Author: Lensyl Urbano

Gravity, the Electromagnetic Forces, and the Inverse Square Law

Calculating the forces between two charged particles (electric force), two magnets (the magnetic force), and two masses (the gravitational force) require remarkably similar equations. But, while electricity and magnetism are directly related (that’s why it’s called electromagnetism), gravity is its own fundamental force. Yet they all depend (inversely) on the square of the distance between the two objects creating the force, so they’re all said to obey some form of the inverse square law.

Gravitational Force (Fg)

The force exerted by two masses on one another is:

F_g = G \frac{m_1 m_2}{d^2}

where:

  • G is the gravitational constant (6.67300 × 10-11 m3 kg-1 s-2
  • m1 and m2 are the masses of the two objects attracting one another.
  • d is the distance between the two objects.

Electrical Force (Fe)

The force exerted by two electrically charged objects on one another (like a proton and an electron), is:

F_e = K \frac{q_1 q_2}{d^2}

where:

  • K is the electrical constant, sometimes called Coulumn’s constant (8.9876 × 109 N m2 C-2
  • q1 and q2 are the sizes of the charges (in Coulumbs) of the two objects attracting one another.
  • d is the distance between the two objects.

Magnetic Force (Fm)

The force exerted by two magnets on one another, is:

F_m = \mu \frac{p_1 p_2}{d^2}

where:

  • μ is a constant, (a little simplified)
  • p1 and p2 are strengths of the magnetic poles of the two objects attracting one another.
  • d is the distance between the two objects.

The magnetic force is a little more difficult to give a single equation for, because you need to factor in the shape of the magnets.

Inverse Square Laws

In addition to gravity, electric, and magnetic forces, light (which is electromagnetic radiation) and sound also obey inverse square laws.

The Last 100 Years: World History as Seen from the U.S.

YouTube user derDon1234 has compiled an interesting video montage of historical events over the last 100 years. derDon1234 makes some interesting choices about what to show — condensed into 10 minutes — but it’s a valuable perspective, with some fascinating and poignant video. It’s worth a look.

Magnetism

Notes:

Magnetic field of a bar magnet; shown by the alignment of iron filings. Image by Black and Davis (1913) via Wikipedia.

What creates magnetic fields? (36.3)

Magnetic fields are created by moving electric charges. At the atomic level:

  • In atoms, most of the magnetic field comes from the spinning of electrons (remember electrons have a negative charge).
  • In most elements, the magnetic fields of electrons pair up and cancel each other out
  • Only certain elements, which have a few electrons that don’t pair up, can form magnets:
    • Iron (Fe), Nickel (Ni) and Cobalt (Co) are the common magnetic elements.
    • Iron is the most powerful magnetic element. It has 4 electrons whose magnetism are not canceled out (because of their arrangement in their electron shells)
    • Some rare earth elements are also naturally magnetic.

Magnetic Domains (36.4)

Each iron atom has a very small magnetic field, but when a bunch of them line up they add to each other to create a stronger field.

A region with a bunch of lined up atoms is called a magnetic domain.

Microscopic view of the grains that make up a magnet. Each grain has a magnetic field that, if oriented in the same direction, makes for a strong magnet. Each magnetic grain is called a magnetic domain. (Image by Gorchy (2005) via Wikipedia).

If all the magnetic domains line up you have a strong magnet. If they’re all randomly arranged, you don’t have a magnet at all; they cancel each other out and it’s unmagnetized).

Unmagnetized iron (left) and magnetized iron (domains aligned) (right). Adapted from image by Theresa knott at en.wikibooks.

Magnetic Poles (36.1)

Magnets have two poles: a north-seeking pole, and a south seeking pole; they align with the Earth’s magnetic field.

Like poles repel and opposite poles attract.

The force between the poles depends on the strength of the poles (p) and the distance (d) between them:

F \propto \frac{p_1 p_2}{d^2}

Note how similar this equation is to the force between two charges (Coloumb’s Law; Fc), and the force between two masses (gravitational force; Fg).

Electric fields come from charged particles, which can be separated, but north and south magnetic poles belong to each domain (and even each atom) so they cannot be separated.

  • If you break a bar magnet in half you don’t get a separate north and south poles, you just get two magnets, each with it’s own north and south pole.

Magnetic Fields (36.2)

Magnets that are free to move will align themselves with the magnetic fields around them.

The region around the magnet that is affected by the magnetic force is filled with a magnetic field. In theory the magnetic force goes on forever, but is only strong in a relatively small region.

Compasses, which have magnets that are free to move, will align themselves with the magnetic fields around them. When you’re away from other magnets and electronic devices, compasses align with the Earth’s magnetic field.

So how do you create a magnet?

In an unmagnetized piece of iron, the magnetic domains are arranged randomly. If you place it in a strong magnetic field, the domains will align with the strong magnetic field and the iron will become magnetic.

  • Softer iron alloys will align easier, and stay aligned to make strong, permanent magnets.
  • Any metal with iron in it (like steel cans or filing cabinets) will gradually align their magnetic domains with the Earth’s magnetic field if they not moved for a long time, but the Earth’s field is not strong enough to make a permanent magnet.
  • Stroking a piece of iron with a magnet will also align the domains.
  • Touching a magnet to a paperclip or iron nail will align its magnetic domains and create a temporary magnet, which is why you can use a magnet to hold up a chain of nails. This induced magnetism will only last a short time and eventually (within seconds or minutes) of removing the magnet, the nails will lose their magnetism.
  • Dropping or heating a permanent magnet will shift some of the domains out of alignment, reducing the strength of the magnet.

The Emergence of Capitalism in China

Planet Money recounts the story of the seminal document that, in 1979, sparked the transformation of China’s economy into capitalism.

A key thing to note: the document was a contract, which assigned property rights to individuals (families actually) rather than the collective. And even though the contract could not be legally binding in communist China, the signers had to be confident enough that it would be respected — by each other at the least.

The result of the change was a 5 fold increase in the amount of food produced by the farm.

Despite the risks, they decided they had to try this experiment — and to write it down as a formal contract, so everyone would be bound to it. By the light of an oil lamp, Yen Hongchang wrote out the contract.

The farmers agreed to divide up the land among the families. Each family agreed to turn over some of what they grew to the government, and to the collective. And, crucially, the farmers agreed that families that grew enough food would get to keep some for themselves.

The contract also recognized the risks the farmers were taking. If any of the farmers were sent to prison or executed, it said, the others in the group would care for their children until age 18.

— Kestenbaum and Goldstein (2010): The Secret Document That Transformed China on NPR’s All Things Considered.

Based on the quotes from the story, the market vs. socialist simulation game seems to capture much of the farmers’ real motivations.

Resistance in Circuits

The greater the resistance (R) of a circuit, the less current (I) will flow through it (an inverse relationship). The equation is:

Current = Voltage / Resistance

I = \frac{V}{R}

An actual resistor. This one is 330 Ω with a tolerance of 5% (you can tell from the colors of the bands). Resistors are used to slow down the current moving through a circuit. Image by Nunikasi, via Wikipedia.

Resistors can be actual resistors, lamps, toasters, or anything else that uses electricity. Even the wires have resistance (but we don’t usually consider that since it’s small compared to the resistance of the other stuff).

The total resistance of a circuit depends on the way the resistors are arranged:

  • Series:
    • Add the resistances.
    • R = R_1 + R_2
    • More resistors in series reduce the amount of current in the circuit. (dimmer lights)
  • Parallel:
    • Add the inverse of the resistances to get the inverse of the total resistance.
    • \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}
    • More resistors in parallel increase the amount of current used by the circuit. (brighter lights).
Comparing resistance and current in parallel circuits. Arranging the same resistors in parallel uses 4.5 times as much current (and energy) in this example.

(Notes: Current is a bit like water flowing through a pipe. Resistors are blockages that slow down the flow. Put two blockages, one after the other, and you reduce the flow rate. If you put two blockages into two different channels then though each one slows down flow the total flow is larger.)

As you put in more and more parallel circuits you draw more and more current until you overload the circuit and the wires heat up enough to burn through their insulation. Fuses (or circuit breakers) are used to prevent this; they burn out, breaking the circuit, when too much current goes through them.

Questions:

  • Why are appliances in your house almost always connected in parallel and not in series?
  • Which bulbs will be brighter: two bulbs in series, or two bulbs in parallel?
  • What are the total resistances and the total current that runs through the two circuits in the diagram?
  • What is the total power used by the two circuits in the diagram? Remember, Power = Voltage × Current

Where the Trees Are

Map of Woody Biomass in the U.S. in the year 2000, by the Woods Hole Research Center, via NASA's Earth Observatory.

The Woods Hole Research Center put together this map of “Aboveground Woody Biomass” that essentially shows where the trees are in the U.S.. The map was created using, primarily, satellite imagery. Their website has a nice, interactive, version of the map, and a 3d video flyover of the of southeastern Georgia.

Trees build their woody biomass using carbon from the atmosphere (remember during photosynthesis plants absorb carbon dioxide gas), so these trees are represent stored carbon. If they are burned their carbon is released to the atmosphere. If more trees are planted then they will absorb more carbon dioxide from the atmosphere. This map serves as an inventory of what we have now; a baseline for discussions about what to do about carbon-driven climate change.

The Mississippi River flood plain shows up remarkably well because of it's lack of trees. Flood plains are great for agriculture.