Week 7 Day 2 Concept Question Ampere s Law Correct answer 3 Integrating B around the loop shown gives us is zero Ampere s Law B J n a Group Problem Line Integral of Magnetic Field Concept Question Line Integral Current Enclosed Third Maxwell Equation Ampere s Law B s J n a Ampere s Law The idea Concept Question Ampere s law Choosing Amperian Close Path Calculate the line integral of magnetic eld B ds about a circular path of radius r centered on the wire for a very long current carrying wire Correct answer 2 The integral expression is an in nite sum of the product of the tangent component of the magnetic eld along a small element of the closed path with a small element of the path up to a choice of plus or minus sign Current density J I J n A Current enclosed is the ux of the current density through an open surface S bounded by the closed path Because the unit normal to an open surface is not uniquely de ned this expression is unique up to a plus or minus sign In order to have a non zero line integral of magnetic eld around a closed path there must be current punching through any area with path as boundary B ds 0 for no enclosed current Correct answer 2 Integrating B around the loop in the clockwise direction shown gives us a negative number Ampere s Law holds for all closed paths It is useful to calculate magnetic eld for some sources with enough symmetry in which there exists a closed path such that The magnetic eld on some leg of the closed path is both parallel to path and has constant magnitude on that leg The magnetic eld on some leg of the closed path is perpendicular to path or is zero Applying Ampere s Law Preliminary Steps Identify the symmetry properties of the current distribution Determine the direction of the magnetic eld up to the choice of sign Decide how many different regions of space the current distribution determines For each region of space determined by the current distribution set up B s J A Equate the two sides of ampere s law in order to determine an expression for the magnetic eld Based on the sign of the magnetic eld and the choice or orientation of the path determine direction for magnetic eld A cylindrical conductor of radius R is carrying a current I The current density J is uniform Find the direction and magnitude of the magnetic eld for the regions i r R and ii r R Calculate the magnetic eld everywhere due to an in nite straight wire of radius R carrying a non uniform current density J Group Problem In nite Wire with Uniform Current Density Group Problem Non Uniform Cylindrical Wire Magnetic Field of Solenoid For ideal solenoids B is uniform inside and zero outside Concept Question Co axial Solenoids Correct answer 2 Two co axial very long solenoids are centered on the z axis Both solenoids have n turns per unit length A current runs through the inner turns in a clockwise direction when viewed from above A current of the same magnitude runs through the outer turns in a counter clockwise direction when viewed from above The magnetic eld in the region between the cols points in the positive z direction Ampere s Law In nite Current B is constant and parallel perpendicular or zero current penetrates A sheet of current in nite in the y and z directions of thickness d in the x direction carries a uniform current density J J k Find the direction and magnitude of B as a function of x Group Problem Current Sheet
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