studiot

Perhaps you would like to show mathematically how this claim follows from the mathematics? While you are at it please explain your use of the word 'current' in this context?

Just to pick up on swansont's point; You need to distinguish between translational (accelerating) motion which classically leads to radiation by a charge and rotation or (accelerating) angular motion which does not. I say classically because with the correct boundary conditions in quantum mechanics the translational motion does not lead to radiation.

No We can derive this directly from Faraday's Law of induction which states: The induced voltage is directly proportional to the time rate of decrease of the magnetic field. [math]\frac{{\partial E}}{{\partial x}} \propto - \frac{{\partial B}}{{\partial t}}[/math] Now the OP shows a sinusoidal E wave and B wave and using the same axes we work as follows Let t be the time, [math]\omega [/math] is the angular frequency of the wave k a constant [math]\varphi [/math] is the yet to be determined phase difference between the E wave and the B wave. The E wave may be represented as [math]{E_{\left( {x,t} \right)}} = {E_0}\cos \left( {\omega t - kx} \right)[/math] and the B wave as [math]{B_{\left( {x,t} \right)}} = {B_0}\cos \left( {\omega t - kx + \varphi } \right)[/math] Differentiate the E wave with respect to distance, x [math]\frac{{\partial E}}{{\partial x}} = k{E_0}\sin \left( {\omega t - kx} \right)[/math] and the B wave with respect to time, t and taking the engative [math] - \frac{{\partial B}}{{\partial t}} = \omega {B_0}\sin \left( {\omega t - kx + \varphi } \right)[/math] Faraday's Law tells us these must be equal everywhere so If k is positive, ie the wave is proceeding in the positive x direction, then [math]\varphi = 0[/math] If k is negative, ie the wave is proceeding in the negative x direction, then [math]\varphi = \pi [/math] These are the only allowable values that fit the equations.

I wonder if the area of this triangle is red herring? The area allows distinguishing between two possible cases of the sine rule, but we are using the cosine rule which is unambiguous. The triangle inequality holds for all triangles therefore for all cases of the sine rule.

In what way is it different? Please look here for the geometry https://en.wikipedia.org/wiki/Triangle_inequality

Can you demonstrate how that might be possible? I note you are now saying shift not rotate. The maxima of the electric and magnetic vectors are interdependent so must coincide along the axis of shift.

First I would like to thank imatfaal for reminding me to read the question properly. I was creating some quite complicated equations, because I neglected to read condition 1 properly and thought the triangle area referred to the original triangle. Stupid fool. Anyway I was also trying to avoid imatfaal's solution and produce a purely geometric one (no trigonometry). So here is the beginning of it. Condition2 allows us to instantly determine that neither angle C nor angle B can be obtuse because the altitude AE is said to line on (intersect) BC as shown. Euclid did produce a construction equivalent to the cosine rule and if anyone is interested they might like to replace the trigonometry with this. Until that point here is my version using condition 1.

Did you miss this? I emphasized the 'if' to remind you that the OP specified a plane wave, not a circularly polarised one. It's there, writ large in post#1. Remember also that if you rotate one vector (you rotate the E and B vectors not the axes themselves) you must rotate the other one to maintain a proper relationship. But all this is off topic. In a plane wave, the E vector and the B vector are separately plane waves at right angles. The phase angle between the E wave and the B wave is either zero or pi depending upon the direction of travel of the wave. (This is why some authors show the E wave in phase with the B wave and others show it in antiphase) In all cases the direction of travel corresponds to the cross product of the E and B vectors. Further in a plane wave the E vector and the B vector are identical at every point in the plane, at right angles to the direction of travel.

I would be interested to see the context of this problem, since my first thoughts are that triangle AED is impossible in euclidian plane geometry.

I am not sure what you mean by rotating an axis through 90o. If the electric and magnetic fields are 180o out of phase (in antiphase) then the wave is travelling in the opposite direction (from right to left). The E vector certainly does take on a helical path if the wave is circularly polarised. However that would be off topic since the OP defines a plane wave. The energy or intensity is proportional to the square of E. This theory also has nothing to do with photons. Can't see this as a valid interpretation of a plane wave.

OK so I agree you first figure is correctly oriented in that the wave is travelling in the correct direction for the E and B fields are to be phase as shown. So here is a sketch of the zy plane, preserving your orientation. The point is that the electric vector (shown solid) is the same at every point in the plane as is the magnetic vector and they form a rectangular grid pattern. The electric vectors are parallel to the z axis and the magnetic ones parallel to the y axis. I am worried your circle suggests a different variation, and I can't think of a way to prevent this.

My point is that specifying the interactions alone will not fully pin down the intrinsic nature of the some particles. Properties can vary with shape for the same material. https://mrsec.uchicago.edu/research/highlights/particle-shape-effects My example was the simplest since it refers to the interaction of a particle with a hole (nothing). Long thin particles can still 'wiggle through' even though one dimension is much larger than the hole and much larger than smaller rounded particles that can't pass the mesh (ie are retained.)

I think this is a bit like a fisherman defining fish by the size of net to catch them or a materials engineer defining aggregate particles by the size of sieve that retains them.

Indeed this is true.

As the author of post#5, did you read the rules here? Why should I go rambling over the net to find out what should be summarised in a post?

In Newtonian mechanics, the total mechanical energy of an isolated system is conserved. Energy may be swapped between individual reservoirs of energy. Often these are just Kinetic Energy and Potential Energy. Further energy can be introduced into the system by outside agents, at the expense of removing the isolation constraint.

Yes mass is a better quantity than volume to work with, both for the mechanics of viscous flow and the chemistry of reaction rate kinetics. Each of this will lead to a differential equation. These differential equations will be coupled or simultaneous and you solve them as such. For the chemical kinetics of solution look at the Law of Mass Action https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=law+of+mass+action&gbv=2&oq=law+of+mass+action&gs_l=heirloom-hp.3..0l10.1297.9234.0.9781.26.13.4.9.10.0.204.1436.5j7j1.13.0....0...1ac.1.34.heirloom-hp..0.26.1953.HxTKyxqO_ak

5 posts and I still have no idea what the debate about the Physics of time is about. There is, anyway, far too much Physics (of time) to fit into one thread here so can someone (since the OP is not listed as having returned since starting this thread last November) establish a simple discussion point please?

Just to make wtf's question a little more difficult How many are left if you only leave every 3rd number instead of every second? What about only leaving every 4th number? What happens if we only leave every nth number? And what happens if we let n tend to infinity?

Once again some preliminary work is needed. It is important to realise that transfinite numbers do not obey the same rules of arithmetic as finite numbers. In particular if you remove (subtract) a finite number or a lesser infinity of numbers from an infinite set you still have the original cardinality. So if you remove an infinite subset of R made up of all the rational numbers you are still left with larger infinity of irrational ones. It is far easier, though, to follow the route of the pioneers and construct the number system, starting with the simple counting numbers and adding new types of number as they become necessary and then exploring as fully as possible the properties of these new types of numbers. That way you prove the rationals have the same cardinality as the integers. http://math.stackexchange.com/questions/12167/the-set-of-rationals-has-the-same-cardinality-as-the-set-of-integers.

Of the choices listed I would suggest that Environmental Science offers the least 'learning load' for those with small backgrounds in the Life Sciences. The syllabus seems to allow someone with an analytical mind to discuss topics logically. The other syllabuses on offer seem to lead on to further work in their respective areas and require a good deal of learning of facts, terminology etc that will frankly be baggage to you.

First terminology again, sorry. An 'interval' is a technical term for a set of real numbers which contains every point between the end points. We don't get to pick and choose. For other technical reasons not of interest here we should only apply 'interval' to the real numbers. Another technical point of interest is that intervals come in two types. Closed intervals include their end points, open intervals do not. This may become useful and relevant if this discussion develops. So The set of real numbers [math]\left\{ {x:1 \le x \le 11,x \in R} \right\}[/math] Is an closed interval between 1 and 11 and the set [math]\left\{ {x:1 < x < 11,x \in R} \right\}[/math] Is a open interval between 1 and 11. However the sets [math]\left\{ {x:1 \le x \le 11,x \in N} \right\}[/math] and [math]\left\{ {x:1 \le x \le 11,x \in Q} \right\}[/math] are not intervals since they refer to integers (denoted N) and to rational numbers denoted Q). Both these sets are subsets of the real number interval. The set of integers is finite , the set of rationals is infinite, and has a lower cardinality than the interval set in R. This is just one way to approach the meaning of numbers, however. We realise that we need different types of number to satisfy different equations. For instance consider the equation 2x = 5 There is no integer that satisfies this equation. In fact there is an infinity of such equations, one for each odd number.

This is why I started with the smallest infinity, that of the whole numbers or integers. An infinite set, chosen from the reals, (ie an infinite subset of the reals) can be chosen to only include the whole numbers. This has already be shown to be of smaller size than the set of reals itself.

You may not have realised it but they are all over the place. Take a cube [math]\delta x,\delta y,\delta z[/math] in continuum mechanics. You can consider the electric/magnetic/fluid/heat flux through the cube and use the engineer's most popular equation Input = output plus accumulation to derive all sorts of useful stuff. Or you can look at each face and note that there is (could be) a shear and normal stress associated with each face. Flux or stress is something per unit area [math]\delta x\delta y[/math] etc and it makes no sense to have zero area in the denominator of a definition. But for engineering purposes you still need a table of the stress at a point, the flow at a point etc.

Since wtf has posted such an able discussion of numbers let us return to the other part of your question - that of infinitesimals. Infinitesimals are not numbers and are not really used these days by pure mathematicians. They are, however, of immense use in applied maths where numbers are given significance in some physical sense and called quantities. Infinitesimals are quantities that are finite but small compared to the main bulk of the property or quantity being considered. We can conceive of a sequence of these getting smaller and smaller and calculate what is known as a 'limit' for some compound property ( a quantity made up of more than one infinitesimal) which we regards as the 'value' of that property at a point. A good example is density which is the ratio of mass to volume. We call the density at a point the limit of this ratio as we shrink the infinitesimals of mass and volume. Obviously they can never actually be allowed to reach zero or we would be trying to divide by zero. At one time the differential calculus was predicated upon such a ratio but we adopt a different approach today.