2012 Semifinal Exam Part A 8 Question A3 This problem inspired by the 2008 Guangdong Province Physics Olympiad Two infinitely long concentric hollow cylinders have radii a and 4a.Both cylinders are insulators; the inner cylinder has a uniformly distributed charge per length of +A;the outer cylinder has a uniformly distributed charge per length of-A. An infinitely long dielectric cylinder with permittivity e=Keo,where k is the dielectric constant, has a inner radius 2a and outer radius 3a is also concentric with the insulating cylinders.The dielectric cylinder is rotating about its axis with an angular velocity wc/a,where c is the speed of light.Assume that the permeability of the dielectric cylinder and the space between the cylinders is that of free space,Ho. a.Determine the electric field for all regions. b.Determine the magnetic field for all regions. Solution a.Consider a Gaussian cylinder of radius r and length I centered on the cylinder axis.Gauss's Law states that E dA=Qenc 2πrEL= 入encl E0 入encd E二2rr0 where Aenc is the enclosed linear charge density. The field due to the hollow cylinders alone is therefore 0 r<a Eapplied= 2xrEo a<r<4a 0 r>4a The field within the dielectric is reduced by a factor K,so that in total 0 r<a 入 a<r 2a E= 2TTKEO 2a<r <3a 入 27rE0 3<r<4a 0 r>4a Copyright C2012 American Association of Physics Teachers2012 Semifinal Exam Part A 8 Question A3 This problem inspired by the 2008 Guangdong Province Physics Olympiad Two infinitely long concentric hollow cylinders have radii a and 4a. Both cylinders are insulators; the inner cylinder has a uniformly distributed charge per length of +λ; the outer cylinder has a uniformly distributed charge per length of −λ. An infinitely long dielectric cylinder with permittivity  = κ0, where κ is the dielectric constant, has a inner radius 2a and outer radius 3a is also concentric with the insulating cylinders. The dielectric cylinder is rotating about its axis with an angular velocity ω  c/a, where c is the speed of light. Assume that the permeability of the dielectric cylinder and the space between the cylinders is that of free space, µ0. a. Determine the electric field for all regions. b. Determine the magnetic field for all regions. Solution a. Consider a Gaussian cylinder of radius r and length l centered on the cylinder axis. Gauss’s Law states that Z E dA = qencl 0 2πrEl = λencll 0 E = λencl 2πr0 where λencl is the enclosed linear charge density. The field due to the hollow cylinders alone is therefore Eapplied =    0 r < a λ 2πr0 a < r < 4a 0 r > 4a The field within the dielectric is reduced by a factor κ, so that in total E =    0 r < a λ 2πr0 a < r < 2a λ 2πrκ0 2a < r < 3a λ 2πr0 3 < r < 4a 0 r > 4a Copyright c 2012 American Association of Physics Teachers