Stars have magnetic fields but the energy of those fields...

Stars have magnetic fields but the energy of those fields has generally been neglected in discussing the equilibrium configuration for stars. Justify this for the Sun by estimating an upper limit for the average magnetic field to be given by that in sunspots, which you may take to be about $0.1 \mathrm{~T}$ (the average field is certainly less than this), and computing the ratio of the energy density of the magnetic field to that of the gravitational field.

Key Concepts

Magnetic Energy Density

This is the energy stored in a magnetic field per unit volume. In physics, it is quantified by the expression B²/(2??), where B is the magnetic field strength and ?? is the permeability of free space. It provides an estimate of how much energy a magnetic field contributes to the overall energy budget in a given volume.

Gravitational Energy Density

This represents the energy associated with the gravitational field per unit volume, often estimated in the context of stellar interiors by considering the gravitational binding energy. It is a key parameter in determining the stability and equilibrium configuration of a star, as it dominates the energy balance in stellar structures.

Stellar Equilibrium

Stellar equilibrium involves a balance between various forces and energy contributions, including gravitational forces and pressure from gas and radiation. In this context, magnetic energy is often negligible compared to the dominant gravitational energy, which justifies its neglect when modeling the overall equilibrium of stars like the Sun.

Order-of-Magnitude Estimation

This is a method of approximating values to the nearest power of ten to assess the relative importance of different physical effects. In comparing the energy density of the magnetic field (using sunspot field strengths as an upper limit) to the gravitational energy density, one can show that the magnetic contribution is very small, thereby justifying its omission in equilibrium models of stellar structure.

Transcript

00:02 In this question, we are given that the magnetic field in most of the regions in our galaxy is equal to 10 raised to par minus 6 and 10 raised to par minus 5 cost.

00:22 Okay.

00:24 So for a typical value, we will adopt that the magnetic field is equal to 3 into 10 raised to par minus 6 cost.

00:33 And we have to find out the total energy in the magnetic field and then they are telling us that the diameter of disk is equal to 10 x2 .23 centimeter that is equal to 10 x .2 part 23 and the thickness of this disk is equal to 10 raised to par 21 centimeter that is equal to 10 to bar 19 meter then it tells us that if the radiation power of the stars radiation power of the stars is equal to 10 raise to power 30 10 to power 30 10 to bar 44 earths per second and if we'll convert it into joules per second so we'll get 10 raised to bar 37 joules per second so we have to find out that what would be so what would be the number of years of star light worth is even magnetic energy worth is the magnetic energy worth so this is the given data and we have two parts that we need to solve so let's start solving the first part so for a part that we need to find the energy so for that we know that the energy density is equal to b squared divided by 2 mu not where mu not is equal to 8 .85 into 10 raise to power minus 7 okay so we have the value of magnetic field substituted and we find that the energy density is equal to 3 .6 into 10 raise to power minus 14 joules per meter cube.

03:37 So this is your energy density.

03:42 The next thing that we need to find is that the volume of ccc is pi r square age.

03:59 So this would become this volume will be equal to pi r square 8 that is 5 times 5 into 10 raised to power 21 times 10 to power 19 so the volume is 8 into 10 to power 60 meter cube now the total energy total energy that is in the magnetic field so so that would be magnetic energy equal to 3 .6 into 10 raised to power minus 14 times 8 into 60...

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