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Zorluk: OrtaMagnetism and Earth's Magnetic Field

At a geomagnetic observation post in West Africa, the total intensity of the Earth's magnetic field is 5.0×10−5 T5.0 \times 10^{-5}\text{ T} and its vertical component is 4.0×10−5 T4.0 \times 10^{-5}\text{ T}. What is the horizontal component of the Earth's magnetic field at this location?

  1. 3.0×10−5 T3.0 \times 10^{-5}\text{ T}Cevap
  2. B
    1.0×10−5 T1.0 \times 10^{-5}\text{ T}
  3. C
    6.4×10−5 T6.4 \times 10^{-5}\text{ T}
  4. D
    9.0×10−5 T9.0 \times 10^{-5}\text{ T}

Cevap

The horizontal component of the Earth's magnetic field is 3.0×10−5 T3.0 \times 10^{-5}\text{ T}.
The Earth's total magnetic field BB is resolved into perpendicular components: horizontal (BhB_h) and vertical (BvB_v). Applying the Pythagorean theorem B2=Bh2+Bv2B^2 = B_h^2 + B_v^2 yields Bh=(5.0×10−5)2−(4.0×10−5)2=3.0×10−5 TB_h = \sqrt{(5.0 \times 10^{-5})^2 - (4.0 \times 10^{-5})^2} = 3.0 \times 10^{-5}\text{ T}.

Adım Adım Çözüm

1
Identify the relationship between total magnetic field intensity and its orthogonal components.
The total magnetic field BB, horizontal component BhB_h, and vertical component BvB_v form a right triangle: B2=Bh2+Bv2B^2 = B_h^2 + B_v^2.
The horizontal and vertical components of the Earth's magnetic field are mutually perpendicular.
2
Rearrange the equation to solve for the horizontal component BhB_h.
Bh=B2−Bv2B_h = \sqrt{B^2 - B_v^2}
Isolating BhB_h allows direct substitution of the given values.
3
Substitute the values B=5.0×10−5 TB = 5.0 \times 10^{-5}\text{ T} and Bv=4.0×10−5 TB_v = 4.0 \times 10^{-5}\text{ T} into the equation.
Bh=(5.0×10−5)2−(4.0×10−5)2=(25−16)×10−10=9×10−10=3.0×10−5 TB_h = \sqrt{(5.0 \times 10^{-5})^2 - (4.0 \times 10^{-5})^2} = \sqrt{(25 - 16) \times 10^{-10}} = \sqrt{9 \times 10^{-10}} = 3.0 \times 10^{-5}\text{ T}.
Evaluating the square root yields the exact magnitude of the horizontal magnetic field component.

Anahtar Kavram

Resolution of Earth's magnetic field into mutually perpendicular horizontal and vertical components using vector geometry.
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Konular

Magnetism and Earth's Magnetic Field

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