5.10 A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at 22° with the horizontal. The horizontal component of the earth’s magnetic field at the place is known to be 0.35 G. Determine the magnitude of the earth’s magnetic field at the place.

<p><strong>5.10 </strong>Given:</p><p>Horizontal component of Earth’s magnetic field, <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> </msub> </math> </span> = 0.35 G</p><p>Angle made by the needle with the horizontal plane, <span contenteditable="false"> <math> <mi>β</mi> </math> </span>= 22<span contenteditable="false"> <math> <mo>°</mo> </math> </span></p><p>If the Earth’s magnetic field at that location be B, </p><p>then<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> </msub> </math> </span> = B<span contenteditable="false"> <math> <mrow> <mrow> <mi mathvariant="normal">cos</mi> </mrow> <mo>? </mo> <mrow> <mi>β</mi> </mrow> </mrow> </math> </span></p><p>B =<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mrow> <mrow> <mi mathvariant="normal">cos</mi> </mrow> <mo>? </mo> <mrow> <mi>β</mi> </mrow> </mrow> </mrow> </mrow> </mfrac> </math> </span> = <span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>0.35</mn> </mrow> </mrow> <mrow> <mrow> <mrow> <mrow> <mi mathvariant="normal">cos</mi> </mrow> <mo>? </mo> <mrow> <mn>22</mn> <mo>°</mo> </mrow> </mrow> </mrow> </mrow> </mfrac> </math> </span>= 0.377 G</p><p>Therefore, the strength of Earth’s magnetic field at that location is 0.377 G</p>

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