Question

The magnetic dipole moment of the Earth is approximately \(8.0 \cdot 10^{22} \mathrm{~A} \mathrm{~m}^{2}\). The source of the Earth's magnetic field is not known; one possibility might be the circulation of ions in the Earth's molten outer core. Assume that the circulating ions move a circular loop of radius \(2500 \mathrm{~km} .\) What "current" must they produce to yield the observed field?

Short answer

Answer: The circulating ions in the Earth's molten outer core must produce a "current" of approximately \(4.08 \cdot 10^9 \mathrm{~A}\) to yield the observed magnetic field.

Step-by-step solution

Write down the given values and formula

We are given: - Magnetic dipole moment, µ = \(8.0 \cdot 10^{22} \mathrm{~A} \mathrm{~m}^{2}\) - Radius of the circular loop, r = \(2500 \mathrm{~km}\) Now, remember that the magnetic dipole moment can be given by the formula, µ = IA (where I is the current, and A is the area of the circular loop)

Convert radius to meters

The radius of the circular loop is given in kilometers, convert this to meters since the magnetic moment is given in \(A m^2\). r = \(2500 * 10^3 \mathrm{~m}\)

Find the area of a circular loop

To find the area A of a circular loop, we use the formula: A = πr^2 Substitute the value of r: A = π(2500 * 10^3)^2 ≈ 1.96 * 10^{13} \mathrm{~m}^2

Find the current produced by the ions

We now have the values for the magnetic moment and the area of the circular loop. We can use the formula µ = IA to find the current: I = µ/A Substitute the known values: I = \((8.0 \cdot 10^{22})/(1.96 \cdot 10^{13})\) I ≈ 4.08 * 10^{9} \mathrm{~A}

Write the final answer

Therefore, the circulating ions in the Earth's molten outer core must produce a "current" of approximately \(4.08 \cdot 10^9 \mathrm{~A}\) to yield the observed magnetic field.