2/25/2014 Master Ithttp://www.webassign.net/v4cgi/questions/tutorial_popup.tpl 1/2Master It(a) A conducting loop in the shape of a square of edge length = 0.490 m carries a current I = 9.90 A as inthe figure below. Calculate the magnitude and direction of the magnetic field at the center of the square.(b) If this conductor is reshaped to form a circular loop and carries the same current, what is the value of themagnetic field at the center?Part 1 of 4 ConceptualizeAs shown in the drawing below, the magnetic field is directed into the page from the clockwise current. If weconsider the sides of the square to be sections of four wires of infinite length, we expect the magnetic field atthe center of the square to be less than four times the strength of the field at a point away from a wireof infinite length with current I. Using the BiotSavart law, we can find the field from a current in a wire ofinfinite length, and this tells us thatwhere a is the distance from the wire to the point where the field is measured. Forming the wire into a circlewould not change the magnetic field in the center by a large amount since the average distance from thecenter to the wire is not much different for the circle than for that of the square.Part 2 of 4 CategorizeEach side of the square is a section of a thin, straight conductor, so we will use the solution derived from theBiotSavart law to calculate the magnetic field at the center of the square. We will also use this result fromthe BiotSavart law to derive the equation for the magnetic field at the center of a circular loop ofcircumference carrying current I./2B < 4 ,μ0I2πa42/25/2014 Master Ithttp://www.webassign.net/v4cgi/questions/tutorial_popup.tpl 2/2Part 3 of 4 Analyze(a) We use the equation derived from the BiotSavart law for the field created by a thin straight wire. Eachside contributes a field pointing away (into the page) at the center, so together they produce the followingmagnetic field, where a is the distance to the wire from the center of the square, and is a constant, thepermeability of free space. The result for the four wires is given by the following.Taking the center of the square to be the origin of the xy plane, and taking the positive z axis to be pointingperpendicularly out of the page toward us, we havePart 4 of 4 Analyze(b) Again, for a circular loop, the direction of the magnetic field is into the page. The radius is found from thelength of the wire, so for the radius, we haveFor the magnetic field at the center of the circular loop, taking the origin at the center, we have the followingequation derived from the BiotSavart law.FinalizeWe did indeed find that the magnetic field of the square loop is less than as predicted. Themagnetic fields from the square loop and circular loop are similar in magnitude. The field magnitude fromthe circular loop is about 13% less than the field magnitude from the square loop.μ0B = sin − sin − = + 4μ0I4πaπ4π44 4π × 10−7 T · m/A 9.9 A4π 0.245 m2222 = − 1.62 × 10−5 T = − 22.9 µT .B24 = 2πR,R = 2 /π = 0.312 m.B = = = 1.99 × 10−5 T = 19.9 µTμ0I2R4π × 10−7 T · m/A 9.9 A2 0.312
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