# 3 Testing Coulomb’s law and using vector components

This activity has three parts: a video demonstration of an experiment, an exercise and a video solution. The activity gives you a practical demonstration of electrostatic forces and the opportunity to practise using the vector form of Coulomb’s law. It also encourages you to think about the assumptions in your calculations and possible sources of experimental uncertainty.

## Activity Testing Coulomb’s law and using vector components

Allow up to 1 hour

#### Part 1 Measuring electrostatic forces

Watch Video 1, which shows Open University (OU) academics Sam Eden and Anita Dawes measuring one component of the electrostatic force that a charged sphere feels due to nearby charged spheres.

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Video 1 Filmed experiment: measuring electrostatic forces at the OU.
Transcript

(The data shown on-screen during Video 1 are available in an Appendix to this activity.)

#### Part 2 Calculating electrostatic force components and comparing them with the measured values

Figure 1 shows the -direction and the relative positions of the charged spheres in the filmed experiment.

Figure 6 Annotated still image from Video 1 showing the positions of the test charge sphere, sphere A, and two possible positions for sphere B, as well as the force component .

### (a)

Use the vector form of Coulomb’s law to predict the -component of the force that the test charge sphere feels in the three situations described below. All charges are given in units of nanocoloumbs (). Write your answers in millinewtons () in Table 1.

1. The test charge sphere and sphere A are charged with each. Sphere B is not present.
2. The three spheres are charged with each. Sphere B is in its first position (see Figure 1).
3. The three spheres are charged with each. Sphere B is in its second position (see Figure 1).

## Table 1Calculated force on the test charge.

situation | calculated
--- | ---
i | (A text entry box would appear here, but your browser does not support it.)
ii | (A text entry box would appear here, but your browser does not support it.)
iii | (A text entry box would appear here, but your browser does not support it.)

Interactive feature not available in single page view.

### (b)

Comment briefly on possible reasons why your calculated values may not agree exactly with the experimental values. Consider approximations in your calculations and experimental uncertainties that may have significant effects.

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### Discussion

Keep in mind that the discussion below only addresses a selection of issues; there are various other valid points that you might have raised in your answers.

The key assumption in the calculation is that the charged spheres can be treated as point charges located at the spheres’ centres. At any position outside a spherically symmetric charge distribution, the field produced by the charge distribution is the same as the field that would be produced by a point charge located at its centre. This is a consequence of Gauss’s law (one of the four key laws in electromagnetism known as Maxwell’s equations). However, in this experiment the charge distribution around each sphere will not be perfectly spherically symmetric. This is partly because no manufactured sphere is perfect, but a more fundamental issue is that the electric field produced by one charged sphere will disrupt the spherical symmetry of the charge distribution on another.

Testing the experiment before filming indicated that most significant sources of experimental uncertainty were unwanted forces and charge leakage, as described below.

The test charge sphere can experience unwanted forces (that is, forces other than the electrostatic force due to nearby charged spheres). For example, air flow in the studio and vibrations transmitted from the floor had noticeable effects in the tests before filming.

Charge leakage occurs in the time between charging the spheres and measuring the forces. No insulator is perfect, and the rate at which charge dissipates from the spheres is sensitive to factors such as air humidity and the cleanliness of the plastic rods.

Further sources of experimental uncertainty include:

- The charge transferred to each sphere is known to a limited precision. This depends on the calibration procedure.
- No force meter is perfectly accurate, and the measured forces are displayed to a limited precision.
- Video 1 does not provide information on how precisely the test charge sphere has been positioned to measure the -component of the force that it feels.
- The - and -positions of the spheres’ centres are given to a limited precision, and the video does not provide information about precisely how they are situated on the -plane.

#### Part 3 Applying the vector form of Coulomb’s law – a model solution and discussion of the results

### (a)

Watch Video 2 in which Sam presents model solutions for the three situations described in Part 2(a) of this activity. The video highlights the use of use of displacement vectors, position vectors, unit vectors and vector components.

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Video 2 Using the vector form of Coulomb’s law to determine theoretical force components for comparison with the measurements in Video 1.
Transcript

The experimental results from Video 1 are summarised in Table 2 for the three situations described in Part 2(a). In each situation, was recorded in before and after the spheres were charged. The results from Video 2 are also shown and your calculated values from Table 1 are also shown.

## Table 2 Summary of the results from Videos 1 and 2.

situation | average measured | calculated values from Video 2 | your calculated values from Table 1
--- | --- | --- | ---
uncharged | charged | difference
i | –0.058 | 0.65 | 0.71 | 0.81 | Original location
ii | –0.10 | 0.66 | 0.76 | 0.81 | Original location
iii | –0.057 | 0.99 | 1.0 | 1.1 | Original location

### (b)

Compare the calculated values in Table 2 with the experimental results (in the ‘difference’ column). Does this comparison support the validity of Coulomb’s law and of your method for applying it in situations with more than two charges?

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### Discussion

For brevity, this discussion is limited to the point charge assumption, and the effects of charge leakage and unwanted forces.

### Point charge assumption

The field produced by one charged sphere disrupts the spherical symmetry of the charge distribution on another. This causes the concentration of positive charge to be greatest on the side of each sphere that is furthest from the other positively charged spheres. Hence, treating the charged spheres as point charges located at the spheres’ centres represents an underestimation of the separation of the charge distributions on each sphere.

### Charge leakage

Charge leakage occurs during the time between charging the spheres and measuring the forces. This means that the magnitude of the charge on each sphere at the instant that the force is measured is lower than the that was initially transferred.

It follows that the point charge assumption and charge leakage both cause the calculated forces to be higher than the forces in the experiment. This is broadly consistent with Table 2, where calculated forces are between 6% and 12% higher than the experimental values.

### Unwanted forces

Table 2 shows that force measurements prior to charging the spheres vary over a range of . This provides a first approximation for the variation in the unwanted forces. The variation is close to the difference between the values measured in situations (i) and (ii), which the calculations indicate should be the same.

### Summary

This short discussion indicates that the key approximation in the calculation and the main sources of experimental uncertainty are broadly consistent with the differences between the calculated and measured forces in this activity. Therefore, the experimental results are broadly supportive of the validity of Coulomb’s law and the present method for applying it in situations with more than two charges.

OpenLearn - Electromagnetism: testing Coulomb’s law
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