Find the magnetic field a distance z above the center of a circular loop of radius R, which carries a steady current I (Fig. 5.21).

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Accepted Answer

The field dB attributable to the segment [latex]d\acute{l}[/latex] points as shown. As we integrate [latex]d\acute{l}[/latex] around the loop, dB sweeps out a cone. The horizontal components cancel, and the vertical components combine, to give

[latex]B(z)=\frac{\mu _{0}}{4\pi}I\int{\frac{d\acute{l}}{\eta ^{2}\cos heta} }. [/latex]

(Notice that [latex]d\acute{l}[/latex] and [latex]\eta [/latex] are perpendicular, in this case; the factor of [latex]\cos heta [/latex] projects out the vertical component.) Now, [latex]\cos heta [/latex] and [latex]\eta ^{2} [/latex] are constants, and [latex]\int{d\acute{l}} [/latex] is simply the circumference, [latex]2\pi R[/latex], so

[latex]B(z)=\frac{\mu _{0}I}{4\pi}\left(\frac{\cos heta}{\eta ^{2}} ight)2\pi R =\frac{\mu _{0}I}{2}\frac{R^{2}}{(R^{2}+z^{2})^{3/2}} . [/latex] (5.41)

Question 5.6: Find the magnetic field a distance z above the center of a c...

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