hw 10 - nguyen(jmn727 homework 10 Turner(59070 This...

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nguyen (jmn727) – homework 10 – Turner – (59070) 1 This print-out should have 9 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points An insulating sphere of radius 11 cm has a uniform charge density throughout its vol- ume. 11 cm 6 . 2987 cm 28 . 8103 cm p If the magnitude of the electric field at a distance of 6 . 2987 cm from the center is 74516 . 4 N / C, what is the magnitude of the electric field at 28 . 8103 cm from the center? Correct answer: 18970 . 7 N / C. Explanation: Let : R = 11 cm , E 1 = 74516 . 4 N / C , r 1 = 6 . 2987 cm , r 2 = 28 . 8103 cm , V = 4 3 π R 3 , and ρ = Q V . R r 1 r 2 p Method 1: We know the magnitude of the electric field at a radius r 1 = 6 . 2987 cm (corresponding to a smaller sphere with sur- face area A 1 and volume V 1 ): the magnitude is E 1 = 74516 . 4 N / C. We want to find the magnitude E 2 at a radius r 2 = 28 . 8103 cm, corresponding to a sphere with surface area A 2 and volume V 2 that is larger than the insulating sphere. From Gauss’s Law, we know that since the flux is constant over the sphere, E 1 A 1 = Φ 1 = Q 1 ǫ 0 relating the flux through the Gaussian sphere of radius r 1 to the charge enclosed, Q 1 . We also know Q 1 = ρ V 1 . For the outer sphere (radius r 2 = 28 . 8103 cm), E 2 A 2 = Φ 2 = Q ǫ 0 with Q = ρ V (Not ρ V 2 , as the Gaussian sur- face is larger than the actual physical sphere, and no charge is outside of the sphere.), so E 2 A 2 E 1 A 1 = ρ V ǫ ρ V 1 ǫ = V V 1 . We know the surface area of a sphere is pro- portional to the radius squared, and the vol- ume is proportional to the cube of the radius (in particular: A = 4 π R 2 and V = 4 3 π R 3 ), so E 2 A 2 E 1 A 1 = E 2 r 2 2 E 1 r 2 1 and V V 1 = R 3 r 3 1 . Combining, we get E 2 r 2 2 E 1 r 2 1 = R 3 r 3 1 E 2 = R 3 r 1 r 2 2 · E 1 = (11 cm) 3 (6 . 2987 cm) (28 . 8103 cm) 2 (74516 . 4 N / C) = 18970 . 7 N / C .
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nguyen (jmn727) – homework 10 – Turner – (59070) 2 Note that in this solution, we did not actually need to remember the specific formulae for the surface area and volume of a sphere as the constants cancelled. Method 2: First, calculate the charge density ρ inside the insulating sphere. Select a spherical Gaussian surface with r = 6 . 2987 cm, concentric with the charge distribution. To apply Gauss’ law in this situation, we must know the charge q in within the Gaussian surface of volume V . To calculate q in , we use the fact that q in = ρ V , where ρ is the charge per unit volume and V is the volume enclosed by the Gaussian surface, given by V = 4 3 π r 3 for a sphere. Therefore, q in = ρ V = ρ parenleftBig 4 3 π r 3 parenrightBig . Gauss’ law in the region r < R E (4 π r 2 ) = q in ǫ 0 = 4 π r 3 3 ǫ 0 ρ. Therefore, ρ = 3 ǫ 0 E r For r = r 1 = 6 . 2987 cm, R = 11 cm, we know E = 74516 . 4 N / C. So ρ = 3 ǫ 0 E r 1 . Now we can calculate the magnitude of the electric field at 28 . 8103 cm from the center.
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