4-4 E-field Calculations using Coulomb s Law
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1 1/21/24 ection 4_4 -field calculation uing Coulomb Law blank.doc 1/ field Calculation uing Coulomb Law Reading Aignment: pp xample: The Uniform, Infinite Line Charge 2. xample: The Uniform Dik of Charge. xample: An Infinite Charge Plane
2 1/21/24 The Uniform Infinite Line Charge.doc 1/5 The Uniform, Infinite Line Charge Conider an infinite line of charge ling along the -axi. The charge denit along thi line i a contant value of C/m. Q: What electric field i produced b thi charge ditribution? A: Appl Coulomb Law! We know that for a line charge ditribution that: C d r r
3 1/21/24 The Uniform Infinite Line Charge.doc 2/5 Q: Yike! How do we evaluate thi integral? A: Don t panic! You know how to evaluate thi integral. Let break up the proce into maller tep. Step 1: Determine d The differential element d i jut the magnitude of the differential line element we tudied in chapter 2 (i.e., d d. A a reult, we can eail integrate over an of the even contour we dicued in chapter 2. The contour in thi problem i one of thoe! It i a line parallel to the -axi, defined a x and. A a reult, we ue for d : d a d d ˆ Step 2: Determine the limit of integration Thi i ea! The line charge i infinite. Therefore, we integrate from to. Step : Determine the vector. Since for all charge x and, we find: ( x a ˆ a ˆ ( x a ˆ a ˆ ( xx a ˆ a ˆ a ˆ r-r x + + x ( x + a ˆ + x
4 1/21/24 The Uniform Infinite Line Charge.doc /5 Step 4: Determine the calar x + +, we find: Since ( 2 Step 5: Time to integrate! ( x + + C 1 ( x ˆ x ˆ d ( x x + + d x + + ( x x + + ( d x + + ( a a + d + + a ( d ˆ + x + + ( ( x + ( x x + ( x x 2 + x + ε x +
5 1/21/24 The Uniform Infinite Line Charge.doc 4/5 Thi reult, however, i bet expreed in clindrical coordinate: xx + a ˆ coφx + inφ 2 x + coφx + inφ And with clindrical bae vector: coφ x + inφ 1 ( co φx + in φ 1 + ( co φx φ + in φ φ φ 1 + ( co φx + in φ 1 co + in ( φ φ 1 + -co in + in co ( φ φ φ φ 1 + co + in ( φ( φ ( φ
6 1/21/24 The Uniform Infinite Line Charge.doc 5/5 A a reult, we can write the electric field produced b an infinite line charge with contant denit a: ε â Note what thi mean. Recall unit vector â i the direction that point awa from the -axi. In other word, the electric field produced b the uniform line charge point awa from the line charge, jut like the electric field produced b a point charge likewie point awa from the charge. It i apparent that the electric field in the tatic cae appear to diverge from the location of the charge. And, thi i exactl what Maxwell equation (Gau Law a will happen! i.e.,: v ε Note the magnitude of the electric field i proportional to 1, therefore the electric field diminihe a we get further from the line charge. Note however, the electric field doe not diminih a quickl a that generated b a point charge. Recall in that cae, the magnitude of the electric field diminihe a 2 1 r.
7 1/21/24 The Uniform Dik of Charge.doc 1/5 The Uniform Dik of Charge Conider a dik radiu a, centered at the origin, and ling entirel on the plane. r r Thi dik contain urface charge, with denit of C/m 2. Thi denit i uniform acro the dik. Let find the electric field generated b thi charge dik! From Coulomb Law, we know: S d
8 1/21/24 The Uniform Dik of Charge.doc 2/5 Step 1: Determine d Thi dik can be decribed b the equation. That i, ever point on the dik ha a cordinate value that i equal to ero. Thi i one of the urface we examined in chapter 2. The differential urface element for that urface, ou recall, i: d d d d φ Step 2: Determine the limit of integration. Note over the urface of the dik, change from to radiu a, and φ change from to. Therefore: < < a < φ < Step : Determine vector. We know that for all charge, therefore we can write: ( x a ˆ a ˆ ( x a ˆ a ˆ ( xx a ˆ a ˆ ( x x a ˆ r-r x + + x ( x-x ( + + a ˆ x Since the primed coordinate in d are expreed in clindrical coordinate, we convert the coordinate to get:
9 1/21/24 The Uniform Dik of Charge.doc /5 ( x a ˆ a ˆ ( x a ˆ r-r x + + x + ( x x' x ( ( x coφ ( inφ x Step 4: Determine We find that: 2 2 ( φ ( φ r-r x- co + in + Step 5: Time to integrate! S a d ( x- coφ ( inφ + + ˆ a x ( x- coφ ( inφ + + d dφ Yike! What a me! To implif our integration let determine r along the -axi onl. In other word, the electric field ( et x and.
10 1/21/24 The Uniform Dik of Charge.doc 4/5 ( x,, a a ˆ ax + S ( coφ ( inφ ( coφ ( inφ ( coφ ( inφ a a ˆ a ( co d ˆ a ˆ ˆ x + a a d dφ + + ˆ a ˆ ˆ x + a a d dφ 2 + φ d dφ ( in 2 + φ d dφ 2 + a d dφ + ˆ a 2 + Note that ince: inφ dφ coφdφ The firt two term ( x and are equal to ero. Integrating the lat term, we get: ( x,, 1 if > 2 ε + a 1 if < 2ε + a
11 1/21/24 The Uniform Dik of Charge.doc 5/5 From thi expreion, we can conclude two thing. The firt i that above the dik ( >, the electric field point in the direction, and below the dik ( <, it point in the direction -. What a urprie (not! The electric field point awa from the charge. It appear to be diverging from the charged dik (a predicted b Gau Law. Likewie, it i evident that a we move further and further from the dik, the electric field will diminih. In fact, a ditance goe to infinit, the magnitude of the electric field approache ero. Thi of coure i imilar to the point or line charge; a we move an infinite ditance awa, the electric field diminihe to nothing.
12 1/21/24 An Infinite Charge Plane.doc 1/ An Infinite Charge Plane Sa that we have a ver large charge dik. So large, in fact, that it radiu a approache infinit! Q: What electric field i created b thi infinite plane? A: We alread know! Jut evaluate the charge dik olution for the cae where the dik radiu a i infinit. In other word: lim a ( x,, 1 if > 2 ε + a 1 if < 2ε + a if > 2ε 2ε if < Therefore, the electric field produced b an infinite charge plane, with urface charge denit, i:
13 1/21/24 An Infinite Charge Plane.doc 2/ 2ε 2ε if > if < Think about what thi a! * Firt, we note that the electric field point awa from the plane if i poitive, and toward the plane if i negative. * Second, we notice that the magnitude of the electric field i a contant the magnitude i independent of the ditance from the infinite plane! >
14 1/21/24 An Infinite Charge Plane.doc / The reaon for thi reult i, that no matter how far ou are (i.e., from the infinite charge plane, ou remain infinitel cloe to plane, when compared to it radiu a. We will find thee reult are ueful when we tud the behavior of a parallel plate capacitor.
4-4 E-field Calculations using Coulomb s Law
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