF

x

yxUF yx

),(

;),(

2D case:

)]()([ 12 rUrUW con

Ifrd

dUF

)()( 12

)(

)(

2

1

rUrUdUrdrd

dUrdFW

rU

rUL

z

zyxUF

y

zyxUF

x

zyxUF zyx

),,(;

),,(;

),,(

or

then

2 or 3D cases:

z

zyxUF

y

zyxUF

x

zyxUF zyx

),,(;

),,(;

),,(

Several dimensions: U(x,y,z)

Compact notation using vector del, or nabla:

kz

jy

ix

UF

,

Another notation:rd

dUF

Partial derivative is taken assuming all other arguments fixed

Geometric meaning of the gradient :U

Direction of the steepest ascent;

Magnitude : the slope in that directionU

:UF

Direction of the steepest descent

Magnitude : the slope in that direction F

http://reynolds.asu.edu/topo_gallery/topo_gallery.htm

1)The electric potential V in a region of space is given by

)3(),( 22 yxAyxV where A is a constant. Derive an expression for the electric field at any point in this region.

2)The electric potential V in a region of space is given by

33)(

r

crV

where c is a constant. The source of the field is at the origin. Derive an expression for the electric field at any point in this region.

Exercise 5 p. 52

An electron moves from one point to another where the second point has a larger value of the electric potential by 5 volts. If the initial velocity was zero, how fast will the electron be going at the second point?

Problem 3 p. 45

Electric potential V is a scalar!

An old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hour

Outline

• Area vector

• Vector flux

• More problems

• Solid angle

• Proof of Gauss’s Law

Electric field lines

These are fictitious lines we sketch which point in the direction of the electric field.

1) The direction of at any point is tangent to the line of force at that point.

2) The density of lines of force in any region is proportional to the magnitude of in that regionE

E

Lines never cross.

1R 2R

Density is the number of lines going through an area (N)

divided by the size of the area

21

14 R

NdensityRAt

22

24 R

NdensityRAt

24 r

NdensityranyAt

204

1

r

qE

For a charge q located at the origin

Edensity

It is important that the force is proportional to 21r

Gauss’s Law

The total flux of electric field out of any closed surface is equal to the charge contained inside the surface divided by .0

S

enclosedQSdE

0

What is or flux of any vector, e.g. velocity of a water flow?

SdE

Consider a flow with a velocity vector . Let S be a small area perpendicular to .

v

v

a) The volume of water flowing through S per unit time is vS

S

a)

v

b) Now S is tilted with respect to . The volume of water flowing through S per unit time is

vcosvS

n

S

b)

vnSS

Area vector

is the angle between velocity vector and unit vector

normal to the surface S.

v

n

SvvS

cosFlux:

S S

Flux of electric field E

The flux of E

SdEd

S

s SdE

SdE

Have a great day!

Hw: All Chapter 4 problems and exercisesReading: Chapter 4