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Difficulty: HardMagnetism and Earth's Magnetic Field

At a certain location, the horizontal component of the Earth's magnetic field is 3.0×10−5 T3.0 \times 10^{-5}\text{ T}. If an additional uniform horizontal magnetic field of 4.0×10−5 T4.0 \times 10^{-5}\text{ T} is applied perpendicular to the magnetic meridian, what is the magnitude of the resultant horizontal magnetic flux density experienced by a compass needle at this location?

  1. A
    1.0×10−5 T1.0 \times 10^{-5}\text{ T}
  2. 5.0×10−5 T5.0 \times 10^{-5}\text{ T}Answer
  3. C
    7.0×10−5 T7.0 \times 10^{-5}\text{ T}
  4. D
    1.2×10−4 T1.2 \times 10^{-4}\text{ T}

Answer

5.0×10−5 T5.0 \times 10^{-5}\text{ T}
Because the Earth's horizontal field component acts along the magnetic meridian and the external field is applied perpendicular to it, the two fields form a right-angled triangle. Applying Pythagoras' theorem gives (3.0×10−5)2+(4.0×10−5)2=5.0×10−5 T\sqrt{(3.0 \times 10^{-5})^2 + (4.0 \times 10^{-5})^2} = 5.0 \times 10^{-5}\text{ T}, which represents the true resultant field.

Step-by-Step Solution

1
Identify the vector orientation of the two magnetic fields.
The Earth's horizontal component BHB_H acts along the magnetic meridian (North-South), while the applied field BextB_{\text{ext}} acts perpendicular to it (East-West) at an angle of θ=90∘\theta = 90^\circ.
Magnetic flux density is a vector quantity, so direction matters when combining fields.
2
Apply the perpendicular vector addition formula.
BR=BH2+Bext2B_R = \sqrt{B_H^2 + B_{\text{ext}}^2}
For two vectors acting at right angles (90∘90^\circ), the resultant magnitude is given by the Pythagorean theorem.
3
Substitute the given numerical values into the equation.
BR=(3.0×10−5)2+(4.0×10−5)2=(9.0+16.0)×10−10=25.0×10−10=5.0×10−5 TB_R = \sqrt{(3.0 \times 10^{-5})^2 + (4.0 \times 10^{-5})^2} = \sqrt{(9.0 + 16.0) \times 10^{-10}} = \sqrt{25.0 \times 10^{-10}} = 5.0 \times 10^{-5}\text{ T}
Simplifying the square root yields the exact magnitude of the total horizontal field.

Key Concept

Vector Superposition of Magnetic Fields
Estimated Time:2m 0s
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Topics

Magnetism and Earth's Magnetic Field

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