Potential negy The change U in the potential enegy is defined to equal to the negative of the wok done by a consevative foce duing the shift fom an initial to a final state. U = U U = W F c = F c d
Potential Diffeence and lectic Potential U = U U = W = dw F F dw = F ds = qds F modify Dividing potential enegy by chage gives physical quantity called electic potential (o simply potential) V: that: U = U U = q d s V U = q depends only on souce chage distibution has value at evey point in electic field
Potential Diffeence and lectic Potential U V = V V = = d q s modify W = qv U V = q F dv = q = ds 1 V 1 J/C 1 N/C = 1 V/m The electic field is a measue of the ate of change of the electic potential with espect to position.
lectonvolt lectonvolt it is the amount of enegy gained (o lost) by the chage of a single electon moving acoss an electic potential diffeence of one volt. W = qv 19 19 1 ev 1.60218 10 = C 1 V = 1.6021810 J
Potential Diffeence in a Unifom lectic Field V V = V = ds = ds cos = ds ( ) d V = s V = d U = q V = qd
Paticle in lectic Field V = ds = ds = s = const V = V V = s cos 0 ( ) V V F = q Positive chage moves fom highe to lowe potential egion. Negative chage moves fom lowe to highe potential egion.
quipotential Sufaces V = ds = ds = s = const V = V V = s C C C = s cos 90 = 0 C ( ) V = V C quipotential suface: any suface consisting of continuous distibution of points having same electic potential. quipotential sufaces ae pependicula to the field.
lectic Potential without lectic Field V = d s = 0 V = 0 V = const ( ) In the egion without electic filed thee is no change in electic potential value.
xample 24.1: The lectic Field etween Two Paallel Plates of Opposite Chage battey has a specified potential diffeence V between its teminals and establishes that potential diffeence between conductos attached to the teminals. 12-V battey is connected between two paallel plates as shown in the figue. The sepaation between the plates is d = 0.30 cm, and we assume the electic field between the plates to be unifom. (This assumption is easonable if the plate sepaation is small elative to the plate dimensions and we do not conside locations nea the plate edges.) Find the magnitude of the electic field between the plates.
xample 24.1: The lectic Field etween Two Paallel Plates of Opposite Chage V V 12 V = = = d 2 0.3010 m 3 4.0 10 V/m
xample 24.2: Motion of a Poton in a Unifom lectic Field poton is eleased fom est at point in a unifom electic field that has a magnitude of 4 8.010 V/m, as shown in the figue. The poton undegoes a displacement of magnitude d = 0.50 m to point in the diection of. Find the speed of the poton afte completing the displacement. What will be the speed if the paticle is an electon?
xample 24.2: Motion of a Poton in a Unifom lectic Field v p = = K + U = 0 v 1 2 0 0 2 mv + e V = 2eV 2e( d ) 2ed = = = m m m ( 19 )( 4 )( ) 2 1.6 10 C 8.0 10 V 0.50 m 6 2.8 10 m/s 27 1.6710 kg v e = = ( 19 )( 4 )( ) 2 1.6 10 C 8.0 10 V 0.50 m 8 1.2 10 m/s 31 9.1110 kg
lectic Potential and Potential negy Due to Point Chage V V = d s kq e 1 = ˆ 2 ke = 4π 0 q ds = k ˆ e ds 2 ˆ ds = 1ds cos dscos = d ds = k e q 2 d
lectic Potential and Potential negy Due to Point Chage d 1 1 V V = keq = k 2 V V = keq e q V = 0 at = V = q ke
lectic Potential and Potential negy Due to Point Chages lectic potential esulting fom two o moe point chages could be obtained by applying supeposition pinciple: Total electic potential at some point P due to seveal point chages = sum of potentials due to individual chages V 1 = 4π 0 i q i i
lectic Potential Due to Continuous Chage Distibutions dq dv = ke V = k e dq
xample 24.3: The lectic Potential Due to Two Point Chages V P s shown in the figue, a chage q 1 = 2.00 C is located at the oigin and a chage q 2 = 6.00 C is located at (0, 3.00) m. Find the total electic potential due to these chages at the point P, whose coodinates ae (4.00, 0) m. V P q = ke + q 1 2 1 2 ( 8.988 10 9 N m 2 /C 2 ) = + 4.00 m 5.00 m 3 = 6.2910 V 6 6 2.0010 C 6.0010 C
Obtaining the Value of the lectic Field fom the lectic Potential V = d s dv = ds ds = x d x x = dv dx
Obtaining the Value of the lectic Field fom the lectic Potential ds = dv = d d dv = d x y z V = x V = y V = z
Conductos in lectostatic quilibium 1. = 0 inside conducto 2. Chage esides on suface of isolated conducto 3. at point just outside conducto, pependicula to suface, has magnitude / 0 4. Iegulaly shaped conducto: geatest whee adius of cuvatue smallest
Conductos in lectostatic quilibium Conducting slab placed in extenal field : efoe extenal field applied: fee electons unifomly distibuted thoughout conducto When extenal field applied: fee electons acceleate (to the left in figue) Causing plane of negative chage to accumulate on left suface Movement of electons to left esults in plane of positive chage on ight suface These planes of chage ceate additional electic field inside conducto that opposes the extenal field s electons move suface chage densities on left and ight sufaces incease until magnitude of intenal field = extenal field Result: net field = 0 inside conducto = 0 inside conducto
Conductos in lectostatic quilibium Gaussian suface dawn inside conducto Can be vey close to conducto s suface lectic field eveywhee inside conducto = 0 when in electostatic equilibium lectic field must be zeo at evey point on gaussian suface Net flux though gaussian suface = 0 Conclusion: net chage inside gaussian suface = 0 No net chage inside gaussian suface (which is abitaily close to conducto s suface) ny net chage on conducto must eside on suface = q 0 Chage esides on suface of isolated conducto
lectic Fields and Chaged Conductos in = d = = = 0 0 q = 0 at point just outside conducto is pependicula to suface and has magnitude / 0
lectic Fields and Chaged Conductos V V = d s = 0 The suface of any chaged conducto in electostatic equilibium is an equipotential suface: evey point on the suface of a chaged conducto in equilibium is at the same electic potential. Futhemoe, because the electic field is zeo inside the conducto, the electic potential is constant eveywhee inside the conducto and equal to its value at the suface.
lectic Potential and lectic Field of Chaged Conducto
Suface Chage Density on Chaged Conductio
Suface Chage Density on Chaged Conducto Two sphees vey fa apat connected by a wie: Potential at suface of each sphee q1 q2 V = ke = ke 1 2 equal because connecting wie assues that whole system a single conducto Ratio of electic fields at sufaces of two sphees q 1 k V = = = q 1 V k 1 e 2 1 1 1 2 2 2 1 e 2 2 2 = invese atio of adii of sphees Field stong when adius small Field weake when adius lage lectic field eaches vey high values at shap points 2 < 1 1 < 2 σ 1 < σ 2 Iegulaly shaped conducto: geatest whee adius of cuvatue smallest ( = ) 0
Cavity Within a Conducto Conducto of abitay shape contains cavity ssume no chages inside cavity lectic field inside cavity must be zeo egadless of chage distibution on outside suface of conducto (Gauss law) Field in cavity = 0 even if electic field exists outside conducto vey point on conducto at same electic potential: any two points and on cavity s suface must be at same potential V V d = s
Faaday Cage Faaday cage: conducting mateial, eithe solid o mesh, suounding inteio space Used to potect sensitive electonic equipment Potects you if you ae inside a ca duing a lightning stom Metal body of ca acts as Faaday cage any chage on ca due to stong electic fields in ca on oute suface lectic field inside = 0 Faaday cages: negative effect, i.e., loss of cellphone sevice inside a metal elevato