Components of flux

We used this apparatus to measure components of flux. There was an alternating current in the copper Helmholtz coil which created an alternating flux inside the circle. The thing propped on the pink ruler in the middle is a search coil. The changing flux passes through this coil and induces an emf which is measured by an oscilloscope. We plotted the angle of between the search coil and the Helmholtz coil versus the peak voltage. You get a rough straight line if you plot the sine of the angle.

Spatial frequency: more thinking about wave number

I did a bit of research after posting yesterday. My thinking had given me the right questions. It turns out that a more stripped back version of wave number is called spatial frequency and is just 1/wavelength. It is given the Greek letter xi, which is the one I find hardest to draw. Frequency is 1/T (T is time period) so I teach it as the number of complete waves in one second. 1/wavelength is the number of complete waves in 1 metre; hence spatial frequency. I've been reminding myself about Fourier Transforms: that a Fourier Transform tells you which mix of frequencies make up a signal in real time. Remember the progressive wave y=Asin(kx-wt). w is paired with t. That is frequency with time, the Fourier Transform pair. So k is paired with x. It's beginning to dawn on me that a Fourier Transform of a repetitive pattern in space will tell me the mix of spatial frequencies involved. I'm almost convinced that the reciprocal lattice is the Fourier Transform of the real lattice for this reason. This hints at it https://agilescientific.com/blog/2012/5/1/k-is-for-wavenumber.html I sued this a lot https://en.wikipedia.org/wiki/Spatial_frequency

thinking about progressive waves and wave number

The equation for a progressive wave is given by y=Asin(kx-wt). y is the displacement from equilibrium and A is the amplitude. k is called the wave number and is 2*pi/wavelength. w (omega) is the angular velocity. It is 2*pi/T, where T is the time period. A wave in periodic so w is how fast things go round in a circle, measured in the number of radians covered in a second. So what is k? Well, it must be measured in radians per metre so I'd say that it is a kind of angular gradient. It is about turning a distance into an angle in radians. It's an odd gradient because it is periodic and resets every wavelength. I was thinking about what happens if you choose a fixed position. Take x = 0. Then displacement y = ASin(-wt). The particle just goes down and up on the spot. Now freeze the time. Take time t=0. y=Asinkx. You get the familiar wave shape.

Bassenthwaite Lake catchment area

I have blogged before about water catchment areas having estimated them from maps but this time I found the catchment area value given. Suppose an inch of rain falls over the catchment area - that might be a decent day in Cumbria. That's 0.025m. 350 square km is 350 million square metres so the volume of water falling in the catchment area would be of the order of 10 million cubic metres or 10 billion litres. I found a figure that on average a person in the UK uses 150 litres per day. That would mean the volume of an inch of water in this catchment would sort out the whole UK for a day. So why are we short of water?

Country Walking magazine travels through space: 1000 miles in 2018

On the Country Walking magazine "Walk 1000 Miles in 2018" chart, it claims that we move 137 miles through space every second. The Earth is 93 million miles from the Sun. Assuming a circular orbit, the distance is 2 x pi x radius = 584 million miles a year. In a year there are 365.25 days x 24 hours x 60 minutes x 60 seconds = 31.6 million seconds. 584/31.6 = 18.5 miles in one second. The distance due to the rotation of the earth about its axis is much smaller. The earth also rotates around the Milky Way https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question18.html gives the speed as 828000 km/hr so that means 230 km per second. That's 144 miles in one second. This is probably the source of their statistic but http://www.physlink.com/education/askexperts/ae548.cfm gives a different figure for movement relative to the microwave background radiation. I do love a good statistic!

Perturbation applied to electron wave function in a solid

I've been working on what happens when the positive ions in the lattice of a solid are taken into account in what is know as the Nearly Free Electron approach. Perturbation theory is the idea that you slightly wobble the steady state energy level and see what happens. Here the perturbation is given a value V and is modelled by a Fourier series because it is periodic, rising and falling from one ion to the next. Applying first order perturbation theory you get this for the final change to the energy

 I had to swot up on my maths to work out why the integrals circled come out to zero.

 For the next bit, I read that you only get non-zero solutions for the values of k given.

 I looked up the integral for cos-squared and I'm hoping that my cancelling is allowed.Then end result is 1/2V.

Mini-Beast: Blizzard at Carlisle Bus Station

We're in the grip of another cold snap. I found a wonderful 1 minute film that explains what has happened to the polar vortex http://www.bbc.co.uk/news/av/science-environment-43172539/why-the-uk-is-so-cold I was interested in the idea that as the winds move more slowly the radius of the circulation decreases. This is the opposite of a spinning ice skater, where pulling her arms in makes her spin faster. In her case, angular momentum is conserved. i'm guessing that angular momentum is not conserved in the polar vortex so dissipative forces must be acting. As the air slows down it will become more dense as particles come together. This is part of the explanation of the aerofoil shape and lift on a plane wing. Here the increased density would explain why the block of air sinks.

Hawking in the Westgate Centre

I took this photograph in the new Westgate Centre in Oxford last month. I was sad to hear of his death today. I still haven't finished A Brief History of Time. Every time I try, I get inspired part way through to go and research stuff that I've read. That's how good it is.

My room is a closed system: what can get in?

I shut all the windows and doors. I shut the vents and turned off the air-con. My room is a closed system because particles can't get in or out. But the temperature can change. How so? In a closed thermodynamic system, heat and work can cross the sealed boundaries when particles can't. In other words, energy can cross the boundaries. Traditionally, we say heat and work. At GCSE, an attempt has been made to interpret that. Work has been interpreted as MECHANICAL WORKING and ELECTRICAL WORKING. Both are valid interpretations of work. Heat has been boiled down to HEATING BY PARTICLES (basically conduction) and HEATING BY RADIATION (infra-red would be the obvious example). So I asked my class what could get in and out of my classroom. Good answers named the two interpretations of heat - with the Sun streaming in through the window, one was obvious. I hadn't expected work, because the room doesn't expand or contract. But my smartest added ELECTRICAL WORKING. And she's right! Energy was coming into the room from outside to work the lights. I'd missed that.

Tide clock

This tide clock is in Maryport Aquarium. It's not like a normal clock: it just tells you how many hours before or after high tide it is. In this case it was 3 hours after high tide. One other difference to a normal clock is that it takes more than 12 hours to go round once. Tides are caused by the Moon, which doesn't stay still. It moves on every day so there will be about 12 hours 25 minutes between two high tides.

Rainbow experiment

This is a chemistry experiment beloved of science teachers. Fill a tube with dilute hydrochloric acid and then dribble dilute sodium hydroxide down the side of the tube. the sodium hydroxide solution is more dense so it sinks to the bottom. At the interface, neutralisation occurs producing the rainbow. It has just occurred to me to leave one for a while next time to see if diffusion is discernible. We inject denser copper sulfate solution into the bottom of a tube of water and wait weeks for it to diffuse upwards (well, and for the water to diffuse downwards). I also need to find out if the sodium hydroxide is denser because it has a higher molecular mass or whether it is related to another factor like solubility.

Mackeral tank as lens in Maryport

The circular fish tank acts like a huge lens. If you look carefully at the writing on the emergency exit sign (the green one) you'll notice that it is backwards, which is called lateral inversion. That's because the light rays have swapped sides as they have been bent through the lens. Writing about this has got me wondering about the focal length. Next time I go I need to use the equation 1/u + 1/v = 1/f to calculate it.

Contrails and the Saltire

Having got my head round the idea that the planes over us are into the stratosphere and that this is because there is a lack of turbulence in the lower stratosphere, it then occurred to me that I had noticed vapour trails being dispersed by the wind. I noticed two trails at 90 degrees to each other like the Scottish saltire legend. They both spread laterally which suggests that the lateral spreading is not due to wind or that the winds are at 90 degrees to each other in two close high layers of air. A little research: I had heard the phrase contrail before and now understand that it is "condensation trail" from the water vapour produced by combustion of the fuel. Research suggest that these trails can become cirrostratus cloud but this doesn't exist in the stratosphere. Hence I conclude that these planes may well not quite have been into the stratosphere.

Thinking about Hamiltonians at Hamilton services

Every time we go through Hamilton Services on the M74 I end up thinking Physics.

I know that the town is not connected with the great Irish mathematician William Rowan Hamilton, but it has given me the kick I need to try out a Hamiltonian. The Hamiltonian is a function which describes the total energy of a system and this can evolve over time so that results in mechanics can be calculated. I decided to see what happens in this type of analysis for a falling object. The object is to be of mass m and dropped from height h. First I used the suvat equations of uniformly accelerated motion to calculate an expression for the distance of fall, which I have called x, and for the speed v at that height.

 Then I used these expressions to calculate the kinetic energy T and potential energy V.

 The Hamiltonian turns out to be the constant value mgh which is good because that's what it should be if air resistance is neglected. Maybe the Hamiltonian will be a good analytical tool if we decide to include a dissipative force.

L is the Lagrangian which is the difference between the energies. I have a problem. My further analysis using the Lagrangian is out by a factor of 2. There may be a flaw in my analysis. I'll think about it before writing more about this.

Thinking about the reciprocal lattice

I have been reading about the lattice structure for solid state physics. This was studied by Henry Bragg, born 2 miles from here, and his son Lawrence. They shone X-rays through crystal structures and interpreted the weird patterns that resulted. I was introduced to the idea of the reciprocal lattice. This is a mathematical representation of the true lattice constructed in an abstract mathematical space. I found a film that suggests that the reciprocal lattice is the X-ray diffraction pattern https://www.youtube.com/watch?v=DFFU39A3fPY. Another source says that the X-ray diffraction pattern is a MAP of the reciprocal lattice. Not sure what that means yet. So here's a thought. This is a net curtain with a lattice of threads.

Here's a diffraction pattern shot through the curtain. The implication is that the reciprocal lattice is connected to this new pattern.

Planes in the stratosphere

Having measured an aeroplane at 10km up the other week and having now learned about the stratosphere, the question becomes why do they fly so high. The answer is to do with the temperature inversion in that layer. Instead of the temperature falling as you go up, the temperature actually rises because the ozone absorbs ultra-violet radiation from the Sun. The heating is higher up in the stratosphere. Convection is difficult when the heating is at the top so there is less turbulence due to air currents. That's why planes fly up there.

The sun sets on the Beast

The snow has mostly melted away - it was a warmer afternoon and the Sun finally came out. It was shining through low cloud towards sunset so I was able to get a photograph of the disc that wasn't overexposed. So I was able to prove that the disc of the Sun subtends an angle of about half a degree at the eye (ie half the width of a little finger at arm's length). This is the same as the Moon, which must work because the Moon almost exactly covers the disc of the Sun during an eclipse.

The Beast from the East

Winter has arrived with a vengeance with the storm being called The Beast from the East. I first read about this in The Times newspaper weather column about 3 weeks ago. They noted that there had been a Sudden Stratospheric Warming (SSW) and that when that happens, the high level polar winds that circulate west to east can be affected. The stratosphere is about 9km up in the sky. The aeroplanes we see over Wigton are flying just into the stratosphere. It's an odd place because instead of it getting colder as you go up, the air temperature rises due to ultra-violet heating the ozone layer. I need to read more to understand the physics of this event more fully but I was really impressed that a warning that I read 3 weeks ago with scientific reasons given has turned out to be accurate. Last time this happened was 2013. You can look back at the blog to see that the summer wasn't too bad afterwards! https://www.metoffice.gov.uk/learning/learn-about-the-weather/how-weather-works/sudden-stratospheric-warming  and https://en.wikipedia.org/wiki/Sudden_stratospheric_warming