We are now flying above the Pacific Ocean about 1/3 of the way between Los Angeles, California and Sydney, Australia on our way to Christchurch, New Zealand, and finally, Antarctica. [Editor’s note: ‘now’ was November 17, 2010 at 1:07 am MST]
Welcome to Mrs. Gatch’s Math Classes! So glad you could join us on our Math-pedition where we will definitely be using your math in Antarctica!
One of the primary electrical properties we will be measuring is electrical conductivity (whose units are 1 / ohm-meter or Siemens/meter (S/m)). Conductivity tells us how much current (units: Amps) the ice can carry. We know that metal wires carry current very well. Conductivity tells us how well the ice will carry current. The higher the conductivity, the more current it will carry. The lower conductivity, the less current it can carry. Metals and wet soil generally have high conductivity. Plastics and dry soil generally have low conductivity. This is why people who work on electricity are careful to wear rubber soled shoes and stand on dry ground! Sea ice has relatively high conductivity, because of the salt water stored in it.
(Mrs. Gatch’s 7/8 Green Math Class: Here is where the formula for AREA comes in …)
The opposite of conductivity is resistivity (units: ohm/meter).
Resistivity = 1 / [ Conductivity x Area] . We can measure resistivity as shown in the picture below. Put two metal plates (with Area = Width x Height as shown. Units: square meters) on either end of a block of sea ice. Attach wires to each of these plates. Measure the resistance (units: ohms) between these two wires. Resistance (ohms) = Resistivity (ohms/meter) x Length(meters). So putting this all together:
We will measure
Resistance (ohms) = Length / [ Conductivity x Width x Height]
(Mrs. Gatch’s PreAlgebra Class: Here is where we set both sides of the equation equal and solve for conductivity…. )
We can solve for
Conductivity (1 / ohm-meter) = Length / [Resistance x Width x Height]
You can try this experiment in your classroom, if you like. Cut two pieces of metal that are 2” x 2” . Tin foil will do, although you may have to double or quadruple it to make if stiff enough to hold in the water. Fill a plastic dish about 3” deep with water. Start with distilled water. Put the two pieces of metal into the dish of water, so that they are parallel (like the ends of the rectangle shown here) and about 6” apart. They should be completely immersed in water. Measure the resistance between them by clipping a wire onto each plate, and measuring the resistance between the ends of those wires. Gradually add salt, measuring it with a tablespoon and stirring it until dissolved between each measurement. Increasing the salt increases the salinity of the water. Conductivity increases as you increase the amount of salt (salinity) in the water, so resistance and resistivity will decrease. Graph the resistance (on the y-axis) vs. the amount of salt (on the x-axis).
Next, experiment with the size (area) of the plates and verify that the resistance is approximately proportional to length and inversely proportional to area. To do this, first plot the resistance (on the y-axis) against the length (on the x-axis). Connect the dots of your measurements. You should see approximately a straight line going up as the length goes up. Next, plot the resistance (on the y-axis) against the area (on the x-axis). You should see approximately a straight line going down as the area goes up.
Finally, just for fun, experiment with the shape of the plates. Cut two circular plates that are the same area as the square plates.
(Pre-algebra, here we go again …)
Area of Square plate = Width x Height
Area of circular plate = Pi x Radius squared
Setting the two areas equal gives:
Pi x Radius squared = Width x Height
And solving for Radius gives:
Radius = sqrt (Width x Height / Pi) (remember Pi = 3.1415926)
This is the radius of the circular plate you will need.
Test your circular plates the same distance apart as the square plates. The resistance should be the same. Is it?
Check here to find out more about resistivity http://en.wikipedia.org/wiki/Resistivity.