Question

Suppose that you have a very long cylinder (treat it as infinitely long) with a uniform charge density p (coulombs per cubic metre). The cylinder has a radius a. Let the axis of the cylinder be the 2- axis. The cylinder is rotating about this axis with a constant angular speed w in a counterclockwise direction. @=w2 a. [5 points] What is the current density ✓ at a general point in the cylinder, at a distance r from the ĉ-axis, at an angle 8 about this axis (starting from the i-axis), and with a z coordinate z? b. [5 points] Use Ampère's law to determine the magnetic field outside this rotating, charged cylinder, r > a, where r is the distance from the ĉ-axis. State any assumptions that you are making. Explain carefully which Ampèrian loop you are choosing and explicitly perform the integration about the loop. c. [5 points] Use Ampère's law to determine the magnetic field inside this rotating, charged cylinder, r <a, where r is the distance from the ĉ-axis. State any assumptions that you are making. Explain carefully which Ampèrian loop you are choosing. a

Answer 1

Solution: - . P (Cm3) 2 a ) Aw=w =wi Charge density let de be the Charge in elemental cylinder. consider cylindrical shell of radius r and thiucney do. For this Cylinder, the current along boundary is given as: I. SX (28L)x do () density at (rar) is Ja Sur da 217 Current I Lor ㅠ Here, b ) outside so for I depends only upon r, not in o. The current are circumferential any loop lying cylinder, fun through loop = 0 » & Bodl=0 for 67 8>a. B=0 L passing consider c) For rco the square ampenian loop Of length from distance & from am's The current passing through this B B L loop is, ra Cross sectional view IM 271 8=8 SI Ldo as s Swr XL do SWL (a²-x2) By symmetry, magnetic field B earst only along side IM 271 AD of loop AB CD. Mo Tin Therefore, og Bidi 6 from A SPA +886 +977 No furt carry AB 2T1 - Bx ya Mo Swy (0282) Mofw B .) (2²²) 2 TI un CSScanned with CamScanner

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