An Electrostatics Class Experiment to determine:
- the power in the inverse relationship in Coulomb's law and
- the quantity of charge on a charged rod.
Theory:
A charged rod brought near the pan of a top loading balance will exert an attractive force on the pan. The balance measures the size of this force, which increases as the separation between the rod and the pan decreases.
This observation can be refined into an experiment to determine the power to which the distance, r, is raised in Coulomb's law. The measurement of the force and the separation also enables the calculation of the amount of charge on the rod, expresses in Coulombs of charge.
The magnitude of the attractive force between a point charge and an infinite conducting plane separating by a distance, r is given by:
F = kQ2/4r2
where Q is the size of the point charge.
The simple explanation for this formula is that the attraction is same as if the conducting plane was a mirror, with an image charge of the opposite sign, but the same size, located an equal distance behind the mirror. So the separation of the charges is 2r, giving F = kQQ/(2r)2. See Extension below for a more complete description.
Equipment:
- top loading balance, accurate to 0.01 g. A more accurate balance (to 0.001g) shows too much instability in the last digit if used in this experiment.
- book to raise the balance so that the charged rod on the laboratory jack can be lowered to the level of the pan.
- metal sheet larger than the pan. A sheet of aluminium from which aluminium strips are cut for chemistry experiments is fine, although it can be easily bent and once placed on the pan, the height of the sheet's edge above the bench can vary by a millimetre. Any metal plate should do. If the pan of the balance is flat and at least 10cm square, a sheet is not needed, but some pans are small and have a depression in the middle.
- laboratory jack to adjust the separation of the charged rod and the metal sheet. If a jack is not available, a pulley system to raise and lower the charged rod could be used.
- G clamp to attach the laboratory jack to the bench top so that the jack does not move as the platform is raised or lowered by the control knob on the jack.
- stand, boss head and clamp to hold the charged rod.
- plastic rod and charging cloth.
- metre ruler with a stand, boss head and clamp to support it along side the end of the plastic rod.
Method:
Setting up
- Adjust the height of the balance and the position of the clamp that will hold the charged rod so that the jack can lower the rod to the height of the pan of the balance.
- Adjust the angle of the clamp to hold the rod to ensure the rod will be horizontal when inserted. See note 1.
- Adjust the tightness of the clamp to hold the rod, so that the uncharged end of the rod can be easily inserted and then quickly tightened into place.
- Adjust the height of the jack so that when the charged rod is placed in the clamp, the separation between the metal plate and the rod is about 5 cm.
Taking measurements
- Turn on the balance.
- After the start up mode, when the reading is zero, place the metal sheet gently on the pan.
- Measure the height, h1, the height of the metal plate above the bench.
- Zero the balance.
- Charge the rod with the cloth by repeated rubbing.
- Place the uncharged end of the rod in the clamp so that that end is adjacent to the ruler.
- Measure the height, h2, of the bottom edge of the rod.
- Record the balance reading quickly, see note 2.
- Lower the rod by about 0.5 cm.
- Repeat steps 7 to 9 until the rod is within 1.0 cm of the plate, then use smaller intervals.
Notes:
- Making the charged rod vertical would seem to better approximate a point charge, but when the rod is placed in the vertical position, the size of the attractive force is much less, probably partly due to the charging action putting charge on the sides of the rod, rather than the end. The smaller value for the force also limits the accuracy of the experiment. The other difficulty is that setting and calibrating the initial height of the rod in the clamp is time consuming, which is significant given the comment in Note 2 below.
- The charge will dissipate from the rod. The 'half life' of the charge on the rod is about 9 minutes. So if possible, the full set of measurements should be taken within a few minutes to ensure the charge does not change too much.
Assumptions: The theory assumes the charged rod is a point source and that the metal sheet is infinite.
Calculations: Separation = h2 - h1Force = Reading x 9.8 /1000
Table of measurements: Metal height, h1 = ______cm
Rod height, h2, (cm) / Separation, r, (m) / Balance Reading (g) / Force, F, (N)Analysis: If an Excel spreadsheet is set up for this experiment, a graph of Force against Separation can be generated. A trendline, as a power relationship, can display the values of both the power to which the separation, r, has been raised, as well as the value of the constant.
Value of the exponent: ______Consider number of significant figures.
Value of the Constant: ______Units: ______
The constant is kQ2/4.
Task: Calculate the charge on the rod in Coulombs
Task: Calculate the number of excess charges.
Extensions
- Uncertainty in the value of the exponent
- Adjusting the force measurements for the 'decay' of the charge
- Comparing the strengths of electric, magnetic and gravitational fields
- Possible topics for Practical Investigations
- More complete physics explanation for theory behind the relationship: F = kQ2/4r2
Uncertainty in the value of the exponent
The calculated value for the value of the exponent will be different from the accepted value of 2.
Also not all the digits in the Excel value for the exponent can be justified. The precision of the measurements and the instruments places a limit on how many digits can be used and whether the accepted value of 2 is within the range of the calculated value.
The uncertainties in the measurements resolves these two issues. They can be used to determine the uncertainty in the measured value of the exponent.
This exercise is good practice for data analysis in the Practical Investigation.
In Excel create two new columns, one for the log of the Force and another for the log of the Separation.
Rod height, h2, (cm) / Separation, r, (m) / Balance Reading (g) / Force, F, (N) / Log (Force) / Log (Separation)Draw an Excel graph of Log (Force) against Log (Separation) and obtain a straight line of best fit and the equation for the line. The gradient of the line should be the same value as the exponent in the other graph.
Because of the uncertainties in the measurements, this straight line graph could have been steeper or flatter, that is, there will be an uncertainty in the gradient, the value of the exponent. This uncertainty would normally be obtained from the error bars of the data, but being a log - log graph, that is quite difficult. An alternative and simpler method is needed.
Uncertainties in the measurements.
- Balance reading. Usually for digital displays the uncertainty is half of the last digit, for example, for a reading of 2.46 g, the uncertainty is ± 0.005. However in this experiment, because of the experimental set up, the balance reading fluctuates, so the uncertainty could be double or even larger. This will depend on your set up.
- Separation. The separation is based on two reasonably accurate measurements. Even though the subtraction between the two measurements to obtain the separation doubles the uncertainty, for purposes of making the task easier, it can be assumed that the uncertainty in the balance reading is more significant than that in the separation.
Task:
From the observation of the balance, estimate an uncertainty for the balance readings (F): ______
This uncertainty can then be added to a reading and also subtracted from the reading to give a range within which the actual value lies.
When these extra values for added to a graph for the data points at either end of the data set, an upper and a lower limit for the gradient can be determined.
The figure over the page shows how this can be done. The table, also over the page, will assist with this calculation.
Upper limit and lower limit for the value of the gradient determined from error bar values.
Using data from above , complete this table.
Sep'n(m) / Balance Reading
(g) / Reading
plus F / Reading
minus F / Log
Reading / Log
Reading
+ F / Log
Reading
- F / Log
Sep'n
Min
Sep'n / A / C / E / G
Max
Sep'n / B / D / F / H
Average gradient = (A - B) / (G - H)
Upper limit to gradient = (C - F) / (G - H)
Lower limit to gradient = (E - D) / (G - H)
Typical results
Uncertainty in the balance reading of 0.01 g.
Sep'n(m) / Balance Reading
(g) / Reading
plus F / Reading
minus F / Log
Reading / Log
Reading
+ F / Log
Reading
- F / Log
Sep'n
Min
Sep'n / 0.045 / 0.51 / 0.52 / 0.50 / -0.292 / -0.284 / -0.301 / -1.35
Max
Sep'n / 0.091 / 0.13 / 0.14 / 0.13 / -0.886 / -0.854 / -0.921 / -1.041
Average gradient = (-0.292 - (-0.886)) / (-1.35 - (-1.041)) = - 1.92
Upper limit to gradient = (-0.284 - (-0.921)) / (-1.35 - (-1.041)) = - 2.06
Lower limit to gradient = (-0.301 - (-0.854)) / (-1.35 - (-1.041)) = - 1.79
Gradient = - 1.9 ± 0.1
Adjusting the force measurements for the 'decay' of the charge
It can be useful to measure the 'half life' of the charge on a charged rod. It is approximately 9 minutes, so to obtain a reasonably accurate value, about 30 minutes of observations are needed, which is best done after class.
Place the charged rod in the clamp a cm or two above the metal sheet so as to obtain an initial strong reading and then record the balance reading over time. The trendline of the graph will give the decay constant, . The half life = ln (2) / .
During the main experiment, the time can be recorded as well as the balance reading, each time the rod is lowered.
If the charge on the rod is slowly leaking away, the balance readings will be less due to there being less charge on the rod. The readings can be adjusted upwards for this loss of charge by using the decay equation.
If t is the half life, then the balance reading at time, t, should be divided by (½)t/t to give the adjusted readings.
The adjusted balance readings, will produce a more accurate measure of the value of the exponent.
Note: The measured half life is actually the half life of the readings, rather than that of the charge on the rod. As the electrostatic force is ∝(charge)2, the half life of the charge will be the square root of the reading half life.
Comparing the strengths of electric, magnetic and gravitational fields
If an iron based metal sheet is used, then a comparison between the strength of electric and magnetic fields can be demonstrated.
A magnet placed above an iron based metal sheet will also exert a measurable attractive force on the sheet.
The balance readings for the charged rod and the magnet at the same separation can be compared, and so, the strengths of their respective fields.
If the balance readings are converted into Newtons, the students can then consider the gravitational force and calculate what mass at the same separation would exert comparable forces to the charged rod and the magnet. The balance can be also used to determine the mass of the iron sheet.
Possible topics for Practical Investigations
- How does the size of the attractive force vary with the number of times the rod is rubbed with the cloth?
- Does the size of the attractive force depends on properties of the metal sheet, such as resistivity, thickness and area?
- Is the attractive force affected by an insulating medium between rod and sheet? e.g. a sheet of paper, and if so, does that depend on the number of sheets. Does the dielectric constant of the material make a difference?
- What factors determine the 'Decay' of a charged rod? dimensions and properties of the rod itself? atmospheric conditions such as air temperature, humidity and wind speed?
More complete physics explanation for theory behind the relationship: F = kQ2/4r2
An infinite conducting plane will have a potential of zero. A horizontal line in space with a potential of zero could be obtained if there was a charge of the same size, but opposite in sign on the other side of the line at an equal distance away, to the charge above.