The fact that an electric current gives rise to a magneticfield is the basis for some

Chapter 18, Problem 55

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The fact that an electric current gives rise to a magneticfield is the basis for some electric motors. Amp`eres Lawrelates the magnetic field B to a steady current I. It saysCB dr = kIwhere I is the current4 flowing through a closed curveC and k is a constant. Figure 18.62 shows a rod carryinga current and the magnetic field induced around therod. If the rod is very long and thin, experiments showthat the magnetic field B is tangent to every circle thatis perpendicular to the rod and has center on the axis ofthe rod (like C in Figure 18.62). The magnitude of B isconstant along every such circle. Use Amp`eres Law toshow that around a circle of radius r, the magnetic fielddue to a current I has magnitude given byB = kI2r .(In other words, the strength of the field is inversely proportionalto the radial distance from the rod.)