A New School Year

As a new school year opens it is timely to start a conversation on the fundamentals for successful learning in physics and mathematics. Many students say "I do not understand physics" or "I cannot do maths". What can be done to remedy this? The solution is to go back to basics and pursue steady step by step practice in each subject with regular testing. If a test reveals that a physics concept is not understood or a mathematical technique is not correctly applied then we zoom in on this area and practice it again until it is mastered. Learning involves dialogue between teacher and student with encouragement to aspire to new levels. This is the heart of education. The tutorial method had its origins with Socrates in ancient Greece and is followed today at the best universities in the world. The journey to academic success comes when students embark on a regular program of study and revision and the best time to start the journey is now.

A Boy Playing on the Sea-Shore......

To Physicists and Mathematicians this title invokes images of one person. Isaac Newton. In a farm house in Colsterworth near Grantham in Lincolnshire the principles of natural philosophy were born. Newton's central problem was this: Given that the path of a planet about the Sun is an ellipse and that the radius vector from the Sun to the planet sweeps out equal areas in equal times, what must the law of attraction be? 

I do not know what I may appear to the world, but to myself I have only been like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me

To Michael Faraday and James Clerk Maxwell

There is no better way to finish the first year of conversations on relativity and electromagnetic topics than to talk of the contribution of Michael Faraday and James Clerk Maxwell to the study of electricity and magnetism. The Royal Institution of Great Britain in Albemarle Street in London has an Electrical and Magnetic Museum devoted to Michael Faraday. Faraday performed his investigations at the Royal Institution and his original equipment is on display. Faraday's lines of electric and magnetic force permeate the thinking of physicists and electrical engineers. While at nearby King's College James Clerk Maxwell used Faraday's lines of force in his Dynamical Theory of the Electromagnetic Field, published 150 years ago. Maxwell's four equations, and their solutions, give us a complete description of the electromagnetic field. As an example, imagine that in a rectangular region of space an electric field E exists whose x-component is constant and whose y-component is proportional to the x coordinate. Many physics and engineering students do not realise that a perpendicular time varying magnetic field B must accompany this E field. Why?

As the Earth Turns

The Science Museum in London has a marvellous long swinging pendulum that appears to change its direction during the day. Some books say that the pendulum keeps swinging in the same plane in space. Is this true in London? How do we explain the shift in the position of the pendulum during the day?

“At the north pole the direction of the swing of the pendulum does not change in space, so the pendulum shows the Earth turning beneath it in 24 hours. Here in London it is not so simple; the direction of the pendulums swing moves around the circle in about 30.5 hours. As we travel towards the Equator the direction of swing moves more slowly, until at the Equator no movement is seen. In the southern hemisphere the direction of swing moves in the opposite direction.”

— Description of Foucault's pendulum at the Science Museum, South Kensington, London

Acceleration

Physics and Mathematics students use the equation

x = ut + 1/2at²

to calculate x the distance travelled by an object moving with acceleration a during a time interval t when its initial velocity was u.  The equation tells us that the distance travelled when an object accelerates is equal to the distance travelled when there is no acceleration plus the distance it would travel if it accelerated from rest. The relationship between distance and time for accelerating objects was first deduced by Galileo using the assumption that the passage of time is the same in all reference frames. Special relativity teaches us that at very high speeds compared time intervals are different in the moving and stationary reference frames and so Galileo's equation cannot be used to calculate the distance travelled. Let us suppose that the acceleration continues for a very long period of time so that relativistic speeds are reached. A common misconception is that special relativity cannot be used to solve problems involving accelerating objects. It can, provided we refer the acceleration to the instantaneous rest frame. Imagine that an object accelerates from rest uniformly at 9.8 m/s/s. What distance does this object travel in the laboratory reference frame in 10 years of laboratory time?

9.1 light-years

Mass of Light

One concept that confuses Physics students is whether light has mass and momentum. In everyday life we observe the effects of the energy carried by light. A famous piece of Physics demonstration equipment is the Crookes' radiometer. The vane of the radiometer spins when the radiometer is exposed to sunlight. Does this demonstration show that light has mass and or momentum?

Some Mathematical Meandering III

In our last mathematical digression we studied the Snowplow Problem of Ralph Palmer Agnew, the solution of which using the logarithmic function appears on page 39 of his book Differential Equations (McGraw-Hill 1960).  Let us make the problem even more interesting following a suggestion by Murray S Klamkin that is given in Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers. Here is the Great Snowplow Chase.

One day it started snowing at a heavy and steady rate. Three identical snowplows started out at noon, 1 pm and 2 pm from the same place and all collided at the same time. What time did it start snowing?

11:30 AM

Mass

We often find in the relativity chapters of Physics and Mathematics books the statement that "the mass of an object increases as its speed approaches the speed of light". Is this correct, does the mass of an object actually increase, or is this a means of describing what is happening to a very rapidly moving object in Einstein's four dimensional space-time in terms of familiar quantities using Newton's laws of motion?

Changing E and B

In this conversation we will attempt to go into the depths of students' understanding of electromagnetism and try to unlock, or at least loosen, a common mindset of Science and Engineering students. We have already had a conversation on electric, E, and magnetic, B, fields. Taking this to the next stage, a common sentence found in many Physics and Mathematics textbooks is  "a changing electric field creates a changing magnetic field and vice versa". Is this correct? Does the changing electric field itself actually create a changing magnetic field (and vice versa) ?

E and B

Physics students learn about the electric field vector E and the magnetic field vector B. In most courses these vectors are taught separately with E introduced first as being produced by a stationary charge and B introduced later as being produced by a moving charge. A common mindset is to consider these fields to be different, just because of the order in which the courses are taught, but fundamentally they are not. They are both aspects of the electromagnetic field and become active in particular situations. In SI units E and B have different measurement units and this tends to cloud their difference. In CGS units E and B have equivalent units and so this system is more suited to electromagnetic calculations as the numerical values of the fields can be directly compared. Many physics students do not realise that electromagnetism is a direct consequence of Einstein's special theory of relativity, as Einstein's 1905 paper on relativity was entitled 'On the Electrodynamics of Moving Bodies'. There is no better way of expressing the relationship of electromagnetism to relativity than the following  marvellous quote given by Leigh Page, then professor of mathematical physics at Yale, in an address to a meeting of the American Institute of Electrical Engineers in 1941.

“The rotating armatures of every generator and every motor in this age of electricity are steadily proclaiming the truth of the relativity theory to all who have ears to hear”

Some Mathematical Meandering II

Here is a beautiful problem. This problem starts with a minimum of assumptions and soars to great heights in its simplicity of solution using high school calculus. It is called a "marvellous problem" by Carl M Bender and Steven A Orszag on page 33 of their legendary book Advanced Mathematical Methods for Scientists and Engineers. Now be warned. Once you are captured by this problem its beauty could lead you on mathematical travels to distant places. Here is the Snowplow Problem.

One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing?

11 :23 AM

A New Twist on an Old Problem I

Physics and Mathematics Extension students are taught that, provided resistance forces are neglected, an object released from rest undergoes the same vertical displacement in the same time interval as an object projected horizontally. Now let resistance forces be included. Would it ever be possible for the horizontally projected object to have a greater vertical displacement in the same time interval than the object released from rest?

Einstein's Happiest Thought

A common phrase often seen in textbooks is "Einstein's greatest blunder". Many Physics and Mathematics students are not aware of "Einstein's happiest thought". What was it?

“In 1907 Albert Einstein was sitting in his patent office in Bern Switzerland when he had what he later described as “the happiest moment of my life”-that a person who falls freely under gravity feels weightless.”

— From the Einstein Display at the Science Museum London, December 2015

Feynman and His Father

The great physicist Richard Feynman (1918-1988) is renown among scientists as being a genius. Feynman had lightning insight into how nature works and he was one of the greatest performers of calculations by hand. Nature presents her answers to many physics problems in the form of  integrals and Feynman could evaluate integrals in a few lines using a basket of simplifications and tricks that able physicists and mathematicians could only do in several pages. How is it that such talent is awakened and developed? In his books and his video recordings, now available on YouTube, Feynman indicates how his curiosity was built. It was through conversations with his father. 

Red Shift

One of the most passed on misconceptions in Physics teaching concerns the red shift of electromagnetic waves. Students in science classes are taught about the Doppler effect of sound waves. If a rapidly moving car moves away from us the sound waves that we hear appear to be stretched out and we hear a lower pitch. Students are then taught that when a very rapidly moving luminous object moves away from us the light waves are stretched out and in this case the effect is known as a red shift. However, is the reverse true? If we observe a red shift does it mean that the object making the light waves must be moving away from us? 

Where Does Nuclear Energy Come From?

Physics students in their examination answers often write that "nuclear fission releases energy because mass is converted into energy" .  At a first glance this statement may seem plausible due to Einstein's well known equation. However, upon closer consideration, based on remembering that the protons and neutrons are rearranged into different combinations in the reaction and so their total number does not change, where does the mass that is converted into energy come from?

Some Mathematical Meandering I

It is often refreshing to spend time wandering through the fields of the world of mathematics, looking at the scenery and following the laneways to see how they link-up. Most students of mathematics know that the sine of an angle in a triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. This is written as sin(x), where x is the angle. In a number of applications of physics our calculations require us to to divide sin(x) by x. As this occurs frequently, we give this operation the name of sinc(x). However, at one point this function gets into trouble. When x = 0, sinc(0) is 0/0. Is this equal to 1?

 

The Equation Everyone Knows

Most physics students are aware of Einstein's famous equation,

E = mc²

This equation is often seen on T shirts carrying an image of Einstein.  It is used to name office buildings to highlight the innovative work being carried out inside. In many ways this equation is the public image of relativity. But what does this equation actually mean? Even among physicists, this equation, often called one of the most important equations of the twentieth century, is often misunderstood.