The magnitude of the magnetic dipole moment of Earth is 8.0 ×10^22 J / T. (a) If the origin of

The magnitude of the magnetic dipole moment of Earth is 8.0...

The magnitude of the magnetic dipole moment of Earth is $8.0 \times 10^{22} \mathrm{~J} / \mathrm{T}$. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth's inner core is $14 \mathrm{~g} / \mathrm{cm}^{3} .$ The magnetic dipole moment of an iron atom is $2.1 \times 10^{-23} \mathrm{~J} / \mathrm{T}$. (Note: Earth's inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth's magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

Key Concepts

Magnetic Dipole Moment

This is a measure of the strength and orientation of a magnet's overall magnetic field. In physics, it is often represented as a vector quantity and is used to characterize both permanent magnets and current loops. Its magnitude and direction determine how an object interacts with external magnetic fields, and in problems like this, it provides a basis for estimating the number or alignment of magnetic moments in a material.

Atomic Magnetic Moment

This refers to the magnetic moment that is inherent to individual atoms, arising primarily from the electron spins and orbital motions. When discussing materials like iron, the atomic magnetic moment is crucial because the collective alignment of these atomic moments determines the overall magnetization of the material. This concept is used to relate microscopic properties with macroscopic magnetic characteristics.

Magnetization and Alignment

Magnetization is the process through which the magnetic moments of atoms or ions in a material are aligned, resulting in a net magnetic moment. Perfect or complete alignment, as assumed in the problem, is an ideal case where every atomic magnetic moment points in the same direction, maximizing the material's overall magnetic moment. This concept is central to understanding how individual magnetic contributions add up to yield the bulk magnetic effect.

Volume and Density Relationships

This concept involves the interrelation between an object's mass, volume, and density. In problems where the volume of a magnetized region (like an iron sphere) is calculated, density is used to connect the total mass of the material to its spatial dimensions. The volume of a sphere is determined from its radius, and the density allows one to infer how much material (and thus how many atoms) is needed to achieve a given magnetization.

Transcript

00:01 High in the given problem total magnetic dipole moment of the earth is given as m let it be capital m and that is given as 8 .0 into 10 dash 22 joules per tesla now it is given that this magnetic moment is due to an iron sphere within the core of earth and we have to find the of that iron sphere suppose radius of that iron sphere which is which is responsible for this magnetic moment around the earth its radius is let it be r so its volume will be given by v is equal to 4 by 3 by r cube volume of the sphere.

01:25 Density of the core is given in the problem as 14 gram per centimeter cube.

01:37 So mass of this inner core will become this mass will be given by the expression.

01:51 Mass equals to volume into density.

01:54 So here it will be for volume this is 4x3 pi r cube only and for density that was given as 14 gram per centimeter cube or we can say this is 14 into 1000 kilogram per meter cube now the number of moles in this core will be given by number of moles of iron will be given by n is equal to mass upon molar mass, mass of the iron core divided by molar mass of iron.

02:44 So here it will be n is equal to for mass.

02:49 If we multiply here this 14 with the 4, it will be 56 ,000.

02:56 So here we may write it like 56 ,000 into pi r, cube divided by 3 into molar mass, which is 56 gram or 56 into 10 dash per minus 3 kilogram.

03:14 We will convert it into kilogram, which finally will be written as 10 dash par 6 pi r cube divided by 3.

03:25 The number of moles.

03:27 So number of atoms using abogatros concept.

03:33 Number of atoms of iron in this core will be number of moles into avogadro means n into a .v.

03:44 So here we may write it like 10 dash par 6 into pi into r cube...