A magnetic field exists in most of the interstellar space in...

A magnetic field exists in most of the interstellar space in our galaxy. There is evidence that its strength in most regions is between $10^{-6}$ and $10^{-5}$ gauss. Adopting $3 \times 10^{-6}$ gauss as a typical value, find, in order of magnitude, the total energy stored in the magnetic field of the galaxy. For this purpose you may assume the galaxy is a disk roughly $10^{23} \mathrm{~cm}$ in diameter and $10^{21} \mathrm{~cm}$ thick. To see whether the magnetic energy amounts to much, on that scale, you might consider the fact that all the stars in the galaxy are radiating about $10^{44}$ ergs $/$ sec. How many years of starlight is the magnetic energy worth?

Key Concepts

Energy and Power Conversion

This concept involves comparing a stored energy value with a rate of energy emission (power) to understand time scales. By dividing the total stored magnetic energy by the luminosity (energy per unit time) of the stars, one can estimate how many seconds or years of stellar output the magnetic energy represents, providing a context for the significance of the magnetic energy relative to stellar processes.

Order-of-Magnitude Estimation

This technique is used to obtain a rough approximate value by focusing on the scale of the numbers involved rather than on precise calculations. It’s especially important in astrophysics where many quantities span several orders of magnitude and exact values are difficult or unnecessary to determine.

Magnetic Field Energy Density

This concept involves the energy per unit volume stored in a magnetic field, which is given by the formula u = B^2/(8?) in cgs units. It is fundamental when calculating the total magnetic energy in a region by integrating this energy density over the entire volume where the field exists.

Volume Estimation in Astrophysical Systems

For astrophysical problems like estimating the energy in our galaxy, it's crucial to approximate the volume of the region under consideration. In this case, the galaxy is modeled as a disk, and its volume is approximated by using geometrical formulas, typically the volume of a cylinder (?r^2h). This approach is useful for getting an order-of-magnitude estimate in large-scale systems.

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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