1 l8.022 (E&M) -Lecture 2 Topics: Energy stored in a system of charges Electric field: concept and...

1

l8022 (EampM) -Lecture 2

Topics

Energy stored in a system of charges

Electric field concept and problems

Gaussrsquos law and its applications

Feedback

Thanks for the feedback

Scared by Pset 0 Almost all of the math used in the course is in ithellip

Math review too fast Will review new concepts again before using them

Pace ofectures too fast We have a lot to cover buthellip please remind me

2

Last timehellip

Coulombrsquos law

Superposition principle

September 8 2004 8022 ndash Lecture 1

3

Energy associated with FCoulomb

How much work do I have to do to move q from r1 to r2


4

Work done to move charges

Assuming radial path


Does this result depend on the path chosen

No You can decompose any path in segments to the radial direction and segments |_ to it Since the component on the |_ is nul the result does not change


5

Corollaries

The work performed to move a charge between P1 and P2 is the same independently of the path chosen

The work to move a charge on a close path is zero

In other words the electrostatic force is conservative

This will allow us to introduce the concept of potential (next week)

6

Energy of a system of charges

How much work does it take to assemble a certain configuration of charges

Energy stored by N charges


7

The electric field

Q what is the best way of describing the effect of charges

1 charge in the Universe 1048708 2 charges in the Universe

But the force F depends on the test charge qhellip

define a quantity that describes the effect of the charge Q on the surroundings Electric Field

Units dynesesu


8

lElectric field lines

Visualize the direction and strength of the Electric Field

Direction to E pointing towards ndash and away from +

Magnitude the denser the lines the stronger the field

Properties

Field lines never cross (if so thatrsquo where E = 0)

They are orthogonal to equipotential surfaces (will see this later)


9

Electric field of a ring of charge

Problem Calculate the electric field created by a uniformly charged ring on its axis

Special case center of the ring General case any point P on the axis

Answers

bull Center of the ring E=0 by symmetry

bull General case


10

Electric field of disk of charge

Problem Find the electric field created by adisk of charges on the axis of the disk

Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip

(creative recycling is fair game in physics)


11

E of disk of charge (cont)

If charge is uniformly spread

Electric field created by the ring is

Integrating on r 01048708R

Electric field of a ring of radius r


12

Special case 1 R1048708infinity

For finite R

What if Rinfinity Eg what if Rgtgtz

Since

Conclusion

Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )


13

Special case 2 hgtgtR

For finite R

What happens when hgtgtR

Physicistrsquos approach The disk will look like a point charge with Q=σπ r2

Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)

14

The concept of flux

Consider the flow of water in a river The water velocity is described by

Immerse a squared wire loop of area A in the water (surface S)

Define the loop area vector as

Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S

15

What is the flux of the velocity

It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop

General case (definition of flux)

16

FAQ what is the direction of dA

Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface

Examples

Intuitively

ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo


17

Flux of Electric Field

Definition

Example uniform electric field + flat surface

Calculate the flux

Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop

NB this interpretation is valid for any electric field andor surface


18

ΦE through closed (3d) surface

Consider the total flux of E through a cylinder

Calculate

Cylinder axis is to field lines

The total flux through the cylinder is zero

because

but opposite sign since


19

ΦE through closed empty surface

Q1 Is this a coincidence due to shapeorientation of the cylinder

Clue

Think about interpretation of ΦE proportional of field lines through the surfacehellip

Answer

No all field lines that get into the surface have to come out

Conclusion

The electric flux through a closed surface that does not containcharges is zero


20

ΦE through surface containing Q

Q1 What if the surface contains charges

Clue

Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)

Conclusion

The electric flux through a closed surface that does contain a netcharge is non zero


21

Simple example

ΦE of charge at center of sphere

Problem

Calculate ΦE for point charge +Q at the center of a sphere of radius R

Solution

everywhere on the sphere

Point charge at distance


22

ΦE through a generic surface

What if the surface is not spherical S

Impossible integral

Use intuition and interpretation of flux

Version 1

Consider the sphere S1

Field lines are always continuous

Version 2

Purcell 110 or next lecture

Conclusion

The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed


23

Thoughts on Gaussrsquos law

Why is Gaussrsquos law so important

Because it relates the electric field E with its sources Q

Given Q distribution find E (ntegral form) Given E find Q (differential form next week)

Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ

Is Gaussrsquos law always useful

No itrsquos useful only when the problem has symmetries

(Gaussrsquos law in integral form)


24

Applications of Gaussrsquos law

Electric field of spherical distribution of charges

Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)

Approach 1 (mathematician)

bull I know the E due to a pont charge dq dE=dqr2

bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)

Comment correct but usually heavy on math

Approach 2 (physicist)

bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip

Comment correct much much less time consuming


25



Physicistrsquos solution 1) Outside the sphere (rgtR)

Apply Gauss on a sphere S1 of radius r

2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r


26

Do I get full credit for this solution

Did I answer the question completely

No I was asked to determine the electric field The electric field is a vector magnitude and direction

How to get the E direction

Look at the symmetry of the problem Spherical symmetry E must point radially

Complete solution


27

Another application of Gaussrsquos law

Electric field of spherical shell

Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)

Physicistrsquos solutionapply Gauss

1) Outside the sphere (rgtR)


1) Inside the sphere (rltR)

Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0

NB spherical symmetry E is radial


28

Still another application of Gaussrsquos law

Electric field of infinite sheet of charge

Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ

Again apply Gauss

Trick 1 choose the right Gaussian surface

Look at the symmetry of the problem Choose a cylinder of area A and height +-z

Trick 2 apply Gaussrsquos theorem


29

Checklist for solving 8022 problems

Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved

Look for a way to avoid all complicated integration

Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out

Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)

Box the solution your graders will love you If you encounter expansions

Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex

Donrsquot stop the expansion too early Taylor expansions are more than limitshellip


30

Summary and outlook

What have we learned so far


Concept of electric field E

To describe the effect of charges independenty from the test charge

Gaussrsquos theorem in integral form

Useful to derive E from charge distributon with easy calculations

Next time

Derive Gaussrsquos theorem in a more rigorous way

See Purcel 110 if you cannot waithellip

Gaussrsquos law in differential form

hellip with some more intro to vector calculushellip

Useful to derive charge distributon given the electric felds

Energy associated with an electric field


Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

2

Last timehellip

Coulombrsquos law

Superposition principle


3




4







5

Corollaries





6





7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

3




4







5

Corollaries





6





7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

4







5

Corollaries





6





7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

5

Corollaries





6





7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

6





7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

7

The electric field





Units dynesesu


8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

8





Properties




9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

9




Answers


bull General case


10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

10






11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

11







12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

12


For finite R


Since

Conclusion



13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

13


For finite R




14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

14

The concept of flux





15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

15




16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

16



Examples

Intuitively



17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

17


Definition


Calculate the flux




18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

18



Calculate



because



19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

19



Clue


Answer


Conclusion



20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

20



Clue


Conclusion



21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

21

Simple example


Problem


Solution




22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

22



Impossible integral


Version 1



Version 2


Conclusion



23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

23










24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

24












25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

25







26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

26






Complete solution


27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

27











28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

28




Again apply Gauss





29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

29










30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

30

Summary and outlook







Next time








Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

1 l8.022 (E&M) -Lecture 2 Topics: Energy stored in a system of charges Electric field: concept and...

Documents

Transcript of 1 l8.022 (E&M) -Lecture 2 Topics: Energy stored in a system of charges Electric field: concept and...