Question

A magnetar (magnetic neutron star) has a magnetic field near its surface of magnitude \(4.0 \cdot 10^{10} \mathrm{~T}\) a) Calculate the energy density of this magnetic field. b) The Special Theory of Relativity associates energy with any mass \(m\) at rest according to \(E_{0}=m c^{2}(\) more on this in Chapter 35 ). Find the rest mass density associated with the energy density of part (a).

Short answer

Answer: The rest mass density associated with a magnetic field of magnitude 4.0 x 10^10 T is approximately 1.13 x 10^13 kg/m^3.

Step-by-step solution

Calculating energy density of the magnetic field

Using the given magnetic field magnitude \(B = 4.0 \cdot 10^{10} \mathrm{~T}\), we can calculate the energy density (u) using the formula \(u = \frac{B^2}{2\mu_0}\), where \(\mu_0\) is the permeability of free space, and its value is \(4\pi \cdot 10^{-7} \mathrm{T\cdot m/A}\)). First, let's square the given magnetic field magnitude: \(B^2 = (4.0 \cdot 10^{10})^2 = 1.6 \cdot 10^{21} \mathrm{T}^2\) Now, divide the squared magnetic field by \(2\mu_0\): $u = \frac{1.6 \cdot 10^{21}} {2 \times(4\pi \cdot 10^{-7})} \approx 1.01 \cdot 10^{18} \mathrm{J/m^3}$ The energy density of the magnetic field is \(1.01 \cdot 10^{18} \mathrm{J/m^3}\)

Finding rest mass density associated with energy density

Using the formula given by the Special Theory of Relativity, \(E_0 = mc^2\), we will now find the rest mass density associated with the energy density calculated in step 1. Rearrange the formula to find the mass density (\(\rho\)): \(\rho = \frac{E_0}{c^2}\) Use the value of energy density \(u\) for \(E_0\) (since energy density and rest mass density are equivalent in this case): \(\rho = \frac{1.01 \cdot 10^{18}}{(3.0 \cdot 10^8)^2} \approx 1.13 \cdot 10^{13} \mathrm{kg/m^3}\) The rest mass density associated with the energy density of the magnetic field from part (a) is approximately \(1.13 \cdot 10^{13} \mathrm{kg/m^3}\).

Key concepts

Magnetar

Magnetars represent one of the most extreme environments known in the universe. These celestial objects are a type of neutron star, which are the collapsed cores of massive stars that have exploded as supernovae. Neutron stars are incredibly dense, so much that a sugar-cube-sized amount of neutron star material would weigh about a billion tons on Earth. But what sets magnetars apart from other neutron stars is their extraordinarily powerful magnetic fields.

Magnetic fields of magnetars can be over a thousand times stronger than the typical neutron star's, reaching magnitudes up to and exceeding \(10^{11} \text{T}\). When a magnetar's magnetic field changes, it can emit X-rays and gamma rays, which we can detect from Earth. Their intense magnetic fields are responsible for starquakes, which are crustal shifts that release massive amounts of energy. Understanding the energy density of such magnetic fields helps scientists estimate the forces involved in these spectacular cosmic events and the effects of such fields on the vicinity of the magnetar.

Special Theory of Relativity

The Special Theory of Relativity, developed by Albert Einstein in 1905, revolutionized our understanding of space, time, and energy. One of the most famous outcomes of this theory is the mass-energy equivalence principle, encapsulated in the formula \(E = mc^{2}\). This equation shows that energy (E) and mass (m) are interchangeable; that is, mass can be seen as a form of concentrated energy. Moreover, the energy associated with the mass of an object at rest is known as its rest energy.

The implication of this for objects in intense magnetic fields, such as magnetars, is remarkable. The magnetic field can be thought of as having an associated energy density, and therefore, by extension, a mass density. This bridges the gap between the abstract concept of magnetic fields and the concrete reality of mass and energy as we understand it in the physical universe. It's key in calculations where the conversion of magnetic field energy into mass or vice versa is considered.

Rest Mass Density

Rest mass density is a concept that arises from discussing the distribution of mass in a given volume while the object or substance is at rest. In the context of magnetars and the Special Theory of Relativity, the rest mass density refers to the equivalent mass for the energy density of a magnetic field, assuming that energy and mass are interchangeable. It's a measure of how much mass is represented by the energy contained in a specific volume of space – a somewhat abstract concept that merges the physicality of matter with the energy in fields like magnetism.

When we calculate the rest mass density associated with the energy density of a magnetar's magnetic field, we're essentially determining how dense that energy would be if it were to be converted to mass. This number is tremendously high due to the immense magnetic energy present, offering insights into the profound influences that such fields could have on matter and space-time around a magnetar.