Chapter 25 Electric Potential

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1 Chapte 5 Electic Potential Can we apply the concept of potential, fist intoduced in mechanics, to electostatic system and find the law of consevation of enegy? We can define an electostatic potential enegy, analogous to gavitational potential enegy, and apply the law of consevation of enegy in the analysis of electical poblems. Potential is a popety of a point in space and depends only on the souce chages. It is often easie to analyze a physical situation in tems of potential, which is a scala, athe than the electic field stength, which is a vecto. Can you tell the diffeence between potential and potential enegy? EXT 5. Potential The motion of a paticle with positive chage q in a unifom electic field is analogous to the motion of a paticle of mass m in unifom gavitational field nea the eath. W + U U f U If W EXT >0, wok is done by the extenal agent on the chages. If W EXT <0, wok is done on the extenal agent by the field. i Potential enegy depends not only on the souce but also on the test paticle. Thus it will be moe convenient if we can define a potential function which is function of souce only.

2 5. Potential (II) When a chage q moves between two points in the electostatic field, the change in electic potential,, is defined as the change in electostatic potential enegy pe unit chage, U q The SI unit of electic potential is the volt (). J / C N m / C The quantity depends only on the field set up by the souce chages, not on the test chage. W EXT q q ( f i ) 3 5. Potential (III) We see that only changes in potential, athe than the specific value of i and f, ae significant. It is convenient to choose the gound connection to eath as the zeo of potential. The potential at a point is the extenal wok need to bing a positive unit chage, at constant speed, fom the position of zeo potential to the given point. In an extenal electic field, both positive and negative chages tend to decease the electostatic potential enegy. Which side will a chage paticle dift if it is in the middle of two conducting plates with potential diffeence, highe o lowe potential side? See next viewgaph. 4

3 du d B du q A 5. Potential (I) In mechanics, the definition of potential enegy in tems of the wok done by the consevative foce is U-W c.. The negative sign tells us that positive wok by the consevative foce leads to a decease in potential enegy. Theefoe, the change in potential enegy, associated with an infinitesimal displacement ds, is F c ds qe ds E ds B E ds A Since the electostatic field is consevative, the value of this line integal depends only on the end points A and B, not on the path taken Potential and Potential Enegy in a Unifom Field In a unifom field, the change in potential in moving fom A to B is -E ds. In a unifom field the potential deceases linealy with distance along the field lines. The electic field lines ae pependicula to the equipotentials and point fom highe to lowe potentials, that is, downhill. 6

4 5. Potential and Potential Enegy: Motion of Chages The motion of a chage in an electic field may be discussed in tems of the consevation of enegy, K+ U 0. In tems of potential, the consevation law may be witten as K q It is convenient to measue the enegy of elementay paticles, such as electons and potons, in tems of a non-si unit called the electonvolt ( e.6x -9 J). Accoding to Einstein famous Emc, find the enegy in tems of e fo an electon of est mass 9.x -3 kg, whee the speed of light is 3x 8 m/s. E9.x -3 x(3x 8 ) /.6x Me 7 f i ((5 6 + q / m 6 6 i ) m/s. q.6 9 Example 5. A poton, of mass.67x -7 kg, entes the egion between paallel plates a distance 0 cm apat. Thee is a unifom electic field of 3x 5 /m between the plates, as shown in Fig If the initial speed of the poton is 5x 6 m/s, what is its final speed? v f mv v mv 6 4 /.67 7 )) 0.5 8

5 5.3 Potential and Potential Enegy of Point Chages 9 Example 5. Thee point chage, quc, q-uc, and q33uc ae fixed at the positions shown in Fig.5.3a. (a) What is the potential at point P at the cone of the ectangle? (b) What is the total potential enegy of q, q, and q3?(c) How much wok would be needed to bing a chage q4.5uc fom infinity and to place it at P? ( a) ( b) ( c) U U W p ext q U p U 0.9 J J Choosing 0 at.

6 Example 5.3 In 93, Boh poposed a model of the hydogen atom in which an electon obits a stationay poton in a cicula path. Find the total mechanical enegy of the electon given that the adius of the obit is 0.53x - m. The mechanical enegy is the sum of the kinetic and potential enegies, EK+U. The centipetal foce is povided by the coulomb attaction. ke U ke mv ke F K mv 9 9 (.6 E U + K U ).8 8 J 3.6 e 5.4 Electic Field Deived fom Potential Can we deive a consevative foce fom the associated potential enegy function? Yes! du d E s E ds F c d ds ds qe ds A deivative in which all vaiables except one ae held constant is called a patial deivative and is witten with a instead of d. The electic field is theefoe E x i x j x k

7 Example 5.4 The potential due to a point chage is given by kq/. Find: (a) the adial component of the electic field; (b) the x- component of the electic field. (a) kq d kq E d kq (b) E x + y + z x kq x x Continuous Chage Distibution The potential due to continuous chage distibution can be found in two ways. Fist appoach: diect integation d kdq Second appoach: use electic field k dq B A B E ds A 4

8 Example 5.5 A nonconducting disk of adius a has a unifom suface chage density σ C/m. What is the potential at a point on the axis of the disk at a distance fom its cente. d d kdq, dq σ (πxdx) kσπ dx x + y 0 a x πkσ kσπ + y dx 0.5 a 0.5 [( x + y ) y] πkσ [( a + y ) y] 0 5 E kq ˆ Example 5.6 A shell of adius R has a chage Q unifomly distibuted ove its suface. Find the potential at a distance >R fom its cente. It is moe staightfowad to use the electic field, which we know fom Gauss s law. ( ) ( ) ( ) kq kq d kq The potential has a fixed value at all points within the conducting sphee equal to the potential at the suface. 0 6

9 dw W dq Q 0 kq kq dq kq dq R Example 5.7 A metal sphee of adius R has a chage Q. Find its potential enegy. The potential enegy U/Q is the wok needed to bing the system of chages togethe Conductos Within the mateial of the conducto, the electic field is zeo elsewhee. All points within and on the suface of a conducto in electostatic equilibium ae at the same potential. Fo a displacement ds along the suface of a conducto, we have de ds0, which means E is pependicula to ds. The field lines ae pependicula to the suface. 8

10 5.6 Conductos (II) Shielding Effect Fa fom the sphee, the field patten emains the same: The field lines ae unifom, and the equalpotentials ae planes. Nea the sphee, the equalpotentials must be sphee and field line must adial. The chages in the sphee edistibute themselves so as to ensue that these conditions ae met Conductos (III) Chage Redistibution Suppose two chaged metal sphees with adius R and R ae connected by a long wie. Chage will flow fom one to the othe until thei potential ae equal. The equality of the potential implies that Q R, since Q 4 R σ R Q σ R πr σ We infe that σ /R: The suface chage density on each sphee is invesely popotional to the adius. The egions with the smallest adii of cuvatue have the geatest suface chage densities. 0

11 5.6 Conductos (I) Dischage at Shap Points on a Conducto E ε σ 0 R The above equation infe that the field stength is geatest at the shap points on a conducto. If the field stength is geat enough (about 3x 6 /m fo dy ai) it can cause an electical dischage in ai. How does the beakdown occu in high voltage tansmission line? 5.6 Conductos () Dust Causing High oltage Beakdown The potential at the suface of a chaged sphee is kq/r and the field stength is EkQ/R. So, fo a given beakdown field stength, beakdown voltage is popotional to the adius, B R. The potential of a sphee of adius cm may be aised to 3x5 befoe beakdown. On the othe hand, a 0.05 mm dust paticle can initiate a dischage at 50. A high voltage system must keep at vey clean condition.

12 Execises and Poblems Ch.5: Ex. 37, 43, 48 Pob. 6, 7,, 3

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage