A magnetar (magnetic neutron star) has a magnetic field near its surface of magnitude 2.45⋅ 1010 T2.45· 1010 T . a) Calculate the energy density of this magnetic field. × 1025 J/m3× 1025 J/m3 b) The Special Theory of Relativity associates energy with any mass m at rest according to E0 = mc2𝐸0 = 𝑚𝑐2 (more on this in Chapter 35). Find the rest mass density associated with the energy density of part (a). × 109 kg/m3× 109 kg/m3
Question
A magnetar (magnetic neutron star) has a magnetic field near its surface of magnitude 2.45⋅ 1010 T2.45· 1010 T . a) Calculate the energy density of this magnetic field. × 1025 J/m3× 1025 J/m3 b) The Special Theory of Relativity associates energy with any mass m at rest according to E0 = mc2𝐸0 = 𝑚𝑐2 (more on this in Chapter 35). Find the rest mass density associated with the energy density of part (a). × 109 kg/m3× 109 kg/m3
Solution
a) The energy density (u) of a magnetic field is given by the formula:
u = B² / (2μ₀)
where B is the magnetic field strength and μ₀ is the permeability of free space, which is approximately 4π x 10^-7 T m/A.
Substituting the given value of B = 2.45 x 10^10 T into the formula, we get:
u = (2.45 x 10^10)² / (2 x 4π x 10^-7) u = 9.5 x 10^25 J/m³
b) The rest mass density (ρ) associated with the energy density is given by the formula:
ρ = u / c²
where c is the speed of light, approximately 3 x 10^8 m/s.
Substituting the calculated value of u = 9.5 x 10^25 J/m³ into the formula, we get:
ρ = 9.5 x 10^25 / (3 x 10^8)² ρ = 1.06 x 10^9 kg/m³
So, the energy density of the magnetic field is 9.5 x 10^25 J/m³ and the rest mass density associated with this energy density is 1.06 x 10^9 kg/m³.
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