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## 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 $P_{av}=V_{rms}I_{rms}$.

## Bluebirds flying high

Even if the state of Welsh regional rugby is not good, the same cannot be said of Welsh football. Swansea City FC are doing very well in the English Premiership, lying mid-table and with several big wins this season. They have also just got through to the final of the League Cup, beating Chelsea 2-0 on aggregate last night. And Cardiff City FC are flying high in the Championship, the league below the Premiership.

The Cardiff City football club club badge.

The Championship table as it stood on Saturday (19th January 2013). Cardiff City are 10 points clear at the top of the Championship table with 28 of the 46 games played.

With 28 games played of the 46 in the season, Cardiff City have 60 points, 10 points clear of their nearest rivals. Of course, as anyone who has been following Cardiff City will know, they have failed to gain promotion to the Premiership several times in the last few seasons. Having grown up in Pembrokeshire I don’t have the same tribal loyalty towards Cardiff City that its die-hard supporters have. So most of them would not join me in saying how wonderful it would be to see both Cardiff and Swansea playing in the Premiership. But surely that can only be good for Welsh football and for Wales as a country.

## Murray in Melbourne

As I type this early on Wednesday morning (23rd of January 2013), half a World away in Melbourne Andy Murray is playing in the quarter final of the Australian Open Tennis tournament. He is playing against Jérémy Chardy of France, and Murray has just won the 1st set 6-4. Chardy is currenty ranked 36 in the World, Andy Murray is ranked 3rd. So, on paper at least, it should be a win for Andy Murray. Should he win, he has the unenviable prospect of facing Roger Federer in the semi final on Friday.

As those of you who follow tennins will know, Murray won his first Major a few months ago, when he won the US Open in September 2012. This came after his agonising defeat in Wimbledon, which I blogged about here, but also after his success one month later in August in the 2012 Olympics.

Andy Murray won his 1st Major, the US Open, in September 2012.

Prior to winning his first major, Murray had suffered an agonising series of 4 defeats in major finals. In particular, in Australia where he lost twice in a row. He lost to Roger Federer in the 2010 final, and in the 2011 final he lost to Novak Djokovic. Here is a summary of Murray’s records in Grand Slam finals.

Murray’s record in Majors
Year Tournament Opponent Score
2008 US Open Roger Federer 2-6, 5-7, 2-6
2010 Australian Open Roger Federer 3-6, 4-6, 6-7
2011 Australian Open Novak Djokovic 4-6, 2-6, 3-6
2012 Wimbledon Roger Federer 6–4, 5–7, 3–6, 4–6
2012 US Open Novak Djokovic 7–6, 7–5, 2–6, 3–6, 6–2

Now that Murray has finally won his first Major and got that monkey of his back, many are expecting him to go on and win many more over the next several years. Only time will tell. Certainly having Ivan Lendl as his coach seems to have made a big difference. Lendl won 8 Major finals in his career, but just like Murray he lost his first 4 Gland Slam finals. This experience has probably proved invaluable in advising Murray on how to improve his mental approach and start winning at the final hurdle. Should Murray win in Melbourne, he will be the first man in history to follow up his 1st major victory with a 2nd victory.

## +++++UPDATE+++++

Murray won easily, 6-4 6-1 6-2.

## The 22nd amendment to the US Constitution

This last Monday (21st of January 2013), the 44th president of the United States, Barack Obama, was sworn into his 2nd term in office. No matter how popular Obama is in 2016, he cannot stand for a 3rd term of office due to the 22nd Amendment to the United States Constitution. This was introduced in 1951, and as a consequence of this amendment to the Constitution, no US President since then has been allowed to be elected to more than 2 terms of office (strictly speaking they could serve 3 terms if they were Vice President and their first term came about through the incumbent President leaving office).

Barak Obama being sworn into office on 21st of January 2013.

In fact, of the 44 US Presidents, only one has served more than 2 terms of office, namely Franklin Delano Roosevelt, who was US President from March the 4th 1933 until his death on the 12th of April 1945, winning elections in 1932, 1936, 1940 and 1944. How many presidents have been elected to two terms of office rather than just one? And how many presidents have never been elected to office, but became President through the death or resignation of the incumbent?

The 2013 US Presidential inauguration.

## Presidents who have been elected to only one term

Here is a table of the presidents who have been elected to only one term. Note, in some cases they actually served two terms, e.g. Lyndon B. Johnson, who became President when John F. Kennedy was assassinated in November 1963, but then won his own election in November 1964 and served one full term as an elected President. Also note that this table does not include presidents who were never elected to the office but served a term as President due to the incumbent President leaving office through death or resignation.

Presidents who have been elected to one term
Name Year of inauguration after being elected Comments
John Adams 1797 2nd President of the U.S.A. Served his full term.
John Quincy Adams 1825 6th President of the U.S.A. Served his full term.
Martin van Buren 1837 8th President of the U.S.A. Served his full term.
William Henry Harrison 1841 9th President of the U.S.A. Died on his 32nd day in office from complications from pneumonia, the shortest presidential term in US history.
James K. Polk 1845 11th President of the U.S.A. Served his full term.
Zachary Taylor 1849 12th President of the U.S.A. Died 16 months into his term of office of natural causes.
Franklin Pierce 1853 14th President of the U.S.A. Served his full term.
James Buchanan 1857 15th President of the U.S.A. Served his full term.
Rutherford B. Hayes 1877 19th President of the U.S.A. Served his full term.
James Garfield 1881 20th President of the U.S.A. Was assassinated 200 days into his term of office. He was the 2nd US President to be assassinated.
Grover Cleveland 1885 22nd President of the U.S.A. Served his full term.
Benjamin Harrison 1889 23rd President of the U.S.A. Served his full term.
Grover Cleveland 1893 24th President of the U.S.A. The only US President to be elected to two non-consecutive terms of office.
Theodore Roosevelt 1905 26th President of the U.S.A. Came into office in September 1901 upon William McKinley’s assassination. Won the Presidency in his own right in 1904. The youngest US President in history, sworn into office when 42 years of age.
William Howard Taft 1909 27th President of the U.S.A. Served his full term.
Warren G. Harding 1921 29th President of the U.S.A. Collapsed and died in August 1923.
Calvin Coolidge 1925 30th President of the U.S.A. Came into office upon the death of Warren Harding, won the Presidential election in his own right in 1924.
Herbert Hoover 1929 31st President of the U.S.A. Served his full term.
Harry S. Truman 1949 33rd President of the U.S.A. Came into office upon the death of Franklin Delano Roosevelt in April 1945. Won the Presidential election in his own right in 1948.
John F. Kennedy 1961 35th President of the U.S.A. The youngest President to be elected to office, 43 years old when inaugurated. Was assassinated by Lee Harvey Oswald in November 1963.
Lyndon B. Johnson 1965 36th President of the U.S.A. Came into office upon the assassination of John F. Kennedy. Won the Presidential election in his own right in 1964.
Jimmy Carter 1975 39th President of the U.S.A. Served his full term.
George H. W. Bush 1989 41st President of the U.S.A. Served his full term.

As you can see from the above table, there are 23 presidents who have been elected to only one term. [note: I have put Grover Cleveland in this table even though he was elected to two terms of office, as he served two non-consecutive terms. He counts twice in the list of the 44 Presidents to-date.]

## Presidents who have been elected to two terms

The following is a list of the Presidents who have been elected to two terms. Note, this list does not include presidents like Theodore Roosevelt, who served two terms but was only elected to his 2nd term, his 1st term as President arose through the death of William McKinley.

Presidents who have been elected to two terms
Name Years of inauguration after being elected Comments
George Washington 1789 and 1793 The 1st President of the United States. Served his full two terms.
Thomas Jefferson 1801 and 1805 3rd President of the U.S.A. Served his full two terms.
James Madison 1809 and 1813 4th President of the U.S.A. Served his full two terms.
James Monroe 1817 and 1821 5th President of the U.S.A. Served his full two terms.
Andrew Jackson 1829 and 1833 7th President of the U.S.A. Served his full two terms.
Abraham Lincoln 1861 and 1865 16th President of the U.S.A. Assassinated in April 1865, just over 1 month into his 2nd term of office, by John Wilkes Booth.
Ulysses S. Grant 1869 and 1873 18th President of the U.S.A. Served his full two terms.
William McKinley 1897 and 1901 25th President of the U.S.A. Was assassinated in September 1901, 6 months into his 2nd term of office. He was succeeded by his Vice President, Theodore Roosevelt, who became the youngest person to serve as US President.
Woodrow Wilson 1913 and 1917 28th President of the U.S.A. Served his full two terms.
Dwight D. Eisenhower 1953 and 1957 34th President of the U.S.A. Served his full two terms.
Richard M. Nixon 1969 and 1973 37th President of the U.S.A. Resigned in August 1974 after the Watergate scandal, some 18 months into his 2nd term.
Ronald Reagan 1981 and 1985 40th President of the U.S.A. Served his full two terms.
William Jefferson Clinton 1993 and 1997 42nd President of the U.S.A. Served his full two terms.
George W. Bush 2001 and 2005 43rd President of the U.S.A. Served his full two terms.
Barack Obama 2009 and 2013 44th President of the U.S.A. Just starting his 2nd term.

As you can see from this table, there are 15 presidents who have been elected to two terms.

## Presidents never elected to the office of President

There are also presidents who have never been elected to office. They are Vice Presidents, who are made President when the incumbent leaves office (usually through death, but once in the case of Richard Nixon, through resignation).

Presidents who have never been elected President
John Tyler 1841 The 10th President and the 1st President to never be elected as President, he was made President when William Henry Harrison died in office and served only 1 term.
Millard Fillmore 1850 The 13th President and the 2nd President to never be elected as President, he was made President when Zachary Taylor died in office and served only 1 term.
Andrew Johnson 1865 17th President, was made President upon the assassination of Abraham Lincoln. Also the 1st President to be impeached, he was acquitted by 1 vote. He is generally considered the worst President in US history.
Chester A. Arthur 1881 21st President, came to office upon the assassination of James A. Garfield.
Gerald Ford 1974 The 38th President, and so far the only President to come to office due to the resignation of the incumbent (Richard Nixon).

There are 5 presidents in this final list.

## Franklin Delano Roosevelt

Franklin Delano Roosevelt (FDR) is unique amongst US Presidents in that he was elected to office a record 4 times. It was this that led to the Constitution being amended to limit presidents to only being allowed to be elected a maximum of two times.

Franklin Delano Roosevelt at his 1st inauguration in 1933.

So, in summary, 23 presidents elected once, 15 elected twice, 5 never elected and FDR elected 4 times, which makes a total of 44.

Who do you consider to be the greatest US President? And the worst?