Physics 1502: Lecture 4 Today s Agenda

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Charles Walsh
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1 1 Physics 1502: Today s genda nnouncements: Lectues posted on: HW assignments, solutions etc. Homewok #1: On Mastephysics today: due next Fiday Go to masteingphysics.com and egiste Couse ID: MPCOT33308 Labs: egin next week Today s Topic : nd of Chapte 21: Gauss s Law Motivation & Definition Coulomb's Law as a consequence of Gauss' Law Chages on Insulatos:» Whee ae they? Chapte 22: lectic potential Definition How to compute it

2 Kal Fiedich Gauss (17771855) Infinite Line of Chage Symmety field must be to line and can only depend

2 2 Kal Fiedich Gauss ( ) Infinite Line of Chage Symmety field must be to line and can only depend on distance fom line y Theefoe, CHOOS Gaussian suface to be a cylinde of adius and length h aligned with the xaxis. h pply Gauss' Law: On the ends, On the bael, ds = 0 ds = 2πh ND q = λh NOT: we have obtained hee the same esult as we did last lectue using Coulomb s Law. The symmety makes today s deivation easie! x

3 3 Gauss Law: Help fo the Poblems How to do pactically all of the homewok poblems Gauss Law is LWYS VLID!! What Can You Do With This?? If you have (a) spheical, (b) cylindical, o (c) plana symmety ND: If you know the chage (RHS), you can calculate the electic field (LHS) If you know the field (LHS: usually because =0 inside conducto), you can calculate the chage (RHS). pplication of Gauss Law: Spheical Symmety: Gaussian suface = Sphee of adius LHS: RHS: q = LL chage inside adius Cylindical Symmety: Gaussian suface = Cylinde of adius LHS: RHS: q = LL chage inside adius, length L Plana Symmety: Gaussian suface = Cylinde of aea LHS: RHS: q = LL chage inside cylinde=σ

4 4 Insulatos vs. Conductos Insulatos wood, ubbe, styofoam, most ceamics, etc. Conductos coppe, gold, exotic ceamics, etc. Sometimes just called metals Insulatos chages cannot move. Will usually be evenly spead thoughout object Conductos chages fee to move. on isolated conductos all chages move to suface. Conductos vs. Insulatos in = 0 in <

. In these mateials, once they ae chaged, the chages R NOT FR TO MOV.

5 5 Hollow conductos Conductos & Insulatos Conside how chage is caied on macoscopic objects. We will make the simplifying assumption that thee ae only two kinds of objects in the wold: Insulatos.. In these mateials, once they ae chaged, the chages R NOT FR TO MOV. Plastics, glass, and othe bad conductos of electicity ae good examples of insulatos. Conductos.. In these mateials, the chages R FR TO MOV. Metals ae good examples of conductos. How do the chages move in a conducto?? Hollow conducting sphee Chage the inside, all of this chage moves to the outside.

Gauss Law tells us!! = 0 inside a conducto when in equilibium (electostatics)!» Why?

6 6 Conductos vs. Insulatos Chages on a Conducto Why do the chages always move to the suface of a conducto? Gauss Law tells us!! = 0 inside a conducto when in equilibium (electostatics)!» Why? If 0, then chages would have foces on them and they would move! Theefoe fom Gauss' Law, the chage on a conducto must only eside on the suface(s)! Infinite conducting Conducting plane sphee

7 7 Conside the following two topologies:, CT 1 σ 2 ) solid nonconducting sphee caies a total chage Q = 3 µc distibuted evenly thoughout. It is suounded by an unchaged conducting spheical shell. ) Same as () but conducting shell emoved. Q σ 1 1 Compae the electic field at point X in cases and : (a) < (b) = (c) > 1 What is the suface chage density σ 1 on the inne suface of the conducting shell in case? (a) σ 1 < 0 (b) σ 1 = 0 (c) σ 1 > 0, CT 2 line chage λ (C/m) is placed along the axis of an unchaged conducting cylinde of inne adius i = a, and oute adius o = b as shown. What is the value of the chage density σ o (C/m 2 ) on the oute suface of the cylinde? a b λ σ 0 =? (a) (b) (c)

8 8 lectic Potential V Q 4πε 0 R Q 4πε 0 C R R R q path independence equipotentials Oveview Intoduce Concept of lectic Potential Is it welldefined? i.e. is lectic Potential a popety of the space as is the lectic Field? Calculating lectic Potentials Chaged Spheical Shell N point chages lectic Dipole Can we detemine the lectic Field if we know the lectic Potential? Text Refeence: Chapte 22

9 9 lectic Potential Suppose chage q 0 is moved fom pt to pt though a egion of space descibed by electic field. Since thee will be a foce on the chage due to, a cetain amount of wok W will have to be done to accomplish this task. We define the electic potential diffeence as: q 0 Is this a good definition? Is V V independent of q 0? Is V V independent of path? Independent of Chage? To move a chage in an field, we must supply a foce just equal and opposite to that expeienced by the chage due to the field. F elec F we supply = F elec q 0

10 10, CT 3 single chage ( Q = 1µC) is fixed at the oigin. Define point at x = 5m and point at x = 2m. What is the sign of the potential diffeence between and? (V V V ) 1µC x (a) V < 0 (b) V = 0 (c) V > 0 Independent of Path? F elec q 0 F elec This equation also seves as the definition fo the potential diffeence V V. The integal is the sum of the tangential (to the path) component of the electic field along a path fom to. The question now is: Does this integal depend upon the exact path chosen to move fom to? If it does, we have a lousy definition. Hopefully, it doesn t. It doesn t. ut, don t take ou wod, see appendix and following example.

11 11 Does it eally wok? Conside case of constant field: Diect: h Θ C dl Long way ound: C So hee we have at least one example of a case in which the integal is the same fo OTH paths. lectic Potential Define the electic potential of a point in space as the potential diffeence between that point and a efeence point. a good efeence point is infinity... we typically set V = 0 the electic potential is then defined as: fo a point chage, the fomula is:

12 12 Potential fom chaged spheical shell Fields (fom Gauss' Law) < R: > R: V Q 4πε 0 R R R Q 4πε 0 Potentials > R: R < R: Potential fom N chages The potential fom a collection of N chages is just the algebaic sum of the potential due to each chage sepaately. q 1 q 2 1 x 2 3 q 3

13 13 lectic Dipole z The potential is much easie to calculate than the field since it is an algebaic sum of 2 scala tems. q a a θ Rewite this fo special case >>a: q Can we use this potential somehow to calculate the field of a dipole? (emembe how messy the diect calculation was?) ppendix: Independent of Path? We want to evaluate potential diffeence fom to What path should we choose to evaluate the integal?. If we choose staight line, the integal is difficult to evaluate. Magnitude diffeent at each pt along line. ngle between and path is diffeent at each pt along line. If we choose path C as shown, ou calculation is much easie! Fom to C, is pependicula to the path. ie Fom to C, is pependicula to the path. ie q q C

14 ppendix: Independent of Path? valuate potential diffeence fom to along path C. C by definition: valuate the integal: q ppendix: Independent of Path? C How geneal is this esult?

14 14 ppendix: Independent of Path? valuate potential diffeence fom to along path C. C by definition: valuate the integal: q ppendix: Independent of Path? C How geneal is this esult? Conside the appoximation to the staight path fom > (white aow) = 2 acs (adii = 1 and 2 ) plus the 3 connecting adial pieces. Fo the 2 acs 3 adials path: q q 2 1 This is the same esult as above!! The staight line path is bette appoximated by Inceasing the numbe of acs and adial pieces.

15 15 ppendix: Independent of Path? Conside any path fom to as being made up of a succession of ac plus adial pats as above. The wok along the acs will always be 0, leaving just the sum of the adial pats. ll inne sums will cancel, leaving just the initial and final adii as above.. Theefoe it's geneal! q 2 1

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole