# COULOMBS LAW PDF

Any charged object can attract a neutral object. Coulomb's Force Law for point- like charges. Between and Charles-Augustin de Coulomb (French. PDF | We examine the theoretical and experimental foundations of Coulomb's Law and review the various roles it plays not only in electromagnetism and. Today's Concepts: A) Coulomb's Law. B) Superposicon. Electricity & Magnecsm Lecture 1, Slide 1. If you haven't done Prelecture 1 yet, please do so later today.

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Abstract. We examine the theoretical and experimental foundations of Coulomb's. Law and review the various roles it plays not only in. Lesson 9: Coulomb's Law. Charles Augustin de Coulomb. Before getting into all the hardcore physics that surrounds him, it's a good idea to understand a little. Coulomb's Law. Charles Augustin de Coulomb. Before getting into all the hardcore physics that surrounds him, it's a good idea to understand a.

The phases and relative steps are described next. This phase starts by detecting the location of each exit door and also measures the width of each exit door.

We consider each exit door as a sphere with the diameter equal to the exit door width. This configuration must meet each sphere only once.

Because the goal of this step is to only determine the initial values and measures of locations and specifications for each exit door in the environment, to calculate and draw the initiate safe boundaries, we consider our environment to be empty; that is, devoid of agents. At this level, all exit doors located inside the environment will have been detected possibly using a raster map.

We need to have two different measures for each. The width and the central point of each exit door, which is placed on the central straight line that connects two edges of exit doors. This can be done by the cameras and detectors that are installed in the environment.

We may use some techniques to increase the speed of detecting and measuring locations of exit doors such as using a wire frame viewing of the environment. In this paper, we considered the zones as a complete 3D shape.

If an exit door blocked for any reason during monitoring the environment, we must consider its sphere and hence its charge considered to be 0. At this level, we divide our environment into different convex zones. To do this, we use convexity concepts and rules that have previously been used in the neural networks. On the other hand, the environment must be divided into convex shapes such that each default line must meet the edges of the shape only two times.

Each convex area must has at least a simple exit door inside; otherwise, we have to join this zone with the adjacent convex zone which has at least a single exit door. This point consists of determining the largest simple nondirected cycle graph of length where is total number of exit doors located in each zone. This graph has vertices and edges.

Every vertex has degree of 2. We label this graph.

For example, in case of 5 exit doors among the convex zone, the largest simple, nondirected cycle graph has the length equal to 5 and also we have 5 different pairs of adjacent charges. We considered each exit door as a sphere with the diameter equal to the width of each exit door.

Based on the metric measurement, the amount of positive charge for each exit door is given by the following equation: For instance, in case of an exit door with the width of 1 meter, the charge of the mentioned exit door would be At this phase, the process will determine the initial boundaries for each exit door using the initial values gathered in the previous phase. To determine the initial boundaries, this phase will find the point which is located on the line that connects the centers of each pair of exit doors as charges.

The location of this point obtained by considering the width of each exit door, then drawing a virtual vertical page crossing that point. This phase bisects the intersectional boundaries between different vertical pages to reach zones. At this step, we will calculate the amount of charge between each adjacent pair of exit doors. To do this, we assumed a straight line between each pair of adjacent exit doors.

We also assume that the locations of all exit doors are fixed and that the charges for each exit door are positive. We will put a positive charge on the straight line between each adjacent exit doors. The amount of the positive charge is given by the following equation: The assumed that the positive charge we used between each pairs is the average value for them, using this strategy guarantees that the positive charge will stay somewhere between two fixed charges and not beyond them.

At this step, the process will find the equilibrium location for each positive charge that is placed on the straight line between each pair of positive charges. All exit doors as positive charges are fixed. The positive charge will locate on the straight line and closer to the smaller positive charge. Since the two charges around the positive charge are positive, they will push the positive charge away from themselves. In such a case, the positive charge will stay closer to the smaller positive charge because the grater positive charge has a larger exert force than the smaller one.

The equations demonstrated in the three electric charges in equilibrium are used to determine the location of the mentioned positive charges. At this step, the process finds that the geometric center i. The center of mass, or centroid of a 2D shape, is the intersection point of all straight lines that divide the shape into two areas with equal moment about the line.

The centroid point is the arithmetic mean of all intersection points. The next step is to draw a line from that location to all positive charges and continue that line until it reaches an environment. This line separates the environment into two different areas such that each of them belongs to a different exit door.

The process continues until all zones are specified for each exit door. There should be several sensors, cameras, and detectors installed at different locations in the environment in order to be able to determine the location of each exit door and especially the location of the agent set at each moment.

These facilities estimate the status of each exit door in each instant in terms of evacuation ability rate. These devices will send pertinent data for agents in terms of their size and an estimate of their movement speeds to the central unit in order to classify analyzes and makes decisions.

This phase generally gathers and analyzes the data obtained by different sensors and detectors installed inside the environment. The crowd is dynamic and is constantly changing locations. The rate of movement is more unpredictable when agents are faced with an emergency situation. The process at this step will be to detect each individual position in the environment.

## Coulomb’s law – problems and solutions

One way to determine the positions is by using a grid. In order to make decision about the boundaries in real time, the sensors should be fast enough to determine individual locations and to send them to the central processing unit for analysis. The sensors and detectors should be able to determine the specifications of each individual such as body size and movement rate. For instance, based on the agent movement rate, the detector might determine the health status of each individual.

Having these measures we are able to assign an accurate value as a charge to each agent. At this step, the process will determine the amount of charge for each individual located in the environment based on physical specifications. The environment may consist of a number of general convex zones inside. Each general convex zone must consist of at least a single exit door. In case of having a convex area without an exit door, we will join it to with its neighboring zone that contains at least a single exit door.

The relations are shown by the following equation: Each zone has its own set of agents. Each agent may have a different situation in terms of the physical status.

A sample set of agent set in a sample general convex zone is shown by the following equation: Here is the th zone and is the th agent that belongs to th zone.

The key features that we considered in this paper were age, sex, and health status. The amount of charge for each agent is given by the following equation: Because all given values of the equation are negative, the final result of is always smaller than zero.

One of key features related to the physical specification of agents that we considered in this paper is the age of each individual. The ranges of ages are varied from a place to another place and it depends on the usage of the environment and determination is based on the average age of the majority of people in the environment. For example, the usage of values in a kindergarten is different from a conference room because in a kindergarten the majority of people are children so we may bind the normal value to the group of ages below than 10 years old whereas in a conference room, because the average range of age is between 20 and 40, we need to bind the normal value to this group of age.

In our case study, we focused on a night club station, which consists of adult people as the majority range of ages inside, so we have to bind the normal values of to the ages that are placed between 10 and 40 years old. Based on the ages of agents in the environment, we used Table 1 to assign values to calculate the charges of each agent. The other key feature in terms of calculation of agent charges is the gender of individuals. The values that may be used for each gender are different from situation to situation.

In this paper, we assumed the default value, of for the males. In case of having only one gender in our environment, we have to consider default value for it. The amount of charge related to the gender of each agent is shown by the following Table 2. The third physical key feature that we considered in this paper is the health status. The health status may vary from a place to place and it depends on the environment usage and is determined based on the health status of majority of the people are the environment.

## Chapter 1 Coulomb's Law.pdf

In this paper, we considered having only two options: Healthy and Disable. In some places, like hospitals or elder houses, there should be other options available in order to have a better estimate of the charges for each agent. The amount of charge related to the health status of each agent is shown by the following Table 3.

Apart from the physical specifications of each agent, focusing on the status of each exit doors is essential. We determine the amount of charge for each single exit door based on its situation at each time instant. Different situations for the status of each exit door may vary based on other options that are related to the usage of the environment, as well as the location of each exit door. We always used the default rate for the best situation of exit door when it is usable, reliable and can evacuate people to its full capacity.

In this paper, we used Table 4 to determine the safety rate for each exit door based on the status of it. We assumed the following values for each group of safety rates for exit doors.

To obtain the new values for charges of exit doors, we have to consider previous amount of charge for each exit door, and the latest safety status of each.

The total amount of positive charge is shown by the following equation: If the exit door completely is blocked or not usable, we have to consider its charge as 0 as shown in: The cameras and detectors will determine the safety rates of each exit door and send their status to the processing unit.

In such situations, we have to remove the exit door from our environment and reassign its zone to the other ones that are still usable. The exit door will not be considered in forming the largest nondirected simple graph.

In order to determine the new value of each exit door charges, we need to consider all agents that are belong to that exit door at the moment. The new amount of charge for each exit door is shown by the following equation: The key feature of calculating the new charges for each exit door is based on 19 which is to expand the zone of those exit doors that have the smaller number of people inside and they are also are usable and stable.

For example, assume having an exit door with 10 people in its zone and the adjacent exit door with the smaller width with only 3 people in its zone. Equation 19 will expand the area of the exit door with the smaller number of people.

For the next round of processing, we might consider many of the people that belong to the bigger exit door for the smaller one. Regardless of the already mentioned features, there are many other features that may exist in the environment that should be mentioned while determining each zones and boundaries.

Determining exit door status is necessary, especially in emergency situations. In the case of blocked exit doors for such reasons as smashed walls or people who block the exit door by pushing or shoving each other, the reliability of the exit door can be significantly decreased.

In such cases, the amount of positive charge of exit door will reduce if its reliability decreases. We called the reliability factor for each exit door as safety rate of that exit door. At the initialization phase, the safety rate for each exit door that is ready to use is set to. This rate will change based on the new environmental information gathered by sensors based on each exit door status.

The safety rate is shown by the following equation: To make the decision and updating the safe boundaries for each of the general convex zones, having the values described in the previous third phases is essential. The process of determining the boundaries for each exit door should continue updating by gathering new data from different installed sensors at each moment. Having a reliable and real-time hardware in order to detect and determine the different physical status of the exit doors, people status, and locations is essential in forming the safe boundaries in a reasonable time.

The process refreshes the results all the time to redirect to the second phase after reaching and completion of the third phase. Having the safe boundaries, which is the result of the 4th phase, helps people to make a better decision.

This produces lower risk and hence better results in terms of evacuating people out of danger in emergency situations. We selected the Station nightclub environment. On Thursday, February 20, , at The Station nightclub located at Cowesett Avenue in West Warwick, Rhode Island a fire accident occurred, which was the fourth deadliest nightclub fire in American history.

More than people lost their lives because of it. In the beginning, the fire ignited flammable sound insulation foam in the walls and then it spreads on ceilings surrounding the stage. Initially, there were about people inside before the fire incident. Some of them were injured and about 32 escaped uninjured. Based on the what cameras and sensors were installed inside the environment recorded, growing billowing smoke and blocked one of exit doors quickly made escape impossible because of limiting the vision site.

Based on some assumptions about the percentage of people who were spread in the environment and their physical specifications, we form the new zones. To apply the strategy, we consider only the map of empty building at first step to form the zones and then regarding the crowd distribution, we form the new safe boundaries.

Figure 6 shows a general view of the building shown in Figure 6. In order to increase the speed of processing and also to simplify the map, we consider the wire frame view for it. Showing the map in frame view also helps to distinguish between objects and people easier.

Figure 7 shows the frame view of Figure 6. At this level, because of determining the initial safe boundaries, we only focus on exit doors.

In order to apply our strategy, at the first step we have to determine the exact locations of each exit door and also the width of each. This task would be done by using a raster technology and will perform and send to the processing unit by detectors and cameras that are installed in the environment.

We also need to determine the general convex zones as well. Based on the environment map, we have generally two convex zones as shown by: The first general convex zone consists of four exit doors and the second zone consists of a single exit door as it shown by: We use the metric measurement in our paper and, therefore, of five exit doors available in the environment, the width of exit doors 1, 3, 4, and 5 all equal 1 meter, and exit door 2 is equal to 2 meters.

We have four exit doors in this zone hence the largest simple non directed cycle graph has the length of 4. To form the mentioned graph, we need to connect the central points of each exit doors together through straight lines. This diagram must meet each exit door only once.

Figure 9 shows the largest simple nondirected cycle graph of length 4 crossing all exit doors in the first convex zone. At the next step, we need to find the centroid point of the 2D shape formed by the mentioned graph.

We also need to find the equilibrium points between each adjacent pairs of charges. To do this, we need to have the values of the adjacent pairs of charges. Based on our strategy, we assumed all exit doors have the positive charges and also they are fixed in their places.

To find the equilibrium location, we use a positive charge that its amount is equal the average of the adjacent pairs of charges. The mentioned positive charge is placed on the straight line between the pairs of charges, and it is closer to the smaller charge.

In case of having the same amount of charges, the positive charge will locate in the middle of pairs of charges. Figure 10 shows all areas needed for the first zone when there is no any individual available in the environment. At this step, we will calculate the charges of the agents are available in the environment and will update the safe boundaries based on their distribution in the environment.

The process of gathering information about the physical specifications of the agents and their locations is done by sensors and detectors that are installed in the environment. This data will then be processed by our method. Of people that we assumed are available in the environment, we consider people are located in the first zone and 30 people are places in the second zone.

We also assumed that in each area, half of the people are male and the other half are female. We considered that all people in our environment have normal health status.

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We consider in each zone, the ages range is between 20 and 40 years old. We assumed that all exit doors are open all the time and safe to use with their full evacuation capacity which means no blocking will happen in the environment during the experiment. We apply our strategy in two different modes. When the distribution is the same and when it is not.

Equation 26 shows the collections of the agents in each zones: Based on our assumption for the first mode, there are males and females available in the first zone.

All of them have normal health statuses and are between 20 and 40 years old. Figure 11 shows the new safe boundaries based on the normal distribution of the crowd in the first general zone. Considering the mentioned distribution may be useful in many places such as theater saloons or conferences rooms.

In such places, because of the kind of usage of environment, the distribution of the crowd is equal for all areas inside, whereas in many other places, such as night clubs or hallways, because the crowd distribution is always subject to significant changes, we need to consider a more accurate and realistic pattern in our environment. In the following mode 2, we will consider having a random crowd distribution based on the different locations that are available in the environment.

In this mode, we consider a rational pattern of crowd distribution based on the usage of the environment and also the different locations available in the general convex zone. In the first zone, we have the highest concentration of the crowd available because of its usage. It consists of a Raised Platform and a Dance Floor. These areas have the most attractive options that can potentially cause the present people to gather in them.

The number of people that each exit door supports is varied based on the area occupied by each. In some cases, because of changing the safe boundaries, some people may belong to a new exit door which means in order to have a safe evacuation they have to be guided through the new exit door that they belong to.

Based on the mentioned values shown in Table 9 , we redraw and form the new safe boundaries based on the new values of charges for exit doors. We assumed that all exit doors are stable and reliable during experiments. In the real word, in order to have better decisions about forming safe boundaries, considering the status of the exit doors is essential.

If for any reason an exit door is blocked completely or it is not reliable anymore in terms of evacuating the crowd, our strategy is not applicable. In such cases, the amount of charge for exit doors will be 0 and it will not conform to the largest simple nondirected graph anymore.

The output of our tool for supervising people is rapid identification of exit doors for groups of people fleeing danger. Use of this mechanism can quickly decrease errors committed by exiting individuals. The technology for how the supervisory people in the control room will use our model to guide people is beyond the scope of this article. Detecting people automatically is not trivial and remains to be explored in future studies.

To be effective, the detectors and sensors should also be able to determine the specifications of each individual such as body size and movement rate. These are a few challenges to be addressed in the future. Despite challenges, our methodology yields strategies for guiding people who are trapped in an indoor public space, at dangerous locations, to be most rapidly evacuated.

Advances in Artificial Intelligence. Table of Contents. Journal Menu.

Abstract This paper focuses on designing a tool for guiding a group of people out of a public building when they are faced with dangerous situations that require immediate evacuation. Introduction The gathering of a group of people at the same location and time is called a crowd. Related Work Anyone living in a populated, gregarious world has experienced the effects of crowds. The Main Attributes for the Environment Generally, based on its usage, each environment will consist of many different groups of objects; such as obstacles i.

The Main Attributes for Exit Doors We considered two general attributes for each exit door that are zones and boundaries. Figure 1: The and zones and angels related to exit door 1 and 2. Figure 2: Charges with a same signs repel each other and charges with a different signs will attract each other. Figure 3: Two charges and which are placed at the distance of.

Figure 4: Figure 5: Darker arrows are electrostatic forces and lighter arrows are reaction based on the Newton 3rd Law. Table 1: The amount of charge related to the age for the th agent that belongs to th zone.

Table 2: The amount of charge related to the gender for the th agent that belongs to th zone. Table 3: The amount of charge related to the health status for the th agent that belongs to th zone.

Table 4: The safety rates for exit doors based on their situation at moment. Figure 7: The frame view of the Figure 6. Table 5: The zones and the amount of charges for each exit doors based on their width. Figure 8: Figure 9: The largest simple non directed cycle graph of length 4 crossing exit doors in the first convex zone.

Figure Table 6: New safe boundaries based on the normal crowd distribution. Table 7: The percentages of the people who occupy each zone. Table 8: The percentage and the number of people for each zone based on the random distribution assumption.

Table 9: The zones with the old zones and new zones after applying the new random crowd distribution. The new safe boundaries based on the random crowd distribution.

References F. Cannavale, H. Scarr, and A. Singer, C. Brush, and S. Gergen, M. Gergen, and W. View at Google Scholar R. Johnson and L. Carver and M. Scheier, Attention and Self-Regulation: Fenigstein, M. Scheier, and A.

Prentice-Dunn and R. Paulus, Ed. View at Google Scholar G. View at Google Scholar T. Korhonen and S. Sharbini and A. Singh and A. Peacock, J. Averill, and E. View at Google Scholar Y. Zhao and S. Perez-Delgado and D. Braun, S. Awab Sir 76 Example 1: Since V is a function of only x we change partial differentiation into normal thus we have. Find the potential and also show the variation of V with respect to x.

Putting these values in Eqn. Two different dielectrics with and forms two different capacitors in series. Consider top and bottom area of each capacitor is A. Two capacitances can be obtained as: Awab Sir 76 For region 1: Voltages are. To find E we have. Integrating once again. To find A and B apply known voltages in region 2 to eqn.

But the values of A and B will be different. Let us see now what is uniqueness theorem. Qn with surfaces s1. Inside the closed surface s0 there are number of charged conducting bodies Q1. Statement of uniqueness theorem: One of the method known as method of images is explained in next section.

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But in complicated problems some other methods of solution must be used.. Let us define a new difference potential: Awab Sir 76 Uniqueness Theorem: In the previous sections we have seen that a relatively simple problems can be solved by direct integration. V1 — V2 is everywhere zero. So the right side of equation v is also zero.

## Notebook CL - Coulombs Law.pdf - Before you start working...

Awab Sir 76 … iv The divergence theorem allows us to replace the volume integral on the left side of above equation by the closed surface integral over the surface surrounding the volume.

The implication of the uniqueness theorem is that a solution of an electrostatic problem satisfying its boundary conditions is the only possible solution.. If the gradient of Vs i. It follows that Vd throughout the volume is zero.

The surface integral on the L. The divergence theorem is: Consider a large surface s0. When R is very large both V 1 and V2 can be thought of as potentials due to point charges here conducting surfaces are treated as point charges since size of it will be very small as compared to R of a very large sphere.

It gives differential magnetic field intensity. Of the two possible normals that normal is chosen which is in the direction of progress of a right handed screw turned from d through the smaller angle to the line from the filament to P. We are interested in finding field intensity H at point P. Consider a filament through which current of I amp is passing. Awab Sir 76 4. It is inversely proportional to the square of the distance from the filament to point P.

In vector notations. Magnitude of dH at point P is proportional to product of current. The Steady Magnetic Field: Awab Sir 76 Differential current elements have no separate existence. All elements making up the complete correct filament contribute to and must be included.

The summation leads to the integral form of Biot-savart law as. The direction of dL is from A to B. Given points A 1. Find dH at C. The line integral of around a single closed path is equal to the current enclosed by that path.

To verify the law. Showing different closed path www. The best well known source is infinite current filament placed along z-axis. When the field vector is present in a single continuous media then at every point in amedia. Path A and B enclose the conductor. Similar to this when the field goes from one media to second media.

Awab Sir 76 Consider the conductor shown in Fig. When light travels from one media to another. Consider a small rectangular volume positioned across the interface such that it is half in each media. The normal components of and are and present at the top and bottom of the volume. Instead of rectangular volume any closed surface can be considered. Since magnetic flux lines are continuous we have. These types of problems are solved by knowing the relation between the tangential and normal components of the fields in two media.

At the top: The closed surface consists of six surfaces. Figure shows a boundary between two isotropic homogenous linear medias with permeabilities The Gaussian surface and the closed path is constructed at the boundary between medium 1 and medium 2 as shown. The effect of it is the contribution of all sides except top and bottom is zero and now LHS consists of only two integrals.

Thus L. As in electrostatics. Awab Sir 76 To determine boundary conditions Suppose the field in one media is given and we require to calculate field in second media. To obtain these boundary relations consider a rectangular path half in each media with width and height as shown in figure.

Since width of the rectangle is. This current is the surface current K flowing along the interface. If no current along interface. Thus LHS reduces to. Awab Sir 76 At the bottom: In vector form.

Now the integrals can be solved as follows: Let and are tangential components of and in medium 1 and medium 2 respectively. We have Amperes work law — In the figure. To obtain normal component. Two homogenous. The emf is included in a loop when the magnetic flux is changing in the vicinity of it. Since for static field. This relation stands because the work done in a closed path is equal to zero. The emf induced in a loop is expressed is expressed in terms of electric field as.

These are summarized below. In order to get clear understanding between time varying and static fields. The line integral of H around a closed path is equal to the current enclosed by that path. The electric flux through the closed surface is. The line integral of H around a single closed path is given as. The total electric flux crossing the closed surface is equal to the total charge enclosed by that surface.

The reason for this is. Thus electric flux line have start and end points. In case of electric field. The magnetic field lines are always www.

Awab Sir 76 Then the Gauss law for electric is expressed mathematically as. The incoming flux. This is not the case for magnetic lines. Due to which a closed surface in the presence of these lines will have same number of incoming and outgoing flux lines. The total magnetic flux crossing the closed surface is equal to zero.

There is no starting and end point. Awab Sir 76 closed in nature. The total current crossing the closed surface is equal to zero. Mathematically it is given as. Instead of node. In practice field always vary with time. All other laws require modification. Using Divergence theorem. Only Gauss law. Awab Sir 76 The fundamental property of electrical charge is that it can neither be created nor destroyed. The current coming out from any volume can be thought of as a rate of decrease of charge in that volume.

Consider a volume v located inside a conducting media. If a charge disappears from one point it must reappear at another point.

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The current density is a vector having the direction of current flow. The displacement current density is denoted by Jd or Jdisp. Consider now the unit of. Suppose we add an unknown term to i. Thus is also a current density and is called as displacement current density. As we have expressed current as surface integral of current density J. Awab Sir 76 5. Substituting this value for C.

The voltage V is applied across the parallel combination. Fringing of the field is neglected. Awab Sir 76 Thus. The constant voltage across a resistor produces a continuous flow of current of constant value given by.

Now consider the resistor and capacitor elements. Inside each element the electric field E equals. The current flows through the capacitor only when voltage across it is changing i.

The nature of the current flow through the resistor is different from that through the capacitor. It is also equal to i 1 divided by the cross-sectional area A. In a closed path loop the electric potential emf is developed due to time varying magnetic field in the vicinity of that closed path. The line integral of around a closed path is equal to the current enclosed by the path. Differential form: In converting to the differential form from integral form.

Both sides of the above equation are volume integrals. And the surface integral of over the surface s bounded by C is replaced by itself as shown: The total flux crossing the closed surface is equal to the total charge enclosed by the closed surface Mathematically.

Applying divergence theorem to convert surface integral on the left hand side to volume integral Then from equation 5. In case of magnetic field the total outgoing flux magnetic is equal to zero Mathematically.

Table a: This equation results from the fact that the magnetic flux lines are continuous.

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Using divergence theorem to convert surface integral to volume integral equation 5. So for free space.

Depending upon the media involved in the problem we should change these equations. In normal state good conductor will not have any charge temperature effect is neglected giving In good conductor the conduction current J is greater than displacement current and can be neglected.

Dielectric also is a medium without charges and thus there is no charge density For a good dielectric. Due to absence of charges there is no conduction through space giving zero conduction density.

Free space: Free space is a space without charges. Awab Sir 76 So. It is given below: For a free space: For good dielectrics: For good conductors: Table d:Figure 7: On page 75 , the authour compares the electric and gravitational forces: However, he did not generalize or elaborate on this.

Awab Sir 76 4. The force given by equations 1. Using Divergence theorem. Hibran Sabila Maksum. Are you now wearing a wrist strap that is connected to ground? Applying divergence theorem to convert surface integral on the left hand side to volume integral Then from equation 5.

Since they have access to global views, we believe supervisory people in the control room can use our simulation tools to determine the best courses of action for people.