2 4πε ( ) ( r θ. , symmetric about the x-axis, as shown in Figure What is the electric field E at the origin O?

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1 p E( r, θ) = cosθ 3 ( sinθ ˆi + cosθ ˆj ) + sinθ cosθ ˆi + ( cos θ 1) ˆj r ( ) ( p = cosθ sinθ ˆi + cosθ ˆj + sinθ cosθ ˆi sinθ ˆj 3 r where the trigonometric identit ( θ ) vectors ˆr and cos 1 = sin θ has been used. Since the unit ˆθ in polar coordinates can be decomposed as rˆ = sinθ ˆi + cosθ ˆj θˆ = cosθ ˆi sin θ ˆj, the electric field in polar coordinates is given b E( p r, θ ) = cosθ ˆ + sin ˆ 3 r r θ ) and the magnitude of is E 1/ p E = ( Er + Eθ ) = 3 ( 3cos θ + 1 ) r 1/.13.5 Electric Field of an Arc A thin rod with a uniform charge per unit length λ is bent into the shape of an arc of a circle of radius. The arc subtends a total angle θ, smmetric about the x-axis, as shown in Figure.13.. What is the electric field E at the origin O? Solution: Consider a differential element of length d = dθ, which makes an angle θ with the x - axis, as shown in Figure.13.(b). The amount of charge it carries is dq = λ d = λ dθ. The contribution to the electric field at O is 1 dq 1 dq 1 d de= rˆ = cos i sin j = cos i sin r ( ˆ ˆ λ θ θ θ ) ( θˆ θj ˆ) 35

2 Figure.13. (a) Geometr of charged source. (b) Charge element dq Integrating over the angle from θ to + θ, we have θ λ sinθ E i j i+ j i ( ˆ ˆ λ cos sin ) ( sin ˆ cos ˆ) 1 λ θ 1 1 = dθ θ θ θ θ ˆ = = θ θ We see that the electric field onl has the x -component, as required b a smmetr argument. If we take the limit θ π, the arc becomes a circular ring. Since sinπ =, the equation above implies that the electric field at the center of a non-conducting ring is zero. This is to be expected from smmetr arguments. On the other hand, for ver smallθ, sinθ θ and we recover the point-charge limit: 1 λθ ˆ 1 λθ ˆ 1 E i = i = Q ˆi where the total charge on the arc is Q= λ = λ( θ ) Electric Field Off the Axis of a Finite od A non-conducting rod of length with a uniform charge densit λ and a total charge Q is ling along the x -axis, as illustrated in Figure Compute the electric field at a point P, located at a distance off the axis of the rod. Figure

3 Solution: The problem can be solved b following the procedure used in Example.3. Consider a length element dx on the rod, as shown in Figure The charge carried b the element is dq = λ dx. The electric field at P produced b this element is Figure dq 1 λ dx de= rˆ = sin ˆi+ cos ˆ r x + ( θ θ j ) where the unit vector ˆr has been written in Cartesian coordinates: rˆ = sinθ ˆi+ cosθ ˆj. In the absence of smmetr, the field at P has both the x- and -components. The x- component of the electric field is de x 1 λ dx 1 λdx x 1 λx dx = sinθ = = x + x + x + 4 πε ( x + ) 3/ Integrating from x = x1 to x = x, we have E x λ x xdx λ 1 x du λ = u 4 πε = = ( x + ) u + x 3/ x 3/ 1/ x + 1 λ 1 1 λ = = x 4 x1 πε + + x + x1 + λ = ( cosθ cosθ1) x + Similarl, the -component of the electric field due to the charge element is 37

4 de 1 λ dx 1 λdx 1 λdx = cosθ = = x + x + x + 4 πε ( x + ) 3/ Integrating over the entire length of the rod, we obtain E λ x dx λ 1 θ λ = = cosθ dθ = sinθ sin 4 πε ( x + ) x 3/ 1 θ1 ( ) θ1 where we have used the result obtained in Eq. (.1.8) in completing the integration. In the infinite length limit where x1 and x +, with xi = tanθi, the corresponding angles are θ 1 = π /and θ = + π /. Substituting the values into the expressions above, we have E x 1 λ =, E = in complete agreement with the result shown in Eq. (.1.11)..14 Conceptual Questions 1. Compare and contrast Newton s law of gravitation, F = Gmm r, and Coulomb s law, F kq q r / e = 1.. Can electric field lines cross each other? Explain. 3. Two opposite charges are placed on a line as shown in the figure below. g 1 / The charge on the right is three times the magnitude of the charge on the left. Besides infinit, where else can electric field possibl be zero? 4. A test charge is placed at the point P near a positivel-charged insulating rod. 38

5 How would the magnitude and direction of the electric field change if the magnitude of the test charge were decreased and its sign changed with everthing else remaining the same? 5. An electric dipole, consisting of two equal and opposite point charges at the ends of an insulating rod, is free to rotate about a pivot point in the center. The rod is then placed in a non-uniform electric field. Does it experience a force and/or a torque?.15 Additional Problems.15.1 Three Point Charges Three point charges are placed at the corners of an equilateral triangle, as shown in Figure Figure.15.1 Three point charges Calculate the net electric force experienced b (a) the 9. µ C charge, and (b) the 6. µ C charge..15. Three Point Charges A right isosceles triangle of side a has charges q, +q and q arranged on its vertices, as shown in Figure

6 Figure.15. What is the electric field at point P, midwa between the line connecting the +q and q charges? Give the magnitude and direction of the electric field Four Point Charges Four point charges are placed at the corners of a square of side a, as shown in Figure Figure.15.3 Four point charges (a) What is the electric field at the location of charge q? (b) What is the net force on q?.15.4 Semicircular Wire A positivel charged wire is bent into a semicircle of radius, as shown in Figure Figure

7 The total charge on the semicircle is Q. However, the charge per unit length along the semicircle is non-uniform and given b λ = λ cosθ. (a) What is the relationship between λ, and Q? (b) If a charge q is placed at the origin, what is the total force on the charge?.15.5 Electric Dipole An electric dipole ling in the x-plane with a uniform electric field applied in the + x - direction is displaced b a small angle θ from its equilibrium position, as shown in Figure Figure.15.5 The charges are separated b a distance a, and the moment of inertia of the dipole is I. If the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion. What is the frequenc of oscillation?.15.6 Charged Clindrical Shell and Clinder (a) A uniforml charged circular clindrical shell of radius and height h has a total charge Q. What is the electric field at a point P a distance z from the bottom side of the clinder as shown in Figure.15.6? (Hint: Treat the clinder as a set of ring charges.) Figure.15.6 A uniforml charged clinder 41

8 (b) If the configuration is instead a solid clinder of radius, height h and has a uniform volume charge densit. What is the electric field at P? (Hint: Treat the solid clinder as a set of disk charges.).15.7 Two Conducting Balls Two tin conducting balls of identical mass m and identical charge q hang from nonconducting threads of length l. Each ball forms an angle θ with the vertical axis, as shown in Figure Assume that θ is so small that tanθ sinθ. Figure.15.9 (a) Show that, at equilibrium, the separation between the balls is q r = πε mg 13 1 (b) If l = 1. 1 cm, m= 1. 1 g, and x = 5.cm, what is q?.15.8 Torque on an Electric Dipole 19 An electric dipole consists of two charges q 1 = +e and q = e ( e = C ), 9 separated b a distance d = 1 m. The electric charges are placed along the -axis as shown in Figure Figure