Determining the Capacitance by Measuring the Energy Stored in the Capacitor with Increasing Voltage - Lab Report Example

Abstract

This experiment was conducted to determine the characteristics and properties of a simple capacitor. This was achieved by connecting the apparatus as described below and then evaluating the corresponding changes in two phases. The first phase included determining the relationship between the voltage supplied from a power source and the corresponding changes in the amount of the energy stored. The results from this phase allowed for a graph based on the values determined. The second phase evaluated the relationship between the distance separating the plates and the resulting capacitance. This experiment determined that there exists a linear relationship between the pd and energy stored. It also determined the existence of an inverse relationship between the distance separating the plates (conductors) and the capacitance therein. Based on the graph, the experiment allowed one to determine the distance or the capacitance when one of them is provided.

Introduction

The experiment discussed in this report analyzes capacitance by determining the relationship between an increase in voltage and the corresponding amount of energy stored as a result of the same. The experiment was set up as indicated below. The corresponding changes in the energy stored were determined and recorded whenever a change in the voltage was initiated. The relationship between the two simultaneous values was determined. An electric charge placed on an object results in the creation of an electric field. This field facilitates repulsion or attraction of other charges which exist around the object (Golden, 1972). If charges of equal magnitude and opposite signs were placed on two parallel conducting surfaces, the result would be an even spread of the charges on the surfaces due to their capacity as conductors. There exists an attraction force between the charges on the opposite parallel plates. Introducing charge to the plates results in stored charge (Campbell and Childs, 1935).

This particular experiment explored the energy stored, capacitance, by analyzing the increasing potential difference between the two conductors. A joint relationship exists whereby the voltage, which represents the amount of energy stored, varies directly as the conductors, U, and inversely as the charge, Q, placed on them. The above relationship can be illustrated as below.

A relationship of the form has been developed from the results of multiple experiments investigating the same (Enhancing Capacitance, 2011). The variable that quantifies the relationship between the two quantities is known as the capacitance. Capacitance has been found to vary directly as the amount of charge and inversely as the potential difference applied to the conductors.

Where C indicates the capacitance, Q is the charge and V is the voltage. Defining capacitance as above makes it independent of the voltage applied to the conductors and the resulting charge. These aspects make capacitance be dependent on the geometric properties of the conductors. These properties include the distance between the conductors, the shape the conductors take, the area of the conductors, and other physical constants. Capacitance facilitates determination of the potential difference applied between two conductors if the amount of charge existing between the two is known (Knudsen, 1953). The experiment conducted involved setting up a simple capacitor using conductors of equal area separated by a known distance. Capacitance in this context is determined using the relation below.

Where A is the area of the conducting plates, d represents the distance which separates the plates. represents the permittivity of free space which is a constant whose value is

Capacitors have been developed and modified to decrease in size and increase the amount of energy they store. They form the basis of development of the majority of the electronics in existence (Walker, 1990). The capacitors in use today have a minimal separation distance between them while the area is maximized and compressed to fit in a small space.

The current experiment is set up to determine the relationship between the potential difference, the amount of energy stored and the distance of separation of the conductors when the all other factors are maintained as constants. It was conducted on a simulation of the lab as indicated in the instructions. The simulation was downloaded from https://phet.colorado.edu/en/simulation/capacitor-lab.

Materials and methods

Simulating this experiment required materials which were connected as below

Figure 1. Connections in the experiment when V=0

The potential difference was then increased gradually and the corresponding values in the energy stored bar recorded. The results were recorded in a tables of results. The distance separating the two plates was then varied and the corresponding capacitance resulting from the same observed as below and the distance noted.

Figure 2. The experiment with d was varied to make

Results

Using the results from the simulation, we generated a graph of E against V2 which is shown below. The minimum point in the graph was point (0, 0) and the maximum values are (2.25, 1). The graph was plotted using the V2and the values taken by E based on the existing hypothesis.

Figure 3 graph of E against V2

Discussion

The capacitor has been described as a very common component in electronics. It is a device which is made for the sole purpose of storing electrical energy. This role is, in essence, used to develop the definition of the term capacitance which is explained as the ability of a device, a capacitor in this context, to store energy in the form of electric charge. The ability to store the charge serves as a vital tool in electrical and electronic circuitry. Examples of the functionalities it serves are time delay circuits, tuning radio receivers, power factor correction and many more (Golden, 1972).

The general capacitor consists of two conductors, which are molded into plates which are placed very closely together with an insulator between them. The insulator in the capacitor varies with the models. The current experiment utilized air as the medium although it comes with the limitation whereby the distance separating the two is limited since air may allow conduction to take place. This scenario may make the capacitance effect expected to be ineffective and therefore nonfunctional.

When a capacitor is attached in a circuit, one of the ends is connected to the upper plate while the lower plate is connected to a different terminal of the power source. Closing the switch in that scenario makes the two plates to have different charges since the first plate has an increase in the number of negative electrons (Enhancing Capacitance, 2011). Positive charges flowing through from the positive terminal of the power source attract the negative electrons in the plate. The negative electrons drawn result in a shortage in the number of the negative electrons matched, and therefore they result in an increase in the number of positive electrons in the plate. The plate, therefore, acquires a positive charge.

The lower plate, however, being connected to the negative terminal has an outflow of positive electrons attracted to the negative charge flowing into the plate. The saturation of the positive electrons, formerly evenly paired to the negative electrons in the plate, results in a deficit of the same in the plate. The negative electrons are more making the plate acquire a negative charge (Campbell and Childs, 1935).

This makes the two parallel plates to have opposing charges. The difference in polarity between the two plates results in a difference of potential between them. This difference in potential limits capacitance in the same way in which it is limited by the battery voltage. The only limit that the voltage supplied by the power source inhibits capacitance is in the amount. The potential difference between the two plates, once it equals the voltage at the power source, prevents more charge from being placed on the capacitor. The amount of charge that a capacitor can store is therefore limited by the amount of voltage supplied by the source. A fully charged capacitor is one in which the potential difference between the plates is equal to the potential difference supplied by the source.

Opening the switch after the plates have been charged has no effect on the charge stored since the electrons held in either plate have no alternative path to flow out. This hypothesis was confirmed by the lack of change in the capacitance after the battery was disconnected. If a conductor closed the distance between the two plates of the capacitor, the electrons would get a path to flow making the charges be neutralized on both ends. This aspect would destabilize the number of charges stored and, therefore, result in the discharging of the capacitor (Walker, 1990). This is a very vital factor considered when developing capacitors since the medium put as an insulator should be made such that the electrons have no way of discharging the capacitor even if the numbers are increased.

Several factors have been found to affect the amount of energy that capacitor stores. These factors have been found to be three main ones. The first aspect that influences capacitance is the surface area of the plate. The surface of the plate is the point utilized by the electrons to store the charge in capacitors. The size of the plate gains value from the stature it takes regarding facilitating more charge. Large sizes and surface areas of conductors have more charge as compared to smaller sized materials of the same type. When the materials are discharged, the electrons in a material are evenly distributed making larger materials to have more electrons since they have a larger surface area (Walker, 1990). A large number of electrons in large sized materials translate into more electrons attracted when the switch is opened and therefore a larger number of free and charged electrons. This aspect, in essence, translates into higher charges for the plates and therefore a higher capacitance. The relationship between the two can be expressed mathematically as below

The other two factors affecting the same are plate spacing and the dielectric material. This experiment limited the use of dielectric materials leaving the space between the two plates with air only. The area of the plates in the experiment was also maintained at a constant. The factor under investigation in this essay would, therefore, be the distance between the two plates. The distance between the plates has been discussed to have an inverse influence on capacitance. The effect of the charges on one another when the plates are closer is bound to have a larger effect. When the plates are moved further apart, the influence they hold over each other is limited since the distance has been increased lowering the capacitance (Knudsen, 1953).

The experiment was conducted in two phases. The first phase involved evaluating the corresponding changes in the charge stored as a result of variations in the potential difference. The experiment determined that a direct relation exists between the two where by the pd varies directly as the amount of energy stored as a result of the same. A best fit graph was drawn using the results obtained facilitating the determination of the gradient as indicated in the results. The gradient was then applied to the standard formula to determine the capacitance. From the standard formula, the capacitance was determined to be, which was reflected by the capacitance measured when the distance between the plates was maintained at 10mm.

When the value of the distance was varied based on the target capacitance, the standard results matched those from the calculations for the same variables. The calculations provided a distance of as the one required to ensure that the capacitance attained was. Adjusting the distance based on the corresponding changes in the capacitance in a bid to manually determine the distance needed for the same capacitance resulted in a distance of The two results were very similar with the differences possibly arising from minute errors in the graphs and truncating during calculations.

From the two phases, two aspects became clear. Both of the cases provided results which quantified reports by other similar studies. The first major aspect which is vivid from the first phase of the experiment is the existence of a relationship between the voltage (pd) and the amount of the energy stored in the two plates. This relationship was indicated in two aspects. The amount of charge indicated as measured was increasing with the increase in the voltage. The two plates also portrayed maximum charge graphically by indicating the number of positive and negative charges indicated on the two plates.

The other relationship visible from the experiment was the dependence between the capacitance and the distance separating the two plates. Reducing the distance between the two plates indicated an increase the capacitance. When the distance was at 10.0mm, the capacitance was. Decreasing the distance to 5.5mm resulted in a capacitance whose value was, an increased value as compared to the former value. These results indicated that an increase in the distance separating the two plates resulted in a decrease in the capacitance and vice versa. This property matched the hypothesis proposed by the other studies further quantifying them.

The experiment further justified the independence of the capacitance from the direct influence of the potential difference and the stored energy. The capacitance remained constant regardless the changes in the voltage and the resulting changes in the amount of energy stored. This finding was supported by the reports from the initial studies. The only factor that directly influenced the capacitance was the distance separating the plates.

Understanding capacitance is a complex undertaking which would be limited if we limited our experiment since it only targeted a simple capacitor. This experiment leaves much more information unknown regarding the other properties of capacitors. One area whose blank the experiment leaves blank is how different types of materials influence the performance of a capacitor. The experiment also does not integrate the chance to evaluate how the capacitance is affected if other mediums are installed as intermediaries between the two plates. The arrangement of capacitors in series and in parallel is also another factor which may be considered for future studies.

Conclusion

Many studies have been proposed and undertaken in a bid to understand the relationship between voltage, capacitance and the factors that affect capacitance. The theories presented have had academic backing and informed support from experiments. The factors that interrelate with each of the two have been discussed in the discussion section of this report. This experiment was setup to determine the relationship between the energy stored in a conductor as a result of connection to a power source. The experiment determined that a linear relationship exists between the potential difference at the source and the amount of energy stored in conductors connected to the source. The other aspect investigated by the experiment was the factors that affect capacitance in a simple capacitor. The results of experiment supported findings by other studies regarding the properties of a simple capacitor.