Wisdom of the Week

28/04/2014

Quote of the Week

"Every sentence I utter must be understood not as an affirmation, but as a question."

Niels Bohr

Puzzle of the Week

The Card Tournament

Some people meet for a round-robin card tournament, where every person plays every other person once. Wallace has to leave after only a few hands, sitting out the remainder of the tournament. A total of 59 hands are played at the tournament. How many hands did Wallace play before leaving?

The card game in question is a two-player game, and no person played with the same opponent more than once. No one sat out any hands besides Wallace.

Equation of the Week

Ampere's Circuital Law

The curl of the magnetic flux density is equal to (permeability x current density) plus (permeability x permittivity x rate of change of electric field)

The fourth and final of Maxwell’s equations. As with the previous three, the equation is general, applying in all cases. The time dependence term again means this equation is dealing with dynamics. The equation relates the magnetic field around a loop to the electric current passing through it. It essentially says that both electric currents and changing electric fields will produce spatially varying magnetic fields. In a similar way to the previous equation, if you know the current density J and electric field E, you cannot calculate the magnetic flux density B, only its curl. However, combining this with Gauss’ Law for Magnetism, which tells us that the divergence of B is zero, then B can be calculated. The final term in the equation permits electromagnetic waves to propagate in free space (vacuum). Again similar to the previous equation, this equation is telling us also that wherever we have a magnetic field with non-zero curl (which is everywhere, since magnetic fields do not diverge, and also a perfectly uniform field does not exist in reality), then there must be a corresponding electric current or changing electric field of some sort.

Now that the four Maxwell’s equations are complete, it is worth mentioning that electric and magnetic fields appear to be intricately related, which may strike you as a little odd when considering other fields (such as gravity) that are seemingly completely independent of one another. One may ask the question, why are these two fields so strongly related? The real reason is that they are not really separate fields. They are part of the same field, called the electromagnetic field. Consider this as an example: A stationary point charge (say an electron) will have an electric field surrounding it, but no magnetic field. If this charge is then put into motion (for simplicity let’s say it travels past you at constant speed), it will “create” a magnetic field. However, motion is relative. So what happens if the charge remains stationary, and you (the observer) moves past it at constant speed instead? You will still observe a magnetic field, because there is relative motion. This implies that two different people could both be observing this charge; one of them standing still, the other moving; and only one of the observers would see a magnetic field. Contradictory? Only if electric and magnetic fields are different entities. If they form part of an electromagnetic field instead, then it makes sense that it is possible for different observers to see it differently (as in the example given).

(Previous Weeks)

21/04/2014

Quote

"We do not stop playing because we grow old, we grow old because we stop playing."

Benjamin Franklin

Puzzle

Red and Black Cards

You have a jokerless deck of 52 cards. You thoroughly shuffle them, divide them into two stacks of 26 cards each, and then check the contents of each pile. If you repeat this process 1000 times, how many times could you expect the number of red cards in one pile to match the number of black cards in the other one?

ANSWER (select with mouse): 1000 times. The number of black cards in one half-deck will always match the number of red ones in the other half-deck.

Equation

Faraday's Law of Induction

The curl of the electric field is equal (and opposite in polarity) to the rate of change of magnetic flux density.

The third of the four Maxwell's equations of electromagnetism. Again this is a general equation (applying in all geometries), this time dealing with dynamics, as there is a time dependence term in the equation. A changing magnetic field (in time) is always accompanied by a spatially varying electric field (and vice versa - see next week's equation). Faraday's Law of Induction gives this relationship explicitly. It tells us that for every point in space, if we have an electric field with a non-zero curl, then we will have a magnetic field that is changing with time that can be directly calculated. Note that this cannot be reversed: if we know how the magnetic field changes in time, we cannot calculate the electric field from this equation, only its curl (however, if we also know the divergence then we can calculate the original vector - we can find the divergence from Gauss' Law). The curl of a vector field tells you about how the field is circulating around any given point (the curl of a vector field is also a vector field itself, so putting in some coordinates will tell you how much the original vector field is circulating about that point). Vector fields that have a finite, non-zero circulation are referred to as non-conservative. This means that the integral of the vector field between two points in space is NOT path independent. In the case of a non-conservative electric field, the implication is that the electric field can no longer be described as the gradient of electric potential (because the curl of a gradient is always zero, by definition), and a more complicated description is required. The most important thing to note from this, is that electrostatic potential (voltage) only has a useful meaning when you do NOT have a time-varying magnetic field present.

14/04/2014

Quote

"We cannot solve our problems with the same thinking we used when we created them."

Albert Einstein

Puzzle

The Two Child Problem

Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?

Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

ANSWER (select with mouse): Probability Mr. Jones' children are both girls is 1/2. Probability Mr. Smith's children are both boys is 1/3.

Equation

Gauss' Law for Magnetism

The divergence of magnetic flux density is equal to zero.

General equation for magnetostatics, describes the behaviour of magnetic fields in static systems (no time dependence). This is the second of the four Maxwell’s Equations that describe electromagnetism. It simply states that magnetic fields do not diverge, or to put it differently, magnetic field lines always form closed loops (we say that the vector field is solenoidal). As a consequence of this, to every “north” there must be a “south”. There is no such thing as a magnetic “charge”, often referred to as a magnetic monopole. Instead there are only magnetic dipoles. Another consequence of this is that for waves, the magnetic field is required to be perpendicular to the direction of wave propagation. The equation is general, and therefore valid for as long as the coordinate system you are using is valid. You may of course choose your coordinate system (for example, Cartesian, spherical polar, …), but the important thing to note is that it is not possible to cover the whole universe all with the same coordinate system. For this reason, it has been hypothesised that there are parts of the universe where this equation might break down, and hence magnetic charges or monopoles might exist. Needless to say, such a case has never been observed.

07/04/2014

Quote

"There is a better way for everything - find it."

Thomas A. Edison

Puzzle

Bill and Ben

Bill and Ben's combined age is 91.

Bill is now twice as old as Ben was, when Bill was as old as Ben is now.

How old are they?

ANSWER (select with mouse): Bill is 52, Ben is 39

Equation

Gauss' Law

The divergence of (permittivity x gradient of electric potential) is equal (and opposite in polarity) to the charge density.

General equation for electrostatics, tells us the behaviour of electric fields in static systems only (no time dependence). This equation is one of the four Maxwell's Equations, which give a complete description of electromagnetism (and hence light). This equation is general, meaning it applies in all cases, there are no specific criteria required for it to be valid. It allows for the calculation of the electric potential V (and consequently the electric field E = -grad V) if certain material properties are known. Can only be solved analytically in very simple geometries, otherwise numerical methods required.

In order to use it, you need to know the electric permittivity (epsilon) of each different material in your system, and also the charge density (rho) for each material (as a function of space if necessary). Note that if the permittivity is a constant in a given material, then it can be moved outside the divergence operator, making the equation more simple to solve. Further, if no charge is present, and there is only one material, the equation simplifies even further to yield Laplace's equation, stating that the curvature of electric potential is equal to zero. In that case, the permittivity doesn't matter, and the solution would be a linear gradient of electric potential between two boundaries at fixed potential (for example, two electrodes).

As Gauss' Law is a general equation, it can be used wherever you wish to calculate the electric potential or field. This could for example be to find the potential around a point charge, or near a charged surface. A lipid bilayer might have some charges present, in which case you could use this equation. Perhaps you are applying a potential across a membrane with a set of electrodes. Gauss' Law can also be used here to find where the largest voltage drops occur etc. The possibilities are endless!