Question

A horizontal desk surface measures \(1.3 \mathrm{m} \times 1.0 \mathrm{m} .\) If Earth's magnetic field has magnitude \(0.44 \mathrm{mT}\) and is directed \(65^{\circ}\) below the horizontal, what is the magnetic flux through the desk surface?

Short answer

Question: Calculate the magnetic flux through a horizontal desk surface with dimensions \(1.3 \mathrm{m} \times 1.0 \mathrm{m}\), given Earth's magnetic field magnitude as \(0.44 \mathrm{mT}\) and the angle below the horizontal as \(65^{\circ}\). Answer: The magnetic flux through the desk surface is approximately \(4.914 \times 10^{-4}\,\mathrm{Wb}\).

Step-by-step solution

Identify the given information

We are given the following information: - Dimensions of the desk surface: \(1.3 \mathrm{m} \times 1.0 \mathrm{m}\) - Earth's magnetic field: \(0.44 \mathrm{mT} = 0.44 \times 10^{-3}\,\mathrm{T}\) - Angle below the horizontal: \(65^{\circ}\)

Find the component of the magnetic field that is perpendicular to the surface

The magnetic field component perpendicular to the surface can be calculated by taking the vertical component of the magnetic field. This is because the horizontal component is parallel to the surface and doesn't contribute to the magnetic flux. To find the vertical component, multiply the magnetic field magnitude by the sine of the angle below the horizontal: \(B_\perp = B \times \sin{(\theta)}\) \(B_\perp = 0.44 \times 10^{-3} \times \sin{(65^{\circ})}\) \(B_\perp \approx 0.378 \times 10^{-3}\,\mathrm{T}\)

Calculate the area of the desk surface

To calculate the area of the desk surface, simply multiply the length and width: \(A = 1.3 \, \mathrm{m} \times 1.0\, \mathrm{m} = 1.3\, \mathrm{m^2}\)

Calculate the magnetic flux through the surface

Now that we have the magnetic field component perpendicular to the desk surface and the area of the desk surface, we can calculate the magnetic flux. The magnetic flux (\(\Phi_B\)) can be calculated using the formula: \(\Phi_B = B_\mathrm{\perp} \times A\) \(\Phi_B = (0.378 \times 10^{-3}\,\mathrm{T}) \times (1.3\, \mathrm{m^2})\) \(\Phi_B \approx 4.914 \times 10^{-4}\,\mathrm{Tm^2} = 4.914 \times 10^{-4}\,\mathrm{Wb}\) So the magnetic flux through the desk surface is approximately \(4.914 \times 10^{-4}\,\mathrm{Wb}\).