Fairy music in Grizedale Forest

One of the installations in Grizedale Forest involves a key that goes through a tree. You have to turn the key 10 times and then music comes out of a speaker in the ground a few metres away. The key clearly turns a generator in the brass disc on the other side of the tree but why 10 turns before you get music? My guess is that you are charging a capacitor from the generator and that when it is charged enough, it can power the speaker circuit. Further evidence for this is that it plays for a while after the key stops turning - presumably whilst the charge runs down.

Periodic pattern in aeroplane contrail

My attention was caught by this contrail. Note the very regular pattern along the far edge. Why is it so regular? Is it due to some periodic motion in the engine? I'd think not guessing that engine frequencies would be too high. And why is it only along one edge? If I see something like this again I need to try to get some idea of the wavelength of the recurring pattern by measuring it against the plane itself.

Lateral inversion jigsaw box

Mrs B realised that the problem she was having with her jigsaw was that the picture on the box was back to front. In other words, it was a mirror image. A mirror causes lateral inversion. Lateral means sideways and inversion means upside down. We don't know whether it was deliberate or not.

Automatic light

 In the circuit above, the battery voltage is shared between the resistance box on the left, which was set to 1000 Ohms, and a light dependent resistor on the right, clipped to the blue board. With light shining on the LDR, its resistance is much lower than 1000 Ohms so it gets less than half of the battery voltage. That's not enough energy to light the LED which is clipped across the LDR.

I put my hand over the LDR to make it dark. The resistance shot up and it got more than half of the battery voltage, enough to light the LED.

I found a wonderful podcast

Just this evening I found out that the physicist and write Sean Carroll is producing a podcast where he talks to some top people. Go here https://www.preposterousuniverse.com/podcast/

Kronecker delta

I'm beginning to get my head round the way that matrices can be written by setting out individual elements within them rather than by writing out the matrix itself. I now realise that when matrices can approach infinitely big sets, then it's just not possible to write them out. In matrix maths, the identity matrix I acts like the number 1 in ordinary arithmetic. I is a diagonal matrix with 1 down the diagonal and 0 everywhere else. So when we don't want to draw matrix, the Kronecker delta symbol is helpful. It = 1 when i=j (so any place that would be on the diagonal of a matrix) but 0 when i and j are not the same numerical value. Sources make it clear that the Kronecker delta is not a matrix but where I see it used, it enables things to be done with individual matrix elements that would require the identity matrix in matrix maths.

Sundials, witches and the eye of God: time in Pendle

 I was interested to spot the small sundial perched on top of the pillar at Newchurch in Pendle. The backstone is not in line with the rest of the church. I think that they have made it face due south to make the angles easier, unlike the one in Radley that I've posted about before. It shows that the church is only approximately west-east in alignment.

 The time theme also plays out with this being the birthplace of Jonas Moore, one of the founders of the Greenwich Observatory which later became the centre of our time system.

And the eye of God? Well, that on the tower in the top picture, left of the bottom of the star decoration. Make of it what you will.

Carboniferous sandstone on Pendle Hill

We climbed Pendle Hill. I was intrigued by the description of the rock as "Carboniferous sandstone". I have normally heard "carboniferous" applied to limestone. It also occurred to me to wonder if the label which was about the UK being in tropical conditions with swamps and coral seas actually applied to the whole world. The answer appears to be "no" if this is anything to go by. https://en.wikipedia.org/wiki/Carboniferous The rocks we found on Pendle were indeed sandy. See particularly the path which is made of rocks dug out of the trench to the side of it. The close up shows large visible grains. Mostly the colour is far paler than the desert sandstones of the the Eden and Wigton area. The suggestion is that these sandstones were deposited in a river delta http://www.kabrna.com/cpgs/stratigraphy/regional_map.htm

LED pavement lights in Blackpool

We found these LED devices on the promenade sea wall by Blackpool Pleasure Beach. They seemed to be worked by small solar panels. The LED I posted about a couple of weeks ago was running 0.02A at 1.5V which is a power of 0.03W or 30mW. There are 6 LEDs so let's estimate 180mW. In peak sunlight, the Sun reaches about 1600 Watts per square metre. On a dull day in Winter maybe 800 Watts per square metre. The solar cell can't be bigger than 2cm x 3cm or 6 x 10^-4 square metres. That would then generate 0.48W in Winter. But the cell doesn't drive the LEDs directly because you want them to come on after dark. It must charge a battery. Then we'd be into calculating Amphours charging versus Amphours discharging.

Making a model of the Levi-Civita Epsilon Tensor

One of the things I have found hard in Quantum Mechanics and in the build up to General Relativity has been the multiple indices used in the notation for multiple dimension spaces. I understand 2-dimensional matrices in the Real number space but have trouble intuiting in higher dimensions. So I try to build models that relate back to the ideas I understand to help me to think that the new idea is "like" a simpler idea in some way. So it was that I came to make a model of the 3-dimensional version of the Levi-Civita tensor. Essentially, it acts in some way like a matrix that assigns a 0, a 1 or a -1 to any multiplication. Obviously, if it is a 0, then the total multiplication is a zero. I used i, j and k coordinates. In my model, i goes to the right, j goes away and k goes up. For any coordinate ijk, you get a zero if any two coordinates are the same. If ijk is 123, 231 or 312 you get a 1. For 321, 213 or 132 you get a -1. You get two none zero entries per vertical or horizontal plane - one is 1 and the other is -1 so it is not quite symmetric - we say anti-symmetric. Now I can visualise it, I need to find out where and how it is used. This was my inspiration https://en.wikipedia.org/wiki/Levi-Civita_symbol#Three_dimensions

Prevalent and Dominant Winds on Hartlepool North Beach

In his book The Pebbles on the Beach, Clarence Ellis makes a distinction between prevalent and dominant winds with reference to their effects on waves. In our latitudes, the prevailing winds come from the south west due to a combination of global-scale convection cells in the air and the Coriolis Effect of the turning planet. On the Atlantic Coast, these winds have miles of ocean to blow over before reaching us, creating huge waves that also travel in from the south west. Here on the east coast, the prevailing wind is coming off the land and cannot therefore be responsible for the waves that break on the shore. When I took the picture on the east coast, the wind was coming from behind me and the waves were coming towards me. For this reason, Ellis would say that the dominant wind would be north east because it is the only one that can cause waves to break on our east coast.

My Very Energetic Maiden Aunt Just Swam Under North Pier

This is North Pier at Hartlepool, seen from South Pier. It reminded me of the mnemonic in the title of this piece, which is used to remember the order of the planets from the Sun. I always preferred it as a mnemonic because it included the Asteroid Belt (A for Aunt!) The problem now is that Pluto has been downgraded to being a "dwarf planet". Pluto belongs to a second belt of small objects left over from the formation of the Solar System which is called the Kuiper Belt. Pluto's problem was that several objects were found that were almost as big as it but wouldn't be called planets, so it had to be downgraded. Find out more about the Kuiper Belt here https://en.wikipedia.org/wiki/Kuiper_belt or listen to Lord Bragg of Wigton's Radio 4 programme about it https://www.bbc.co.uk/programmes/b08g7ttx I was thinking that the mnemonic should be changed to include K for Kuiper Belt but My Very Energetic Maiden Aunt Just Swam Under North Key should really be Quay for it to work.

Experiment to show that water is an insulator

This is one of my favourite experiments. We put ice into a boiling tube and then inserted a piece of gauze to hold the ice down. We then filled up the tube with water. Normally the ice would float - hence the gauze. We gently heated the water at the top of the tube and it boiled. The oddity is holding the tube in your hand to heat it rather than using test tube holders. You'll notice from the picture that we have boiling water at the top but the ice at the bottom has not melted. How is that possible? Thermal energy normally spreads best in fluids by convection. As water is heated, it expands, becomes less dense and floats up. So thermal energy is carried upwards. By heating the tube in the way shown, thermal energy transfer by convection is limited to the very top of the tube. The other way of transferring thermal energy here would be conduction ("heating by particles"). If the tube were metal, you'd burn your hand. But the molecules in water are not joined so they find it hard to pass on the thermal agitation hence thermal energy would take a long time to pass down the tube by conduction. Water is an insulator because it is a bad conductor.

Physics in Advent

The amazing Physics Advent Calendar is back. http://www.physics-in-advent.org/ to get it in English. Toggle switch for language in the top left hand corner of the home page if you want it in German...

Wigton Rugby and the wind turbine

We went up to see Wigton Rugby beat St Benedict's this afternoon and spotted the display for the wind turbine. It will have been up for 10 years next April. It has generated nearly 6000kWh per year. That will be nearly £1000 of electricity per year. It is only a small turbine but it will have paid for itself and more.

"Deep time" at Red Wharf Bay

In his introduction to the re-print of Clarence Ellis's book The Pebbles on the Beach, Robert Macfarlane says that geology gives him a sense of what he calls "deep time". Ellis was born on Anglesey and it was on Anglesey that I got the sense of "deep time". We found these alternating layers of limestone and sandstone. The former was laid down in tropical seas and then the land rose, leaving rivers to deposit the sand and swamps the black mud shale deposits. Human history goes back in the thousands of years but I get the impression that each layer must represent a time longer than that. I'd be interested in finding out how long a time each layer represents and how you would work it out.

Getting an IV characteristic for a red LED

 We used a potential divider to change the potential difference across a red LED. We measured the current through it for each voltage. To obtain negative voltages, we swapped the connections round on the power supply.

The graph looks like this:

There is no light for negative voltages but even with positive voltages, nothing happens until around 0.7V and even then we were getting current of 0.02A with 1.5V.

Contact resistance

We were doing an experiment to find the EMF and internal resistance of a cell by clipping in different sizes of resistor. The idea is that the resistance changes the current and the current heats the internal resistance, changing the output p.d. If you plot output p.d.against current, the y-intercept is the EMF and the internal resistance is - gradient. But with the 1 Ohm resistor I was asked if the direction of the resistor affected the current like it would with a diode.

The answer is that it doesn't, but that there is extra resistance added into the circuit by the way that the clips grip the wire. It is unavoidable and changes in size every time you reattach, hence the seemingly different reading. It is a small resistance but might be a reasonable fraction of 1 Ohm and thus affect the current readings for very small resistances.

A simple cell

We made simple cells by soaking filter paper in a strong brine solution and connecting up two metals of differing reactivity on either side of the paper. Here we had nickel and copper. I'm working on an explanation as to how the differing reactions mean that more electrons are pushed by one side than by the other.

Wind in the wire: Abergwyngregyn

It was a windy day when we walked under the wires and there was a steady roar that I worked out was the sound of the wind in the wires. I'll estimate 100Hz. Let's assume that it has set up the first harmonic of a stationary wave. The equation for calculating the frequency is:

Where L is the length of the wire (1000m as measured on the map), T is the tension of the wire and mu is the mass per metre. The wire is aluminium with steel core. Let's round up the density of aluminium to 3000 kg per cubic metre and estimate circular cross-section of 4cm diameter. Cross-sectional area = 0.001 square metres and mass per unit length of 3kg. That would give a tension of 120 billion Newtons. Breaking stress for the steel is 7.9 x 10^8 Newtons per square metre. This figure would exceed that so some of my estimates or assumptions must be wrong.

Diffraction at RSPB Conwy

This was going to be a piece about how the ripples on the water help us to see the invisible gusts of wind. Each gust sent a set of short wavelength ripples across the surface - an interesting real-time visualisation tool. But it didn't photograph well. However, I did notice that as well as the high frequency noise on the surface, there were also lower frequency waves crossing the pond.You can see them in the clearer water beneath the clump of reeds in the water in the middle of the picture. Why are they suddenly visible there? The answer seems to be that their wavelength is closer to the size of the clump so they are able to bend into the sheltered space behind the clump by diffraction. The much shorter wavelength noise isn't able to diffract and so goes straight past.

Variable resistor controlling a bulb

 Here we took a rheostat that has 3 terminals and only connected 2 to make a variable resistor. Above, the variable resistor is set to zero so that all of the potential difference is across the bulb. But below, the variable resistor is on its highest resistance setting. You can tell because the current reading in the left hand meter has gone down. There is still a potential difference across the bulb, shown by the other meter. The bulb is off so you would have though that the potential difference would be zero. That's the problem with a variable resistor as a way of controlling a bulb. No matter what you do, the potential difference of the bulb will never go down to zero. If you want that when, for example, doing IV characteristics, you have to connect up all 3 terminals to wire the rheostat up as a potentiometer.

Short circuit

 We set up a circuit with two bulbs in it. The ammeter reading was 1.00A. Note that I'm holding a spare unconnected lead.

Then the lead was connected to the other side of the bulb. That bulb went out straight away and the current went up to 1.39A. The spare wire is called a SHORT CIRCUIT. The increased current can be dangerous because current is what heats in a circuit. The extra heat can sometimes set fire to the insulation and cause house fires.

Interesting design of car park light at Dobies

This design of light is at Dobies at Dalston. It seems to be designed to reduce light pollution. All of the light should be directed downwards. The silvered surfaces are also dimpled. I assume this is the ensure diffuse reflection so that an image is not focused anywhere below.

Thermistor experiment

Today we put a thermistor into hot water with a thermometer and watched the resistance as the water cooled down. The resistance went up. I now understand the band theory idea better after learning some solid state quantum mechanics. Single atoms have clearly defined electron energy levels but as you bring atoms closer together to bond them, the electron energy levels cannot merge. Instead, for neighbouring atoms, one has its level nudge slightly up and the other has it nudge slightly down. On a grand scale, this results in a lot of almost but not quite identical electron energy levels. They form bands because they are grouped around the same energy. In semi-conductors, there is a clear gap between the valence band for bonding and the conduction band. Thermal energy is enough to move some electrons up from the valence band to the conduction band. More conduction electrons means higher current so we conclude that the resistance has fallen. The converse is true for cooling.

Shadows in the sky

 In the picture above, there is a thin dark band stretching down and right from the edge of the cloud. It is a shadow of the cloud as the Sun was away over to the left at the correct angle. Consider the photograph below. The shadow appears on the solid surface. There is no evident shadow visible in the air between us and the concrete. I have seen higher clouds project shadows onto lower levels so it suggests that tenuous water vapour in the air above the Solway might be substantive enough to scatter light from the Sun back to us, making it obvious where there is no light getting through to be scattered. It's like seeing a shadow projected onto smoke.

Considering eigenvalues and eigenvectors

I'm reading this wonderful book again. I have been stuck for a long time trying to get an understanding of what a state actually means in Quantum Mechanics. I started with eigenvalues and eigenvectors for spin. If the spin operator acts on |u> then it remains unchanged. So |u> is an eigenvector. The eigenvalue is the multiplier - in this case it is +1. Notice that applied to |d> you get the same vector |d> but multiplied by -1. So |d> is an eigenvector with eigenvalue -1. The significance of the eigenvalues is that whatever state the system is in, you only ever measure it out as +1 or -1. So I picked another state vector and multiplied it out. I showed that it could be written in terms of the eigenvectors |u> and |d>. However if the system is in this mixed up state and you measure, you still get either +1 or -1. These are the eigenvalues. I have been wondering what the significance of the eigenvectors are.

 Susskind says that this component state vector tells us the probability of measuring +1 or -1. I experimented myself with the maths, so what is below might be wrong but I think that the expectation value is the inner product squared. I get a probability of 1/2 which is what I expected. I think I have uncovered the significance of the orthogonal basis. The zero result for removes it from the equation. Orthogonal products give either 1 or 0. 

I have struggled with degenerate states. Different eigenvectors but with the same eigenvalue. So only one value is measured. The eigenvectors are not orthogonal but you can construct an orthogonal basis by putting together linear combinations of them. I am trying to get this in writing whilst I am thinking about it and may have to publish corrections later.

Maths on In Our Time

Melvyn Bragg is Lord Bragg of Wigton and through his Radio 4 programme In Our Time has been a great friend of Physics. Recently I have been enjoying back issues on Maths. Try these: https://www.bbc.co.uk/programmes/b09gbnfj and https://www.bbc.co.uk/programmes/b00dshx3 I have been trying to learn about Hilbert Spaces to understand Quantum Mechanics and the second programme gave me an insight into Hilbert's character.

Learning about Hemititian Hamiltonians

Today I have learned that a Hermitian matrix like a Hamiltonian is real on its diagonal but has complex conjugates the other way. Leonard Susskind says that any operator L can be made up up components of the spin operators and the identity. a, b, c and d are real.

 Dagger means the complex conjugate of the transpose. I then did an example with numbers.