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Physics Lab Formal Report

Capacitance

by Robert Chapin

with Jesse Roe

and instructor Michael Gastner

Section #051

Friday, October 12, 2001

Introduction

Chapter 3, Capacitance, contains laboratory experiments designed to explore the relationship between voltage and the amount of charge stored in an object. These experiments involve measuring electrical properties of capacitors in series, in parallel, while charging, discharging, and at varying widths between the surfaces. Hands-on experience and resulting data should provide insight into the nature of capacitance.

Theory

In order to charge an object, a certain amount of energy is required to transfer charge to that object. The energy per unit of charge is called voltage. Given a certain voltage, charge can be transferred to an object until the amount of energy that is required to add more charge exceeds the energy potential. A derived unit is useful for expressing the capacity of charge (in Coulombs) that can be transferred to an object per unit of voltage (in Volts). Therefore, a unit of capacitance called the Farad exists, and is defined as C = Q/V.

A capacitor comprised of two parallel surfaces will have a capacitance equal to

8.85 ρF/m, times the area of one of the plates, divided by the distance between them. When sharing the charge applied to one capacitor with a second capacitor, charge is conserved, therefore V_f * (C₁ + C₂) = V_i * C₁. When discharging a capacitor through a resistor, V(t) = V₀ * e^-t/RC. When charging a capacitor through a resistor, V(t) = V_f V_f * e^-t/RC.

Experiments

3.5.2: Charging a Capacitor

This experiment required a 9V battery, a voltmeter, and voltage a follower that were assembled in this way: The battery and voltage follower ground contacts were connected to the volt meter ground, while the voltage follower output was connected to the positive terminal of the volt meter. To measure the voltage across the 0.033 μF capacitor, I connected one end of the capacitor to the positive lead from the voltage follower, and connected the other end to the ground.

To charge the capacitor, I touched the positive probe from the 9V battery to the ungrounded side of the capacitor. The voltmeter displayed 8.90 V after removing the battery probe. Therefore, the charge on the capacitor was 0.033 μF * 8.90 V = 0.294 μC.

3.6.1: Measuring Unknown Capacitance

This experiment was the same as the previous one, except that after the capacitor was charged, a second one was connected to it in parallel. When I connected the second capacitor, the measured voltage dropped to 6.87 V. So, theoretically, the capacitance of the second capacitor was C₂ = C₁V_i/V_f C₁ = 0.033 μF * 8.90 V / 6.87 V 0.033 μF = 0.00975 μF. The second capacitor was actually rated at 0.01 μF, and the voltmeter was precise enough to measure the drop in voltage accurately within (0.00975 0.01) / 0.01 = -2.49%. Such small error could have been caused by the multi-meter or irregularities of the capacitors themselves.

I repeated the experiment using one capacitor of unknown rating, and one of 0.033 μF. The voltage dropped this time to 8.24 V, which was unsatisfactory. Repeating again with a 0.0033 μF capacitor, the voltage read 5.43 V. This greater change in V indicated that the capacitances were of the same order of magnitude, allowing for greater accuracy in the following calculations. C₁ = 0.0033 μF and V_f = 5.43 V. So, C₂ = 0.0033 μF * 8.90 V / 5.43 V - 0.0033 μF = 0.00211 μF.

3.6.2: Variable-Gap Capacitor

This experiment measured the capacitance between two parallel plates that were connected directly to a capacitance meter. The distance between the plates was increased at prescribed intervals, and I recorded the capacitance for each interval. (See page 10 for these data.)

Data from the variable width capacitor experiment were graphed on log-log scales. The resulting scatter plot was very linear in shape. (See page 7.) To find the slope of this line, I divided the difference between the capacitance and distance coordinates after taking the log of each. I selected two points that fell exactly on the trend line, and used those points to find an equation for the line.

Point One = (log[2.4], log[13]) = (0.380, 1.114)

Point Two = (log[0.3], log[101]) = (-0.523, 2.004)

Slope = m = [(1.114 2.004) / (0.380 + 0.523)] = -0.9859

Using the general form of a line where the axes are labeled x and y, m is the slope, and b is a constant equal to the y-intercept:

y = (m)(x) + b

log C = (m)(log d) + b

log C = (-0.9859)(log d) +b

Substituting the coordinates of Point Two where C is the capacitance and d is the distance between the plates:

2.004 = (-0.9859)(-0.523) + b

b = 1.4888

So, an equation for the trend line is:

log C = (-0.9859)(log d) + 1.4888

To compare this result with the current capacitance theory, I had to manipulate equation 3.2 so that it was in the form of y = mx + b.

C = kε₀A/d

k is equal to one, ε₀ equals 8.85 ρF/m, and A is the surface area of one of the capacitors disks. For consistency with lab measurements, I converted all of the lengths to centimeters and all of the capacitances to picofarads. So, using ε₀ = 8.85 * 10^-2 pF/cm, and r = 10 cm;

C = ε₀A/d

log C = log(ε₀A) log(d)

log C = log(8.85 * 10^-2 * 10² * π) log(d)

log C = (-1.0000)(log d) + 1.4441

Finally, this equation is comparable to my own, and illustrates a small margin of error.

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Biology 100 – Concepts of Biology

Spring, 2006

Microscopy

One of the fundamental tools of biology is the microscope. Its purpose is simple: to enable one to view tiny objects. Many living organisms are too small to be seen with the unaided eye. In multi-cellular organisms, individual cells are also too small to see without a microscope. In many areas of the sciences microscopes are a fundamental tool with which to test hypotheses. There are many kinds of microscopes. In this exercise we will concentrate on the compound light microscope.

Drawings help you to remember.

It is often necessary to obtain a written record of what is observed with a microscope. The simplest way to do this is to make a drawing of what you see. In this exercise, you will practice sketching what you see through the microscope. Drawing forces you to examine objects more carefully than you would otherwise.

It is often necessary to estimate the size of the object you are viewing. In this exercise you will learn how to estimate the size of a microscopic object.

Caution! The eyepieces will easily slip out of these microscopes. Be careful when removing the dust covers to assure that the eyepieces do not fall out.

I. Parts of the Compound Microscope

Your instructor will explain the parts of the microscope and their functions. Below is a diagram of the Olympus CX-31 compound microscope, which is the microscope we will be using.

A. Eyepiece (ocular)

B. Objectives (4x, 10x, 40x)

C. Stage

D. Substage condenser

E. Field Lens

F. Fine focus knob

G. Coarse focus knob

Your instructor will point out additional parts of the microscope including: substage condenser iris diaphragm, mechanical stage knobs, main power switch, light intensity knob. Be sure you are conversant on the parts of the microscope. Make special note that each microscope has a pointer in the right eyepiece and a micrometer in the left eyepiece.

Magnification

The magnification of a compound microscope ranges from about 100x to about 1000x, depending on the lenses used. The total magnification of a microscope is easily calculated with the following formula.

Magnification of the eyepiece x magnification of the objective lens being used

For technical reasons, 1000x is about the maximum magnification one can obtain with a compound microscope and still have good resolution, i.e., sharpness of the image. Magnification of the objective lens is 10x.

II. Setting up Koehler Illumination on the compound microscope

Setting the adjustments of a compound microscope such that the optimal resolution is obtained is called setting up "Koehler Illumination." It takes only a minute or so to do this.

1. Turn on the microscope. Set the light intensity knob so it is comfortable for your eyes. Place a prepared slide on the stage.

2. Start with the 4X objective lens (blue ring) in place. Use the mechanical stage knobs (right side below stage) to orient the prepared slide so that it is centered below the objective lens.

3. Use the course and fine adjustments to focus on the slide. Now switch to the 10X objective lens.

4. While looking through the microscope use the knurled ring on top of the field lens to close the field diaphragm. You will see the edges of the diaphragm close in the field of view.

5. Focus the edge of the field diaphragm by raising or lowering the substage condenser (black knob on left side under stage). The substage condenser should now be near the upper end of its range of travel. Refocus the specimen.

6. Open the field diaphragm until the light fills the field of view.

7. Open the diaphragm of the substage condenser (front of scope beneath stage); now close it until the image of the specimen just begins to darken. If you close it further the contrast will increase but your resolution (ability to see detail) will decrease.

Guidelines for the Safe Use of the Compound Microscope

1. Always carry the microscope with two hands.

2. Never touch the lens surface.

3. If any of the lens need cleaning, alert your instructor and she/he will show you how.

5. Always observe any specimen first under low power (4x). When you are ready to

switch to high power (10x or 40s), do so carefully, watching the lenses as you rotate the revolving nosepiece. Remember, the high power objective is longer than the low power objective.

6. If you spill any liquid on the stage, wipe it up immediately.

7. When unplugging the microscope, pull on the plug, not the cord.

8. When finished,

a. remove the slide from the stage,

b. carefully wrap the cord around the base of the microscope,

c. cover the microscope with the dust cover.

III. Practice with the Compound Microscope

The letter 'e'.

1. Obtain a small piece of newspaper with text that contains a lower case letter 'e' Cut out the 'e' and place it on a clean microscope slide. Place a cover slip over the newspaper. You have just made a ‘dry mount’ (so called because there is no water).

2. Place the slide on the microscope stage so that the 'e' is in normal reading position. Observe the slide under low power. Draw the 'e' as it appears in the microscope.

3. What is the relationship of the orientation of the object to that of the image?

4. While looking through the eyepiece, slowly move the slide to the left. Which way does the

image appear to move?

Move the slide to the right. Does the image move the direction you expected it to move?

While looking through the eyepiece, slowly move the slide away from you on the stage. Which way does the image move?

What do you think will happen if you move the slide toward you on the stage? Try it.

Practice moving the slide and focusing until these actions become comfortable. When you are finished, discard the letter 'e' and the cover slip, but save the slide for later use.

HYPOTHESIS: What hypothesis have we tested by observing the letter 'e'?

Prediction. Did we make a prediction? What was it?

Colored threads.

1. Obtain a prepared slide containing three pieces of thread positioned so that they cross at a common point. Observe the slide under low power and locate the point at which the three threads cross. Adjust the diaphragm lever and focus. Can you focus on all three threads simultaneously?

2. Without moving the slide, carefully switch to high power. Compare the depth of focus with that under low power. Which is greater?

3. Switch back to low power. Adjust the focus until the slide and the low power objective are barely separated (about 2 mm apart). Look through the eyepiece; all threads should be out of focus. Adjust the fine focus until the first thread comes into focus. Note its color. Continue focusing until the second, and finally, the third threads come into sharp focus, noting their respective colors.

What color was the first thread to come into focus? Second? Third?

From your observations, which thread do you think is on top? On the bottom?

Your instructor will tell you the correct order of the threads. Is it what you predicted? Is the

image inverted vertically?

4. What hypothesis(es) have we tested with the colored thread exercise?

HYPOTHESIS: What hypothesis have we tested by examining the colored threads?

Prediction. Did we make a prediction? What was it?

IV. Measurement

Sizes of microscopic objects are usually given in microns (abbreviated µm;

1000 µm = 1mm). How many microns in a meter?

These microscopes have an ocular micrometer (a little ruler) in right eyepiece. These ocular micrometers are calibrated as indicated below.

Objective lens Smallest division on micrometer

4x 25 µm

10x 10 µm

40x 2.5 µm

Practice with the microscope

1. Thickness of hair

Pull a hair from your head and make a dry mount. Place the slide on the stage, find the hair with low power, and then switch to high power. Sketch what you see and estimate the diameter of your hair. Develop an hypothesis about your hair and test it by examination with the microscope.

2. Human blood cells

Obtain a prepared slide containing preserved human blood cells. Find a white blood cell. The white blood cells will be much rarer than the red blood cells, and will be larger and stained purple. Estimate the diameter of a human white blood cell.

3. Other prepared slides

There may be other prepared slides available for examination.

4. Diversity of life in pond water

The purpose of this exercise is to introduce you to the variety of microscopic organisms. Do not worry about trying to identify what you see; just look and enjoy.

Making a wet mount: Obtain a clean microscope slide and cover slip. Place a drop or two of pond water onto the slide and carefully lower the cover slip onto the slide, edge first.

Your instructor will demonstrate!

Examine your slide. When you find something interesting, show your neighbor and instructor. Develop a hypothesis about an organism that you find on the slide and test it by making additional observations.

What color are most of the organisms you see?

If organisms are clear (colorless), what can you do to increase your ability to see them with the microscope?

Sketch two organisms that you see in the space below.

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Diffusion and Osmosis

Mary Messner

Cells maintain a constant internal environment, a process called homeostasis. In a constant environment enzymes and other cellular components can operate at optimum efficiency. The ability to selectively exchange materials with the environment is one component of the cell’s homeostatic mechanism. Ions and molecules, such as sugars, amino acids and nucleotides, must enter the cell, and the waste products of cellular processes must leave the cell. Regardless of the direction of movement, the common mediator of these processes is the cell membrane, or plasma membrane.

The plasma membrane is a fluid, or mobile, mosaic of lipids and proteins. Ions and molecules cross the plasma membrane by a number of processes. Large particles are engulfed by the membrane, forming a vesicle or vacuole that can pass into (endocytosis) or out (exocytosis) of the cell. Some small, electrically neutral molecules diffuse through the spaces between the lipid molecules of the plasma membrane. Others bind to transport proteins embedded in the plasma membrane and transported into or out of the cell.

Atoms, ions and molecules in solution are in constant motion and continuously collide with each other because of their kinetic energy. As the temperature is raised, the speed of movement of the molecules increases and they collide more frequently and with greater force. An observable consequence of this motion is Brownian motion, an erratic, vibratory motion of small particles suspended in water, which is caused by the collisions of water molecules with the particles.

Diffusion also results from the kinetic energy of molecules. If a small crystal of a soluble substance is added to water, molecules of the substance break away from the crystal surface and enter solution. As a consequence of the collisions with water molecules, molecules of the substance move in a random pattern in the solution, but always away from the crystal, with some moving to the farthest reaches of the solution. This process continues until the molecules of the substance are evenly distributed throughout the water (the solvent). In a general sense, in any localized region of high concentration, the movement of molecules is, on average, away from the region of highest concentration and towards the region of lowest concentration. The gradual difference in concentration over the distance between the regions of high and low concentration is called the concentration gradient. The steeper the concentration gradient, the greater the rate of diffusion. The rate of diffusion is also directly proportional to the temperature and inversely proportional to the molecular weight of the diffusing molecules. In other words, the higher the temperature the faster the rate of diffusion, but larger molecules move slower than smaller molecules at the same temperature. Small, electrically neutral substances diffuse into and out of cells by passing through the spaces between the lipids of the plasma membrane or by dissolving in the lipids or proteins of the membrane. Substances that are large or electrically charged cannot pass through membranes. Membranes that block or inhibit the movement of molecules are called differentially permeable, or selectively permeable. Selective permeability explains the phenomenon of osmosis, the diffusion of water across a membrane under certain conditions. If two solutions containing different concentrations of a solute are separated by a selectively permeable membrane (permeable to water but not to the solute), water will move from the solution with low solute concentration to the solution of high solute concentration. The water flows in this direction because the solution with low solute concentration has a high water concentration and the solution with high solute concentration has a low water concentration. Thus, the water diffuses from a region of high water concentration to a region of low water concentration.

Many ions and molecules important to cells are taken into cells by specific transport proteins found in cell membranes. Facilitated diffusion occurs when such a protein functions as a binding and entry port for the substrate. In essence, the protein functions as a pipeline for the specific substance. The direction of flow is always from high concentration to low concentration. The gradients are maintained because frequently the molecules are metabolically converted to other types of molecules once they enter the cell. For many other molecules and ions, favorable diffusion gradients do not exist. For example, sodium ions are present at higher concentrations outside mammalian cells than inside the cells, yet the net movement of sodium ions is from the inside to the outside of the cell. Likewise, potassium ions are found inside mammalian cells at significantly higher concentrations than outside the cell, but the net movement of potassium ions is from the outside to the inside of the cell. For such molecules and ions, cellular energy must be used to transport the molecules across the plasma membrane. Active transport occurs when transport proteins in the cell membrane bind with the substrate and use cellular energy to drive the “pumping” of the molecules into or out of the cell, against the concentration gradient.

The vibratory movement exhibited by small particles in suspension in a fluid was first observed by the Scottish botanist Robert Brown in 1827. Brown incorrectly concluded that living activity was the cause of this movement, but we now know that Brownian movement results from the collisions between water molecules and small particles (less than 10 micrometers in diameter) suspended in the water.

The term tonicity describes the relative concentration of solvent to solute in two solutions. A solution with the lower solute concentration is said to be hypotonic relative to the other solution. Conversely, the more concentrated solution is hypertonic relative to the first. If the solute concentrations of each solution are equal the solutions are isotonic with respect to each other. It is important to remember that these terms are relative terms, that is, the description of a solution as being hypertonic, hypotonic or isotonic depends on the solution it is being compared to. Traditionally, in biology, the cell is the frame of reference. An isotonic solution has the same solute concentration (and water concentration) as the cell; a hypertonic solution has a higher solute (and lower water) concentration than the cell; a hypotonic solution has a lower solute (and higher water) concentration than the cell. If a cell in a hypotonic solution (low solute concentration) is enclosed in a rigid box, for example a plant cell surrounded by the rigid cell wall, the increasing water pressure inside the cell would cause water to flow back out of the cell towards the area of lower pressure. Eventually, equilibrium would be reached when the flow of water into the cell, due to the concentration differences, equals the flow of water out of the cell, caused by pressure differences. The pressure at equilibrium is called the osmotic pressure. Since all cells contain molecules that cannot cross the plasma membrane, osmosis always occurs when cells are placed in dilute aqueous solutions

The plasma membrane of a cell is selectively permeable because it allows the diffusion of some substances and not others. Small, uncharged molecules diffuse freely across the plasma membrane, but charged molecules and large molecules cannot cross the membrane. The dialysis membrane used in this experiment simulates the activity of the plasma membrane. Osmosis (the diffusion of water) occurs whenever two solutions of different solute concentration are separated by a selectively permeable membrane. The difference in solute concentration between the two solutions determines both the direction and rate of water flow. Water always diffuses from a hypotonic solution to a hypertonic solution; consequently, a cell placed in a hypotonic solution will gain water and a cell placed in a hypertonic solution will lose water. Elodea is a plant that lives in fresh water. In such a hypotonic solution, the large, central vacuole of each Elodea cell gains water and exerts pressure (called turgor pressure), which compresses the cytoplasm and plasma membrane against the cell wall. In this situation the chloroplasts appear to be either evenly distributed throughout the cytoplasm or around the cell perimeter, with the central vacuole visible as a transparent region in the center, depending on the plane of the cell being observed. When an Elodea leaf is placed in a hypertonic solution, water moves out of the central vacuole and the cytoplasm into the surrounding solution. The cell volume is reduced, and the plasma membrane visibly pulls away from the cell wall, a process known as plasmolysis. In this situation, water concentration inside the cell is higher than water concentration outside the cell and the net flow of water is out of the cell.

Molecules, which are in constant motion, tend to move from regions of higher concentrations to lesser concentrations. Diffusion is defined as the net movement of molecules down their concentration gradient. Osmosis is the passive transport (diffusion) of water. In osmosis, water moves through a semi-permeable membrane from a region of higher concentration to a region of lower concentration. The terms hypotonic, hypertonic, and isotonic are used to compare solutions relative to their solute concentrations. The hypotonic side is the side with the larger water percentage and a lower solute concentration. The hypertonic side is the side with the smaller water percentage and a higher solute concentration. It is isotonic when both sides have equal concentrations of solute and water percentages. Cell membrane requires the cell to expend its own energy, the movement is said to involve active transport. Passive movement occurs “down” the concentration gradient, from high to low. Active transport, by contrast, moves from low to high concentration, against the concentration gradient.

Some molecules would move across the membrane by passive means, but they are either too large or have some type of charge that would not allow them to cross. In this case, special protein “channels” offer larger or insulated passageways that make it possible for them to cross. In this case, the proteins facilitate the movement, and the process is called facilitated diffusion. If material crosses the membrane through one of these channels in a high to low direction (with respect to concentration), it is undergoing diffusion. Since it requires a carrier protein to diffuse across the membrane, we say it is undergoing facilitated diffusion. In other words, the protein facilitates its movement. When something crosses a membrane from an area of low concentration to an area of high concentration, it must be assisted and we say it is undergoing active transport. For example, while a person can supply energy to push a rock up a hill, moving a molecule against a concentration gradient (from low to high) means that the cell must apply energy. Often this energy (to assist movement) is in the form of molecules of ATP. The sodium-potassium pump provides an example of active transport in which energy is used to drive the movement of sodium and potassium across the plasma membrane.

Hydrophobic molecules must orient away from water. It just so happens that the phosphate heads are hydrophilic. As a result, the tails point inward toward each other and the hydrophilic heads point outward, toward water. Osmosis is the movement of water (solvent) across a semi permeable membrane. The tonicity of a solution (hypo, hyper, or iso) determine the direction of water travel. No net movement of water occurs between isotonic solutions.

Placing plant and animal cells into a hypotonic solution will result in a net influx of water across the plasma membrane. As a result, the cell increases in size. The animal cell may eventually explode, however, the cell wall of plant cells prevents such a scenario from occurring in plant cells. The concentration gradient molecules move from an area of high concentration to an area of low concentration. In facilitated diffusion, molecules utilize a membrane protein and move along the concentration gradient (high to low). However, in active transport, molecules are transported from an area of low concentration to an area of high concentration via a membrane protein with the utilization of energy in the form of ATP.

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