Physics Courseware Physics II Electric Field and Force
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1 Physics Cousewae Physics II lectic iel an oce Coulomb s law, whee k Nm /C test Definition of electic fiel. This is a vecto. test Q lectic fiel fo a point chage. This is a vecto. Poblem.- chage of µc is place at each cone of a suae 0.m on a sie. Detemine the magnitue of the foce on each chage. Poblem.- Detemine the electic fiel (magnitue at the oigin ue to the chages locate at an. Consie the cooinates given in metes an chage -µc an -µc. To fin the electic fiel we can fist calculate the magnitues: Consie the foces on the chage in the uppe ight cone: ( ( k.4 0. ( ( k. (0. ( ( 0..4 To a these vectos we have to fin thei components: (0,.4 (.4,0 (.7,.7 o o (.cos4,.sin4 (0., N ( 8000V/m 0V/m ( The magnitue of the total vecto will be: + 8,40 V/m Poblem.- Detemine the electic fiel at the oigin of cooinates ue to the chages locate at an. Consie the cooinates given in metes an chages Q 4µC an Q -µc. In you answe wite the vecto as components an fin its magnitue.
2 To fin the electic fiel we can calculate the contibution fom each chage an then a the vectos: Q 4 4,000 V/m Q, V/m ( The vecto pouce by Q at the oigin is ownwa because it is a positive chage, so: (0, 000 Chage Q pouces an attactive foce on a positive test chage, so the vecto will be fom the oigin to point, which is at 4º above the positive -ais. The vecto is: o o (cos 4,cos 4 (77,77 The esultant vecto is: (77, Which has a magnitue of 77 +,78 V/m Poblem 4.- Calculate the net foce on the positive chage on the ight uppe cone of the suae shown in the figue. [Give you answe in magnitue of the vecto]. ( ( 7. N ( (.08 N 7.07 ( (.44 N To a these vectos we nee to wite them as components, so: ( 7.,0 (.08 ( 7.4 (0,.44 o cos4,.08, 7.4 N ing the components, we get: + + ( 4.84,.04 n the magnitue is: ( (.04 o sin4. - N Poblem.- Calculate the electic fiel at point P in the figue: The positive chage pouces an electic fiel pointing to the ight, as shown. With a magnitue eual to: Thee ae thee foces acting on the positive chage as shown in the figue below. The magnitues ae calculate using Coulomb s law. 70V/m The negative chage instea pouces an electic fiel pointing to the left, with a magnitue eual to: 87.V/m ing the vectos, we get 8. V/m pointing to the ight
3 Poblem.- Detemine the magnitue an iection of the electic fiel at a point P miway between a chage -µc an anothe +.µc that ae at a istance Lcm apat. ( 4µ C k( C k µ + ( L Solving fo : ( L ( L 4L ach chage will pouce an electic fiel at point P. The vectos will be: 7. V/m 0.0 an the vecto points to the left an the vecto points to the left too. V/m Since the two vectos point to the left the sum will be: 40. V/m + 7. V/m 7. V/m to the left. Poblem 7.- Detemine the position of a point P between a chage -4µC an anothe +µc that ae at a istance Lm apat whee the electic fiel has the minimum value. Suggestion: Calculate the electic fiel as a function of, then take the fist eivative with espect to an make that eivative eual to zeo. The electic fiel (total value is given by: k( 4µ C k( µ C + ( L Taking the fist eivative we get: k( 4µ C k( µ C + ( L To get the minimum value, the eivative has to be eual to zeo, so: 0. L Poblem 8.- Calculate the electic foce between an alpha paticle an an aluminum nucleus sepaate by a istance of 0. - m Chage of an alpha paticle. - C Chage of an aluminum nucleus C The euation neee is Coulomb s law., with the values of the poblem: (. ( N (0. Poblem.- in the acceleation epeience by an electon in an electic fiel of 0 V/m Chage of the electon e -. - C an mass m e. - kg [Recall Newton s secon law ma] The efinition of electic fiel is the foce pe chage:, so. We also know that ma o a m m With the values given: (. C(0N/C 4.4 m/s m. kg a Poblem.- Given the two chages shown in the figue, at what position is the electic fiel zeo?
4 The electic fiel ue to the positive chage is µc p an the one ue to the negative ( + 4µ C chage is n, but in the opposite iection, so the net fiel will be zeo if the two vectos have the same magnitue: µ C 4µ C k ( + 4 simplifying we get: ( + an taking the suae oot on both sies: + m + Poblem.- Daw aows to inicate the iection of the electic fiel at points, an C ue to the ipole shown in the figue. Thee ae two foces acting on the chage, an : kq (cosθ,sinθ h + kq (cosθ, sinθ h + The sum of the foces is: kq + (cosθ,0 h + The cosine of θ can be witten in tems of an h: kq + +, 0 h h + The maimum foce: Poblem.- In the following aangement fin an epession fo the net foce on the chage ue to the othe two chages Q an fin the value of that makes the foce maimum. / ( h + ( h + / kq/ ( h + ( h + kq kq h + 0 h / ( 0 Poblem.- We want to fin the electic fiel at point P ue to a wie with constant linea ensity of chage λµc/m an length Lm locate a istance 0.m fom point P as shown in the figue.
5 C Nm C C,000N/C C (m Now we consie the iections of these vectos: Consie a iffeential of wie as shown above. It will have a chage λ an it is locate at a istance L+- fom point P, so it will contibute an electic fiel eual to: kλ ( L + The integal that we nee is: L 0 kλ ( L + If you integate, you get: kλ L + L 0, kλ kλ kλl L + ( L + ( ( ( 0,000V / m 0.( + 0. Poblem 4.- Detemine the electic fiel (magnitue an iection at the oigin ue to the chages locate at, an C. Consie the cooinates given in metes an chage of -µc, -µc an C µc. : Since is negative the electic fiel vecto will point towa the chage, so it will be a vetically upwa vecto. (0, 8000N/C : Since is also negative the electic fiel vecto will point towa the chage, so it will be a hoizontal vecto pointing to the ight. (000 N/C, 0 C : C is positive, so the electic fiel vecto will point away fom the chage. It will be a hoizontal vecto pointing to the ight. (000 N/C, 0 We can a these thee vectos: + + C (8000 N/C, 8000 N/C: ( + ( y 8, ,000,00 N/C The iection of this vecto will make 4º with the hoizontal. Poblem.- Detemine the magnitue of the electic fiel at the oigin ue to the two chages an. Consie the positions given in metes an chage of 0µC an -0µC. ist, let us fin the magnitues of the thee vectos using the fomula fo the electic fiel of a point chage: k 8,000N/C ( (,000N/C sketch of the poblem: The magnitue of each electic fiel is:
6 a (0.7 (.0 V m Regaing the iection of the vectos: Notice that pouces an electic fiel pointing ownwas because the foce on a positive test chage at the oigin woul be epulsive. Instea, which is negative, woul pouce an attactive foce on a positive test chage at the oigin, so its electic fiel is a vecto that points to the ight. (.7 V/m, -.7 The magnitue of the vecto is: (.7 V/m + (.7.8 V/m V/m V/m Poblem 7.- The ing shown in the figue has a unifom chage Q an aius R. Detemine the electic fiel at point P, which is locate on the ais of the ing a istance fom its cente. In the same chage istibution, fin the electic potential at point P. To calculate the electic fiel, notice that we can ivie the ing into iffeentials of chage. ach one contibuting an electic fiel eual to: Q Q + R Poblem.- n 8µC chage is place. m fom an ientical 8µC chage. Calculate the electic fiel an the electic potential at the point miway between them The electic fiel pouce by each chage will be the same in magnitue, but one towas the left an one towas the ight, so when you a them togethe you get zeo. lectic potential is a scala uantity, so we nee to a the potential pouce by each chage. The istance that goes in the fomula is the istance fom the point miway between the chages an each chage, so it is 0. m (not. m an the potential is: 8 8 V + 40,000 V Only the component of the vecto in the vetical iection will contibute to the integal. The othe components will give zeo ue to the symmety of the poblem, so: Q + R fte integating we get Q ( + R / Q cosθ + R + R If we want the electic potential at point P, we o not nee to be concene with vectos, we just a (integate all the contibutions: V Q k Q + R V kq + R
7 Poblem 8.- in the net foce on a chage Q8. - C locate at the oigin of cooinates (0,0 ue to a chage. - C locate at (.0m,.0m an anothe chage C locate at (-.0m,.0m. sketch of the poblem: Notice that the two foces fom 0º with espect to each othe: Coulomb s Law gives the magnitue of each foce. Q Qa 0.08 N a Qa (. (. (.0 Since Q an have the same sign, the foce between these chages will be epulsive. On the othe han, is negative, so it will attact Q. Coulomb s Law gives the magnitues of the foces. Q Q Q. - N Q Q Q.8 - N (8. (. ( (8. ( ( Since the two vectos ae othogonal to each othe, we can use Pythagoas theoem to fin the magnitue of the sum: Q Qb 0.07 N b Qb (. (.0 (.0 Regaing the iection of the vectos, notice that Q an a have the same sign, so the foce is epulsive. It will be in the positive Y iection. On the othe han, b is negative, so the foce on Q is attactive an will be in the positive X iection. The vecto foce will be then: (0.08 N, 0.07 N The magnitue of the vecto is: (0.08 N + ( 0.07 N 0.07 N 0. N Q + Q (. N + (.8 N Poblem.-in the magnitue of the net foce on a chage Q. - C locate at position (0,.0 m ue to a chage a. - C locate at the oigin (0,0 an anothe chage b - - C locate at (.0m,.0m.
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