Feeds:
Posts

## Archive for January, 2013

I mentioned in this blog here that I would be on TV talking about the calculation that the Milky Way galaxy contains some 17 billion Earth-like planets.

Here is a youtube video capture of my appearance on the TV show. My apologies that the subtitles lag behind what is being said, and for the subtitles only being a summary of what is said. But at least if will give you a vague idea of what I’m saying if you cannot understand Welsh.

## Galilean Relativity and Electrodynamics

Quite a few months ago now I derived the so-called Galilean transformations, which allow us to relate one frame of reference to another in the case of Galilean Relativity.

$\boxed {\begin{array}{lcl} x^{\prime} & = & x + vt \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & t \end{array} }$

It had been shown that for experiments involving mechanics, the Galilean transformations seemed to be valid. To put it another way, mechanical experiments were invariant under a Galiean transformation. However, with the development of electromagnetism in the 19th Century, it was thought that maybe results in electrodynamics would not be invariant under the Galilean transformation.

## The electrostatic force between two charges

If we have two charges which are stationary, they experience a force between them which is given by Coulomb’s law.

$\vec{F}_{C} = \frac{ Q^{2} }{ 4\pi\epsilon_{0}\vec{r}^{2} }$ where $Q$ is the charge of each charge, $r$ is the distance between their centres, and $\epsilon_{0}$ is the permittivity of free space, which determines the strength of the force between two charges which have a charge of 1 Coulomb and are separated by 1 metre.

Coulomb’s law gives us the force between two charges. If the charges are the same sign the force is repulsive, if the charges are opposite in sign the force is attractive.

## Moving charges produce a magnetic field

If charges are moving we have an electric current. An electric current produces a magnetic field. The strength of this field is given by Ampère’s law

$\oint \vec{B} \cdot d\vec{\l} = \mu_{0}I$ where $d\vec{l}$ is the length of the wire, $\vec{B}$ is the magnetic field, $\mu_{0}$ is the permeability of free space and $I$ is the current. So, if the two charges are moving, each will be surrounded by its own magnetic field.

A wire carrying a current produces a magnetic field as given by Ampère’s law.

## The Lorentz force

If the two charges are moving and hence producing magnetic fields around each of them then there will be an additional force between the two charges due to the magnetic field each is producing. This force is called the Lorentz force and is given by the equation

$\vec{F}_{L} = Q\vec{v}\times\vec{B}$. If $r$ is the distance between the two wires, and they are carrying currents $I_{1}$ and $I_{2}$ respectively, and are separated by a distance $r$, we can write $B=\frac{\mu_{0}I}{2\pi r}$ which then gives us that the Lorentz force $F_{L} = \frac{ I_{1} \Delta L \mu_{0} I_{2} }{2 \pi r }$ and so the Lorentz force per unit length due to the magnetic field in the other wire that each wire feels is given by $\boxed{ \frac{ F_{L} }{\Delta L} = \frac{ \mu_{0} I_{1} I_{2} }{ 2 \pi r} }$. Writing the currents in terms of the rate of motion of the charges, we can write this as

$F_{L} = \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$

The Lorentz force is the force on a wire due to the magnetic field produced in the other wire from the current flowing in it.

## Putting it all together

Let us suppose the two charges are sitting on a table in a moving train. This would mean that someone on the train moving with the charges would measure a different force between the two charges (just the electrostatic force) compared to someone who was on the ground as the train went past (the electrostatic force plus the Lorentz force).

The force measured on one of the charges by the person on the train, for whom the charges are stationary, which we shall call $F$ will be

$F = \frac{ Q_{1}Q_{2} }{ 4 \pi \epsilon_{0}r^{2} }$.

The force measured on one of the charges by the person on the ground, for whom the charges are moving with a velocity $v$, which we shall call $F^{\prime}$ will be

$F^{\prime} = \frac{ Q_{1}Q_{2} }{4 \pi \epsilon_{0}r^{2} } + \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$.

These two forces are clearly different, and so it would seem that the laws of Electrodymanics are not invariant under a Galilean transformation, or to put it another way that one would be able to measure the force between the two charges to see if one were at rest or moving with uniform motion because the forces differ in the two cases.

As I will explain in a future post, Einstein was not happy with this idea. He believed that no experiment, be it mechanical or electrodynamical, should be able to distinguish between a state of rest or of uniform motion. His solution to this problem, On the Electrodynamics of Moving Bodies, was published in 1905, and led to what we now call his Special Theory of Relativity. This theory revolutionised our whole understanding of space and time.

## Will I manage a hatrick?

On Sunday evening (27th of January 2013) the Penarth & Dinas Running Club had its annual prize evening. For the 2nd year in a row I managed to win my category, old farts. Over the course of the year the Club nominates 15 races which count towards the Club Championships, and one’s best 8 races will be chosen if one runs more than 8.

In 2011 I ran 10 races, and my best 8 races gave me a points total of 160 out of a possible 160. This last year (2012), mainly because I ran two marathons, I entered fewer Club Championship races, only doing 8, and my score for these 8 races was 155 points out of a possible 160. In 2nd place in my category was Steve Goodfellow, who got 109 points. In the overall Club Championships, I came 5th with a points total of 127, 10 points behind Malcom Bradley, our exceptional Senior Vet, who seems to defy age and beats many members less than half his age.

Receiving my tropy for the 2011 Club Championships.

Receiving my trophy for the 2012 Championships.

As of today, we have had our 1st Club Championship race, the Lliswerry 8 this last Sunday. I have done very little running the past two months, having what turned into a 7 week break after the Florence Marathon on the 25th of November, so I ran this year’s Lliswerry 8 with only 2 weeks’ training. My lack of fitness showed, I finished in 1 hour 5 minutes 38 seconds, a full 8 minutes and 21 seconds slower than my 2012 time! (but still over 4 minutes quicker than my 2011 time). I was beaten by two others in my category from Club, so I will need to get back into shape pretty quickly to stand any chance of retaining my title for a 3rd year in a row. What better incentive do I need to knuckle down to training after my lay-off? The next Club Championship race is in less than 3 weeks, a 10-mile race in Llanelli. So I’d better get my arse into gear!

## The Challenger disaster

I saw in the paper this morning that today marks the 27th anniversay of the Challenger Space Shuttle disaster, which happened on the 28th of January 1986. 1986 was before the era of 24-hour news, but I vividly remember mid-afternoon programmes being interrupted to bring this sad news, and to show the horrific footage of Challenger exploding a few minutes into launch on that cold Floridian January day.

The explosion of the Challenger Space Shuttle on launch in January 1986 shocked the World.

This is a video of the explosion

In 1967 all 3 Apollo 1 astronauts were killed in a fire in the command module during final testing for launch, but Challenger was the first time NASA had suffered the loss of astronauts during a mission. In the subsequent Presidential enquiry (headed by Bill Rogers, a former US Secretary of State during Nixon‘s administration) many NASA management mistakes were uncovered, and the cause of the accident was traced to O-rings which failed to prevent pressurised hot gas from escaping, which was the cause of the explosion. In a dramatic demonstration of the O-rings’ inability to cope with cold temperatures, the celebrated Nobel Prize-winning Physicist Richard Feynman illustrated on TV how this failure would occur. Here is a video where Feynman summarises his experiences of working on this investigation.

Sadly, the Challenger disaster was not the last in the Space Shuttle program. On the 1st of February, as it was re-entering the atmosphere at the end of a successful mission, the Space Shuttle Columbia burnt up in the atmosphere, again leading to the loss of all 7 astronauts on board. People often forget how dangerous an activity going into space is.

## The greatest try ever?

Exactly 40 years ago today, on the 27th of January 1973, Gareth Edwards scored what is often voted as the greatest try of all time. I was lucky enough to be at the match where this try was scored – a match between The Barbarians and the All Blacks of New Zealand.

Gareth Edwards in the Barbarians match against the All Blacks in January 1973.

My family and I were late arriving, and as we were taking our seats in the South stand of the old Cardiff Arms Park National Stadium, everyone else leapt to their feet as this incredible passage of play culminated in Edwards going over in the corner of our side of the pitch to score.

The entire match is also available here, and if you do take the trouble to watch the entire match you will not be disappointed. It was running rugby from the first to the last whistle. The greatest rugby try ever in possibly the greatest exhibition of running rugby ever.

## You may use dividers, but not on each other

Following my trip down memory lane last week, when I posted the wonderful Rowan Atkinson sketch about hell, I thought I would follow with this equally wonderful one by the same comic genius. I don’t think this was part of the one man show I saw him doing in London in 1986, but he did do it at the Secret Policeman’s Ball amongst other places.

Enjoy!

## Alternating current basics

I have recently been teaching my Physics students the difference between Altnernating Current (AC) and Direct Current (DC). I thought I would try and explain some of what I have been saying here.

## Direct Current

The kind of current we get from a battery is direct current. This means that the current remains constant with time. Many electrical devices such as radios, computers and electronic devices run on DC.

A direct current of 4 Amps. The current stays constant with time.

The power dissipated in a device which has a resistance $R$ when we have a direct current of value $I$ is simply $P=I^{2}R$. So, for example, if our device has a resistance of $100 \Omega$ and we have a current of $4 A$ then the power disippated will be $4^{2} \times 100 = 16 \times 100 = 1600$ Watts or 1.6kW.

## Alternating Current

An alternating current is constantly varying with time. A plot of the current against time looks like this:

An alternating current. This particular plot shows an alternating current which has a peak value of 4 A and a frequency of 0.25 Hz.

The peak current, 4A in this case, can be denoted by $I_{0}$. Mathematically this alternating current can then be described by the equation $I=I_{0}sin(2\pi f t)$ where $f$ is the frequency. The power disippated can be determined by looking at $I^{2}$. Below is a plot of $I$ and $I^{2}$.

A plot of current I and I squared (I x I) for an alternating current.

As we can see from the plot of $I^{2}$, it is also always varying with time. It varies between a maximum of $I_{0}^{2}$ and zero.

If we look closely at the plot of the square of alternating current over half of the AC cycle (so from a time of 0 to 2s in our example), we can see that it takes 1s to reach the value of 16, and a further 1s to come back down to a value of zero. How long does it take to reach a value of 8, half the peak value of 16? To work this out we need to calculate $t$ when $4 \sin\left(\frac{\pi}{2}t\right)= \sqrt{8}$ so when $\sin\left(\frac{\pi}{2}t\right)= \frac{\sqrt{8}}{4} = \sqrt{\frac{8}{16}}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}$. To find $t$ we do $\frac{\pi}{2}t=\arcsin\left(\frac{1}{\sqrt{2}}\right)$ so $\left(\frac{\pi}{2}t\right)=\frac{\pi}{4}$ which gives us $t=\frac{2}{4}=\frac{1}{2}=0.5s$ which is exactly half of the time it takes to reach its maximum value of 16. As the curve is symmetrical, it will be above 8 for exactly half of the time and below 8 for exactly half of the time, so the average value of $I^{2} = \frac{1}{2}I_{0}^{2}$.

We therefore say that the average power disippated in the device is $P_{av} = \frac{1}{2}I_{0}^{2}R$. This leads us to define a new quantity, $I_{rms}$, the root mean square current, which is defined as $I_{rms}^{2}=\frac{1}{2}I_{0}^{2}$ and so $I_{rms} = \frac{1}{\sqrt{2}}I_{0}$. This means we can write the average power dissipated in an AC device as $\boxed {P_{av}=I_{rms}^{2}R}$.

The root mean square current and its related root mean square voltage are often more useful to us than the peak current and voltage. When we say that the voltage from the mains in Europe is 240V, and in the USA is 110V, this is actually the root mean square voltage, not the peak voltage. This is quoted because the average power $P_{av}=I_{rms}^{2}R$ used by any device is also given by $P_{av}=V_{rms}I_{rms}$.