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Chapter 28: Problem 25

Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius \(2.00 \cdot 10^{3} \mathrm{~km}\) through the Earth's molten core. The strength of the Earth's magnetic field on the surface near a magnetic pole is about \(6.00 \cdot 10^{-5} \mathrm{~T}\). About how large a current would be required to produce such a field?

Short Answer

Expert verified
Answer: The required current to produce the Earth's magnetic field is approximately 3 x 10^7 A.

Step by step solution

01

Ampere's law states that the magnetic field (B) produced by a current (I) in a circular loop of radius (r) is given by \[B = \frac{\mu_0 I}{2\pi r}\] Where: - B is the magnetic field strength - I is the current - r is the radius of the loop - \(\mu_0\) is the permeability of free space, which is \(4\pi \times 10^{-7} \frac{T \cdot m}{A}\) #Step 2: Rearrange the equation to solve for the current (I)#

To determine the current needed to produce the given magnetic field strength, rearrange the equation to solve for I: \[I = \frac{2\pi r B}{\mu_0}\] #Step 3: Substitute the given values into the equation#
02

We are given the radius of the loop (r) as \(2.00 \cdot 10^{3} km\) and the magnetic field strength (B) as \(6.00 \cdot 10^{-5} T\). Convert the radius from kilometers to meters: \[r = 2.00 \cdot 10^{3} km \times \frac{10^3 m}{1 km} = 2.00 \cdot 10^{6} m\] Now, substitute the given values and the permeability of free space into the equation: \[I = \frac{2\pi (2.00 \cdot 10^{6} m)(6.00 \cdot 10^{-5} T)}{4\pi \times 10^{-7} \frac{T \cdot m}{A}}\] #Step 4: Calculate the current (I) required to produce the given magnetic field strength#

Calculate the current needed to produce Earth's magnetic field: \[I = \frac{2\pi (2.00 \cdot 10^{6} m)(6.00 \cdot 10^{-5} T)}{4\pi \times 10^{-7} \frac{T \cdot m}{A}} = 3 \times 10^7 A\] Hence, a current of about \(3 \times 10^7 A\) would be required to produce the Earth's magnetic field strength of \(6.00 \times 10^{-5} T\).

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Most popular questions from this chapter

A loop of wire of radius \(R=25.0 \mathrm{~cm}\) has a smaller loop of radius \(r=0.900 \mathrm{~cm}\) at its center such that the planes of the two loops are perpendicular to each other. When a current of \(14.0 \mathrm{~A}\) is passed through both loops, the smaller loop experiences a torque due to the magnetic field produced by the larger loop. Determine this torque assuming that the smaller loop is sufficiently small so that the magnetic field due to the larger loop is same across the entire surface.

A 50-turn rectangular coil of wire of dimensions \(10.0 \mathrm{~cm}\) by \(20.0 \mathrm{~cm}\) lies in a horizontal plane, as shown in the figure. The axis of rotation of the coil is aligned north and south. It carries a current \(i=1.00 \mathrm{~A}\), and is in a magnetic field pointing from west to east. A mass of \(50.0 \mathrm{~g}\) hangs from one side of the loop. Determine the strength the magnetic field has to have to keep the loop in the horizontal orientation.

A toy airplane of mass \(0.175 \mathrm{~kg}\), with a charge of \(36 \mathrm{mC}\), is flying at a speed of \(2.8 \mathrm{~m} / \mathrm{s}\) at a height of \(17.2 \mathrm{~cm}\) above and parallel to a wire, which is carrying a 25 - A current; the airplane experiences some acceleration. Determine this acceleration.

A square loop, with sides of length \(L\), carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of \(i\) and \(L\).

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 culating ions move a circular loop of radius \(2500 \mathrm{~km}\). What 鈥渃urrent" must they produce to yield the observed field?
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