I recently asked for an explanation of Maxwell’s equations. It appears that my real difficulty is with differential equations.
When answering, please assume that I can look up questions pertaining to ordinary integral equations. And keep in mind the goal remains to comprehend Maxwell’s equations and not actually to penetrate the full expression of the field of differential equations.
Do you have any calculus background? LOL I found that regular differential equations could be easily understood as simply the slope (equation of the angle of the line at any particular point) of the graph. Partial differentials are a little bit more difficult to simplify though. If I recall correctly, integrals were comparable to the area under the slope?
Let’s rephrase - this looks like a general translation problem. Generally I think the best way to translate mathematics is just to draw the graph, but I may just be a visual person. What method of learning is best for you? Spatial/visual? Linguistic? Mathematical (obviously not)? Kinesthetic/tactile? Musical? Inter or intrapersonal? Naturalistic?
Simply put, a differential equation is one in which a relationship exists between some quantities and the rate of change of quantities with respect to something else. That sounds vague, but if I make it more definite, I’ll be excluding something.
In the simplest case of a first order linear differential equation, there are terms for one variable, say x, and it’s rate of change with respect to something, say dx/dt. If x is a position, then dx/dt is the velocity, and a simple differential equation could be one relating the position to the velocity:
dx/dt = Ax
Behold, a differential equation.
But it could be you’re relating x to the rate of change of x with respect to another variable, y. Or you could be relating the rate of change of the velocity with respect to time to x, which would make it a second-order equation. Or the velocity might be related to the square of the position x , or any of a number of possibilities. But the constant thing throughout is that you are relating various rates of change of variables to themselves and other variables.
Even if you restrict yourself to linear equations of sets of variables, it can be complex, and there are many methods developed for attacking such problems (Klaatu refers to one of them in the original Day the Earth Stood Still – Variation of Parameters. Aliens know math.) As far as Maxwell’s Equations are concerned, they’re first order linear equations, but they can be a pain. To begin with, you can generate second order equations by combining them.
Look up Differential Equations (Diffie Q’s) online, or in some books on the topic. But you’ll want basic differential calculus first.
Easy to visualize example:
You have a 5 gallon bucket completely full of sea (~3.5% salt) water. To avoid unit conversions, lets say that is 3.5% salt by volume instead of weight. You take a garden hose, flowing fresh (no salt) water at one gallon/minute and run it into the bucket, allowing it overflow. You have your pet monkey stir it constantly with a stick, so that all the water in the bucket is at the same salt concentration. *
Now it should be obvious that if you wait a VERY long time, the water in bucket will have virtually no salt. But what how much salt is in the bucket after 1 minute, 2 minutes, 2.3 minutes, etcetera?
To solve this problem, consider that the 1/5 of the salt in the bucket is overflowing each minute. That is to say that the rate that salt is going out of the bucket depends on how much is there at the time. If we call the amount of salt in the bucket at any given time "x(t) " , The differential equation that discribes this is:
dx(t)/dt=-((1gal/minute)/5gal) * x(t) = -x(t)/5 per minute. (eq 1)
Stated in English this is approximatly “Each minute we lose 1/5 of whatever our current amount of salt was”. This ignores the fact that amount of salt is NOT the same during each minute, it is constantly decreasing. Eq 1 accounts for this, while the version in english does not.
A complete solution to a differential equation requires an initial condition, and in this case that is:
x(0) = 3.5% * 5 = 0.175 gal salt. (eq 2)
If we take the time integral of both sides of eq1, we get:
x(t)= -1/5* ( the integral of x(t)) . Since we have studied calculus, we know that it is only the exponential function (e**x) that is it’s own integral. (this is where “the best way to solve a DE is to already know the answer” comes in) so we get:
x(t)=x(0) * (e**(-t/5))
(sorry for the lack of intermediate steps…I don’t know how to show integrals)
The final solution to our problem is:
x(t)=.175 * e**(-t/5)
X(1)=.143 gallons salt
x(2)=.117 gal salt,
x(2.3)=.110 gal salt, etc.
After a thousand minutes the monkey will become bored and run off, and the amount of salt is:
X(1000)= 242 * 10**-90 gal.
OR pretty much no salt at all unless you subscribe to the theories of homeopathy.
A better English gloss would then be “At every moment, the current speed of salt loss is 1/5 of whatever the current amount of salt is per minute”. Which is of course exactly what the equation says, but don’t sell English short.
Yeah, I got A’s through calc 208 but had major problems with diff eq. Long story. It has been a long time and it looks like I have forgotten much. It’s just as well that I start over again with these maths.
I’m… polymathic? Ha ha, ok, I did best in 208 because I was able to visualize all of the equations and all the details (at the time). Most things relate to music with me, though math doesn’t relate much to music, in that case it is more the other way around. My brain often snags narratives and recalls them word-for-word if I have some reason to remember them- which has a strong song-y aspect to it, though remaining pure language. So what’s best for me? I guess here I am going for heavy explanatory. To see the graphs might bring the ‘aha’ moment.
Maybe someone knows of some free equation-graphing software.
Thanks everyone for your answers. Actually, I need to chew on it. I can’t check into the 'dope every day this week, sorry if that comes across as inattentive.
As it relates to Maxwell… a charge is produced by the changing of a magnetic field. Apparently moving a magnet is the way to do that; putting one on a wheel will cause it to generate a constant charge, calculable with differential equations?
The changing magnetic field doesn’t create charge. It creates an electric field. Electric (and magnetic) fields ARE associated with charge, but they can and do exist without charge also.
There are no charges associated with photons (or electromagnetic waves if you prefer) They consist entirely of magnetic and electric fields expanding and collapsing each other through space. THAT is most of what Maxwell’s great insight was all about.
Possibly an even simpler example of differential equations: Every day, you drive to work along the same route. Parts of your route are on residential streets with a 25 MPH speed limit, parts are on major city streets with a 35 MPH speed limit, part is on the interstate with a 65 MPH speed limit, and there’s a bit at the end in the company parking lot with a speed limit of 10 MPH. So if we know where you are on your route at any given moment, we can also say how fast you’re going (i.e., what the rate of change of your position is) at that moment. The rate of change of your position is a function of your position. And an equation which expressed what that function is would be a differential equation.
But now let’s suppose that your wife knows that you left home 15 minutes ago, and wants to know where you are right now. In other words, she wants to know your position as a function of time. Well, one can get that information, of course, from knowing your speed as a function of position. And the process of getting that information is what’s referred to as solving the differential equation.
Now, of course, this is a very simple differential equation to solve, and would not require any knowledge of calculus, since the function for finding speed given position is very simple (if it’s between here and here, it’s 35 MPH, and so on). But one can construct much more complicated differential equations, as well, some of which require very advanced mathematical techniques to solve, and some of which can only be solved by doing the equivalent of driving along the route and seeing where you are at each time (the solution can’t be written down in a simple mathematical form at all).