Chapter 21: Gauss s Law

Size: px

Start display at page:

Download "Chapter 21: Gauss s Law"

Egbert Morton
5 years ago
Views:

1 Chapte : Gauss s Law Gauss s law : intoduction The total electic flux though a closed suface is equal to the total (net) electic chage inside the suface divided by ε Gauss s law is equivalent to Coulomb s law

2 Gauss s law : intoduction (cont d) Conside a distibution of chage Suound it with an imaginay suface that encloses the chage Look at the electic field at vaious points on this imaginay suface F q q imaginay suface test chage To find out chage distibution inside the imaginay suface, we need to measue electic fields especially on the suface To do that place a test chage of a known chage amount and measue the electic foce F q

3 Chage and lectic Flux lectic fields by diffeent chages q q q outwad flux q q outwad flux q q inwad flux q - q - - q inwad flux q

4 Chage and lectic Flux lectic flux q q q outwad flux q q outwad flux ( ) q ( ) 4 q q When the distance to the suface doubled, the aea of the suface quadupled and the electic field becomes ¼.

5 Definition of electic flux Chage and lectic Flux Fo any point on a small aea of a suface, take the poduct of the aveage component of pependicula to the suface and the aea. Then the sum of this quantity ove the suface is the net electic flux A qualitative statement of Gauss s law Whethe thee is a net outwad o inwad electic flux though a closed suface depends on the sign of the enclosed chage. Chages outside the suface do not give a net electic flux though the suface. The net electic flux is diectly popotional to the net amount of chage enclosed within the suface but is othewise independent of the size of the closed suface.

6 Calculating lectic Flux Analogy between electic flux and field of velocity vecto A good analogy between the electic flux and the field of velocity vecto in a flowing fluid can be found. A (aea) volume flow ate: dv dt υ A φ A A A υ υ velocity vecto (flow speed) A vecto aea that defines the plane of the aea, pependicula to the plane υ volume flow ate: dv dt υ υ cosφ ; A Acosφ ; υacosφ υ A υa A An υ A

7 Calculating lectic Flux Analogy between electic flux and field of velocity vecto A (aea) volume flow ate: electic flux: dv dt Φ υ A A field is pependicula to this plane φ A A lectic field vecto A vecto aea that defines the plane of the aea, pependicula to the plane electic flux: dv dt cosφ ; A Acosφ υacosφ υ A υ A Φ Acosφ A A

8 Calculating lectic Flux A small aea element and flux d Φ da Total flux fo an aea Φ nda dφ da cosφ da da ; da xample: lectic flux though a disk. m A φ 3 A π (. m) Φ 54 N m.34 m Acosφ (. / C 3 N/C)(.34 m )cos3

9 Calculating lectic Flux xample : lectic flux though a cube ˆn 3 ˆn 5 ˆn ˆn L Φ ˆ cos n A L 8 L ˆn Φ ˆ cos n A L L 4 Φ Φ Φ Φ L cos ˆn 6 Φ i i 6 Φ i

10 Calculating lectic Flux xample : lectic flux though a sphee. m q q3. µc A4π da Φ, // nˆ // da q 4πε ( N/C da A N m 9 / C N m ( / C 6 3. C ) (.m) N/C)(4π )(.m)

11 Peview: Gauss s Law The total electic flux though any closed suface (a suface enclosing a definite volume) is popotional to the total (net) electic chage inside the suface. Case : Field of a single positive chage q q A sphee with suface q at 4πε Φ A 4πε (4π ) q q ε The flux is independent of the adius of the suface.

12 Gauss s Law Case : Field of a single positive chage q (cont d) da 4dA 4 vey field line that passes though the smalle sphee also passes though the lage sphee q da da The total flux though each sphee is the same The same is tue fo any potion of its suface such as da 4 dφ da 4dA da dφ Φ q da ε This is tue fo a suface of any shape o size povided it is a closed suface enclosing the chage

13 Gauss s Law Case : Field of a single positive chage (geneal suface) da q n cosφ φ dφ da φ da dacosφ suface pependicula to cos da Φ da q ε

14 Gauss s Law Case 3: An closed suface without any chage inside Φ da Gauss s law Q encl Φ da ; Qencl q i i, ε lectic field lines that go in come out. lectic field lines can begin o end inside a egion of space only when thee is chage in that egion. The total electic flux though a closed suface is equal to the total (net) electic chage inside the suface divided by ε i i

15 Intoduction Applications of Gauss s Law The chage distibution the field The symmety can simplify the pocedue of application lectic field by a chage distibution on a conducto When excess chage is placed on a solid conducto and is at est, it esides entiely on the suface, not in the inteio of the mateial (excess chage chage othe than the ions and fee electons that make up the mateial conducto A Gaussian suface inside conducto Chages on suface Conducto

Applications of Gauss s Law lectic field by a chage distibution on a conducto (cont d) A Gaussian suface inside conducto Chages on suface Conducto at evey point in the inteio of a conducting mateial

16 Applications of Gauss s Law lectic field by a chage distibution on a conducto (cont d) A Gaussian suface inside conducto Chages on suface Conducto at evey point in the inteio of a conducting mateial is zeo in an electostatic situation (all chages ae at est). If wee non-zeo, then the chages would move Daw a Gaussian suface inside of the conducto eveywhee on this suface (inside conducto) The net chage inside the suface is zeo Gauss s law Thee can be no excess chage at any point within a solid conducto Any excess chage must eside on the conducto s suface on the suface is pependicula to the suface

17 Applications of Gauss s Law xample: Field of a chaged conducting sphee with q q 4 πε / 4 / 9 Gaussian suface 3 < : < Q elcl (4π : Gauss's q, A ) Daw a Gaussian suface outside the sphee law : q ε 4π const.on the sphee suface pependicula to the sphee suface : 4πε q 4πε q

18 Applications of Gauss s Law xample: Field of a line chage, da Gaussian suface chosen accoding to symmety line chage density Q encl Φ lλ ( on the cylindical λl π l ) ε πε λ Gaussian suface

19 Applications of Gauss s Law xample: Field of an infinite plane sheet of chage σ : chage density the sheet aea A aea A Q encl σa Φ ( A) two end sufaces σa ε Gaussian suface σ ε

20 Applications of Gauss s Law xample : Field between oppositely chaged paallel conducting plane plate plate b a c S S 4 S S 3 - No electic flux on these sufaces Solution : σa S : A ε outwad flux S 4 : inwad flux σa A ε σ (ight suface) ε (left suface) σ (left suface) ε Solution : At Point a : b : σ σ ε ε c : (ight suface)

Applications of Gauss s Law xample : Field of a unifomly chaged sphee Gaussian suface 3 3 4 chage

21 Applications of Gauss s Law xample : Field of a unifomly chaged sphee Gaussian suface chage density Q π ρ 3 / ) 3 4 ( : ε π ρ A < ) / 3 4 ( ) (4 Q πε ε π ρ π 4 : Q πε 4 ) (4 : Q Q πε ε π <

22 Applications of Gauss s Law xample : Field of a hollow chaged (unifomly on its suface) sphee.3 m.5 m.8 Φ q (4πε N/C da (4π ) ).8 nc q ε Hollow chaged sphee Gaussian suface

23 Chages on Conductos Case : chage on a solid conducto esides entiely on its oute suface in an electostatic situation The electic field at evey point within a conducto is zeo and any excess chage on a solid conducto is located entiely on its suface. Case : chage on a conducto with a cavity If thee is no chage within the cavity, the net chage on the suface of the cavity is zeo. Gauss suface

24 Chages on Conductos Case 3: chage on a conducto with a cavity and a chage q inside the cavity Gauss suface The conducto is unchaged and insulated fom chage q. The total chage inside the Gauss suface should be zeo fom Gauss s law and on this suface. Theefoe thee must be a chage q distibuted on the suface of the cavity. The simila agument can be used fo the case whee the conducto oiginally had a chage q C. In this case the total chage on the oute suface must be qq C afte chage q is inseted in cavity.

25 Chages on Conductos Faaday s ice pail expeiment chaged conducting ball conducto () Faaday stated with a neutal metal ice pail (metal bucket) and an unchaged electoscope. () He then suspended a positively chaged metal ball into the ice pail, being caeful not to touch the sides of the pail. The leaves of the electoscope diveged. Moeove, thei degee of divegence was independent of the metal ball's exact location. Only when the metal ball was completely withdawn did the leaves collapse back to thei oiginal position.

26 Chages on Conductos Faaday s ice pail expeiment (cont d) chaged conducting ball conducto (3) Faaday noticed that if the metal ball was allowed to contact the inside suface of the ice pail, the leaves of the electoscope emained diveged (4) Aftewads, when he completely emoved the ball fom the inside of the ice pail, the leaves emained diveged. Howeve, the metal ball was no longe chaged. Since the leaves of the electoscope that was attached to the OUTSID of the pail did not move when the ball touched the inside of the pail, he concluded that the inne suface had just enough chage to neutalize the ball.

Chages on Conductos Field at the suface of a conducto The electic field just outside a conducto has magnitude σ /ε and is diected pependicula to the

27 Chages on Conductos Field at the suface of a conducto The electic field just outside a conducto has magnitude σ /ε and is diected pependicula to the suface. Daw a small pill box that extends into the conducto. Since thee is no field inside, all the flux comes out though the top. Aq/ε σa/ ε, σ / ε

28 xecises xecise This is the same as the field due to a point chage with chage Q

29 xecise (cont d) xecises

30 xecises xecise : A sphee and a shell of conducto Q -3Q Q Q Fom Gauss s law thee can be no net chage inside the conducto, and the chage must eside on the outside suface of the sphee Thee can be no net chage inside the conducto. Theefoe the inne suface of the shell must cay a net chage of Q, and the oute suface must cay the chage Q Q so that the net chage on the shell equals Q. These chages ae unifomly distibuted. σ inne Q Q Q Q σ oute 4π 4π 4π

31 xecises xecise : A sphee and a shell of conducto (cont d) Q Q Q -3Q Q k Q Q k Q k ˆ ˆ : ˆ : : < < < <

32 xecise 3: Cylinde xecises An infinite line of chage passes diectly though the middle of a hallow chaged conducting infinite cylindical shell of adius. Let s focus on a segment of the cylindical shell of length h. The line chage has a linea chage density λ, and the cylindical shell has a net suface chage density of σ total. σ total λ σ inne σ oute h

33 xecises xecise 3: Cylinde (cont d) The electic field inside the cylindical shell is zeo. Theefoe if we choose as a Gaussian suface a cylinde, which lies inside the cylindical shell, the net chage enclosed is zeo. Thee is a suface chage density on the inside wall of the cylinde to balance out the chage along the line. σ total λ σ inne σ oute h

34 xecises xecise 3: Cylinde (cont d) The total chage on the enclosed potion (length h) of the line chage: λh The chage on the inne suface of the conducting cylinde shell: Q inne λh σ inne λh πh λ π σ total λ σ inne σ oute h

35 xecises xecise 3: Cylinde (cont d) The net chage density on the cylinde: σ total The oute chage density : σ oute σ oute σ total σ inne σ total λ π σ total λ σ inne σ oute h

36 xecises xecise 3: Cylinde (cont d) Daw a Gaussian suface suounding the line chage of adius (< ) πh q ε encl, q encl λh λ πε fo < σ total λ σ inne σ oute h

37 xecises xecise 3: Cylinde (cont d) Daw a Gaussian suface suounding the line chage of adius (>) Net chage enclosed on the line: πh q ε encl, q encl λh Q λh Net chage enclosed on the shell: σ ε σ total total λ πε fo Q πhσ > total λ σ inne σ oute h

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing