Magnetic Field/Strength questions

[Neoxenok]

I'm working on something of a writing project and while I'm a layperson as far as most forms of science goes, I've had my head in science and astronomy since I was very young and I'm familiar with many concepts and such.

Yet, when I came to writing about a magnetic field, I've found that I know far less than I thought I have, so I've come here to ask a few questions of the people here.

First, I need to explain a bit about my 'writing project' which is a science fiction writing project in which a spacecraft generates a very powerful magnetic field of a very particular kind, but all my searches and inquieries have proven to be quite fruitless when attempting to figure out how much energy that a very powerful magnetic field requires.
As such, my first question is just how much energy is in a magnetic field or how much energy in joules does a magnetic field require to operate per tesla of field strength?
The purpose of the question is essentially to answer how much power from the space craft's generator would the field would require to effectively operate.

While I may have found some sources that could have helped me, my lack of understanding of the subject prevents me from understanding what I need to know to properly utilize these equations. Any help in this regard would be tremendously helpful.

My second question is what effect do extremely powerful magnetic fields have on space-time or the fabric of space? When I say 'extremely powerful' I generally mean the equivelent strength of a magnetar's magnetic field and greater.

Any help in these matters would be greatly appreciated.

 

[diazona]

There's a simple formula for the energy density that's actually stored in a magnetic field:
$$u = \frac{B^2}{2\mu_0}$$
(see e.g. Wikipedia). You take the magnetic field strength (in tesla) at a particular point, square it, and divide it by 2μ₀ (where μ₀ is the vacuum permeability), and that gives you the energy per unit volume at that point. To get the total energy stored in the magnetic field in an extended region of space, you would have to integrate the energy density:
$$U = \iiint u \mathrm{d}^3 V = \frac{1}{2\mu_0}\iiint B^2\mathrm{d}^3 V$$

Now, note that that's just the energy stored in the magnetic field itself. The amount of energy required to produce that magnetic field will generally be more, since some energy will be wasted in the production process. How much exactly depends on how the magnetic field is produced.

As far as I know (or can think of now), the only known effect powerful magnetic fields would have on spacetime is the gravitational distortion created by the energy they contain. Basically, according to general relativity, the "warping" of spacetime that we call gravity is related to the value of something called the stress-energy tensor, which is a set of numbers describing the contents of space. One of the numbers is related to (well, basically is) the density of matter, which means that matter causes gravity. The energy density of a magnetic field also enters into the stress-energy tensor, so a magnetic field would also have some gravitational influence, although it's much less than that of normal matter.

 

[Neoxenok]

I'm sorry, when I returned and found your answer, I've found that I neglected to mention a few important details, so I edited my primary post to reflect the missing few pieces of information.

Specifically, I'm trying to equate the power generated by the craft's power source into the craft's artificial magnetic field. Mostly, I've figured that the craft has a budget of about 449,377,589,500,000 joules of power per second before the engines and minor internal power needs and that it needs very slightly over 49,500,000,000,000 joules of power per second to achieve a 100 m/s acceleration for a roughly 2.2 billion kilogram object.

So by leaving out a few trillion joules to spare, I could potentially make use of up to 390 trillion joules per second to make use elsewhere. More to the point, I just need to know if that's enough power for my magnetic field of the target power level at the rate in which power is being generated.

In regards to your first answer, I think I understand what I have to do.

For simplicity's sake, let's say my MF is 1000 tesla, square it (1 million) and then divide that by twice the vacuum permiability.

The vacuum permiability seems to be 2 x 10^-7 Newton per square meter, so double that figure would be what I divide the 1000 tesla squared number by, resulting in:

1,000,000 divided by 4 x 10^-7 is 2,500,000,000,000 but I don't know what form of energy it is (if it's in joules or some other measure of energy).

So, assuming that number were in joules, that would mean I would need 2.5 trillion joules to power a 1000 tesla magnetic field?

From the sound of what you're saying, a field might have a certain amount of energy stored in it (like any permanent or temporary magnet) and I could calculate that amount of energy by taking the energy density and area the field occupies and taking all that into effect, just like figuring out how many asteroids are in the asteroid belt without taking a one-by-one rock count so I could calculate the total mass so I could properly know how much energy I'd need for a certain field strength.

I'm not entirely sure it'll be necessary to calculate the total energy stored within the magnetic field, more or less, I'm just trying to get some idea of a second-by-second energy requirement of the field, but I suppose if the field doesn't constantly loose energy, my generators could potentially need a much lower power requirement than I anticipated because the field is storing energy and loosing it at a comparatively lower rate.

Am I on the right track here?

There's a simple formula for the energy density that's actually stored in a magnetic field:
$$u = \frac{B^2}{2\mu_0}$$
(see e.g. Wikipedia). You take the magnetic field strength (in tesla) at a particular point, square it, and divide it by 2μ₀ (where μ₀ is the vacuum permeability), and that gives you the energy per unit volume at that point. To get the total energy stored in the magnetic field in an extended region of space, you would have to integrate the energy density:
$$U = \iiint u \mathrm{d}^3 V = \frac{1}{2\mu_0}\iiint B^2\mathrm{d}^3 V$$

Now, note that that's just the energy stored in the magnetic field itself. The amount of energy required to produce that magnetic field will generally be more, since some energy will be wasted in the production process. How much exactly depends on how the magnetic field is produced.

 

[Curl]

So by leaving out a few trillion joules to spare, I could potentially make use of up to 390 trillion joules per second to make use elsewhere. More to the point, I just need to know if that's enough power for my magnetic field of the target power level at the rate in which power is being generated.

hold on... you're making this in your garage?

 

[diazona]

For simplicity's sake, let's say my MF is 1000 tesla, square it (1 million) and then divide that by twice the vacuum permiability.

The vacuum permiability seems to be 2 x 10^-7 Newton per square meter,

Actually it's $4\pi\times 10^{-7} N/A^2$ (that's Newtons per ampere squared)

so double that figure would be what I divide the 1000 tesla squared number by, resulting in:

1,000,000 divided by 4 x 10^-7 is 2,500,000,000,000 but I don't know what form of energy it is (if it's in joules or some other measure of energy).

The joule is the SI unit of energy, so if you use SI units (e.g. amperes, meters, Newtons) for everything else, you'll get an answer in joules. Actually, what you calculate here is an energy density, so it'll be in joules per cubic meter.

So, assuming that number were in joules, that would mean I would need 2.5 trillion joules to power a 1000 tesla magnetic field?

0.4 trillion joules (per cubic meter) actually... and not exactly. What it really means is that the magnetic field stores 0.4 TJ (terajoules = trillion joules) of energy in each cubic meter of space that it fills - or equivalently, that it takes 0.4 TJ of energy per cubic meter of space to raise the magnetic field strength from 0 T up to 1000 T. So for one thing, you have to take the volume occupied by the field into account. But also note that this is (roughly speaking) really the amount of energy it takes to create the field. Once it's created, it could theoretically be kept at that strength without expending any additional energy. Of course, it wouldn't be useful; you'd just have a big magnetic field sitting in space not doing anything. I'm guessing that in your story, the ship probably runs some device inside this magnetic field, or maybe uses it to do work in some way, and that may incur an additional energy requirement. It depends on the details.

From the sound of what you're saying, a field might have a certain amount of energy stored in it (like any permanent or temporary magnet) and I could calculate that amount of energy by taking the energy density and area the field occupies and taking all that into effect, just like figuring out how many asteroids are in the asteroid belt without taking a one-by-one rock count so I could calculate the total mass so I could properly know how much energy I'd need for a certain field strength.

Yep, pretty much.

I'm not entirely sure it'll be necessary to calculate the total energy stored within the magnetic field, more or less, I'm just trying to get some idea of a second-by-second energy requirement of the field, but I suppose if the field doesn't constantly loose energy, my generators could potentially need a much lower power requirement than I anticipated because the field is storing energy and loosing it at a comparatively lower rate.

Yeah, like I said, it really depends on what is going on within the magnetic field. Since this is science fiction, that's really up to you - you could say that the magnetic field is used in some process that is unknown to modern (real-world) science, and then just make up a number that fits within the ship's energy budget.

P.S. If these numbers are going to go in your story, note that the unit of power (energy usage per unit time) is the watt, which is one joule per second (1 W = 1 J/s). Engineers on the ship would probably talk about power requirements in terms of gigawatts (billions of watts) or terawatts (trillions of watts).

 

[Neoxenok]

@ Curl

No. I wish. I'm just making sure my sci-fi story is less ridiculously out of whack with science than it has to be.

Actually it's $4\pi\times 10^{-7} N/A^2$ (that's Newtons per ampere squared)

The joule is the SI unit of energy, so if you use SI units (e.g. amperes, meters, Newtons) for everything else, you'll get an answer in joules. Actually, what you calculate here is an energy density, so it'll be in joules per cubic meter.

0.4 trillion joules (per cubic meter) actually... and not exactly. What it really means is that the magnetic field stores 0.4 TJ (terajoules = trillion joules) of energy in each cubic meter of space that it fills - or equivalently, that it takes 0.4 TJ of energy per cubic meter of space to raise the magnetic field strength from 0 T up to 1000 T. So for one thing, you have to take the volume occupied by the field into account. But also note that this is (roughly speaking) really the amount of energy it takes to create the field. Once it's created, it could theoretically be kept at that strength without expending any additional energy. Of course, it wouldn't be useful; you'd just have a big magnetic field sitting in space not doing anything. I'm guessing that in your story, the ship probably runs some device inside this magnetic field, or maybe uses it to do work in some way, and that may incur an additional energy requirement. It depends on the details.

Okay. Well, I've based the size and shape of the ship roughly on the empire state building, though it is four times its mass and twice its actual physical size. I figure the field's shape would mirror the ship's shape (due to being not one but several magnetic field that would be adding to the whole) so that the object itself and all the humans inside wouldn't be affected by the field any more than is necessary to shape the field itself.
So in order to get the area and make things as easy on myself as possible, I'm going to assume that the total volume of the field is twice the empire state building's floor area, except in cubic meters instead of square meters, or 514,422 m^3.

I had a much grander idea for what it would do when I started my story (I thought it would be a major component of its FTL ability), but part of the reason I'm here is because it seemed quite ... implausible. So it's primary function now is simply going to be to protect the crew while it travels at FTL speeds from everything from gasses to micro-meteorites if not actual asteroids and comets in the vast vacuum of space. Either way, I think the field is going to have to be extremely powerful and it will need to be able to power down enough so that the crew won't be trapped inside once activated. I've pegged the maximum field strength to be 800 petateslas so I'm hoping that'll be enough to allow a 2.2 billion gram object to punch through just about anything it would have a reasonable expectation of coming across at 1.5c, unless it's actually unnecessarily powerful.
This is of course combined with heavy radiation shielding of the hull and everyone on the ship has received treatments - both chemical and gene therapy to allow for additional resistance to radiation.

I'm not sure exactly how it'll be going at FTL speeds yet, but perhaps I can mine NASA.gov, these forums, or wikipedia for ideas.

In any case, I think you've given me my answer then, but I'll double check to be certain.

Now, if i take the field strength of 800 petatesla, or 8 x 10^17 tesla, square it (6.4 x 10^35), divide by 4(3.14159265) x 10^-7, which seems to equal (approximately):

5.092958 x 10^41 joules/m^3

Then I take that number and multiply it by the actual number of cubic meters, which I've decided was 514,422 m^3, so simple multiplication results in a grand total of (approximately):

2.6199297 x 10^47 joules of total energy within the magnetic field

Since I have the total energy, this also seems to mean that I can calculate how long it would take for my ship to actually be able to raise the field to its full strength, so if the ship budgeted 390 gigawatts of power (the other half going to the FTL component), then simple division will give me the number of seconds needed to fully energize the field.

The result being: 6.71776855 x10^32 seconds.

Eesh. 21 septillion years. ... I have a newfound respect for magnetars now.
Hmm. Let me try 5000 tesla, which according to math is 10.23 quintillion joules in total strength and would take just over 26 thousand seconds or about 7 hours to fully energize.

I probably could go a fair bit higher than that, but I won't waste space here writing all that out.

Okay. I think I've gotten a far better understanding of what I'm doing now. I'll definitely need to go back and change a few things, but not nearly as much as I thought I would have.

You're right though, gigawatts or terawatts would seem more approrpriate. I think I just got into the habit awhile ago after dealing with figuring out the distance, speed, velocity, acceleration, and all that fun stuff.

P.S. If these numbers are going to go in your story, note that the unit of power (energy usage per unit time) is the watt, which is one joule per second (1 W = 1 J/s). Engineers on the ship would probably talk about power requirements in terms of gigawatts (billions of watts) or terawatts (trillions of watts).

Thanks. You've been an enormous help. Let me know if I missed anything and I'll make adjustments as appropriate.