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.
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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.
Since forces from all other charges on a specific charge are linear, we can apply superposition on each particular charge. The ball was charged with a known charge of static electricity , and a second charged ball of the same polarity was brought near it. 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. Covariant formulation Electromagnetic tensor stress—energy tensor Four-current Electromagnetic four-potential. These facilities estimate the status of each exit door in each instant in terms of evacuation ability rate.
The magnitude of force between two electrical charges is given by Coulomb Law. Experiments conducted by Coulomb showed that the following hold for two charged bodies that are very small in size compared to their separation so that they can be considered as point charge: The magnitude of the force is directly proportional to the product of the magnitudes of the charges.
The magnitude of the force is inversely proportional to the square of the distance between the charges. The magnitude of the force depends on the medium. Thus, if we consider two point charges Q1 C and Q2 C separated by a distance R m in free space, the force is given mathematically as,. Force is a vector quantity and acts along the line joining the two charges therefore the above expression is to be multiplied by a unit vector along the line. Thus, vector form is.
The unit of force is Newton N. The force given by equations 1. Step 1: Draw a neat sketch.
Step 2: Find vector joining these charges and pointing towards the charge on which force is to be determined. For example, while finding force on Q1 i. F1 arrow of the vector points towards Q1, while finding force on Q2 i.
F2, the arrow of the vector points towards Q2. Step 3: Find unit vector in the direction of vector and length of the vector i. Step 4: Force acting on Q2 due to Q1 is. It is directed from Q1 to Q2. In the problem if it is asked to find force on a particular charge due to other charges then the principle of superposition can be used In which, find the force on that charge due to other charges separately and the total force is the addition of all forces.
Obviously this addition is a vector addition. Example 2: A fifth 10 nC positive charge is located at a point 8 cm distant from the order charges. Calculate the magnitude of the total force on this fifth charge for. Dec Solution: In the fig. Illustrating Ex. Coordinates of point M are 0. Hence, position of Q 5 is 0. To find force on Q5 due to Q1: In the figure above it is clear that horizontal components of and are getting cancelled, while horizontal components of is cancelled by.
When we add all four forces the resultant should have only z component. This makes vertical components of each force to be equal. Four like charges of 30 each are located at four corners of a square, the diagonal of which measures 8 mts.
Find the force on a charge located 3 mts above the centre of the square. The answer is. Refer figure. For equilibrium resultant of forces acting on any one charge. We have selected this orientation. Required vectors and unit vectors are: Let us fit the square in the coordinate system such that one corner is at the origin as shown in figure.
Force acting on charge at A due to remaining charges is obtained as: Awab Sir 76 Solution: Given square can be placed in coordinate system as shown in figure. Calculate the point charge. Total force on Q at A must be zero for equilibrium. Awab Sir 76 i. C For equilibrium total force acting on any charge must be zero. That is. Let us place the equilateral triangle as shown in figure. Force on Q at A due to charge at C is. Force on Q at A due to charge at B is.
Let QT is the test charge. Find the relation between Q1 and Q2 such that the total force on a test charge at the point P Two point charges Q1 and Q2 are located at 1. At point P consider an differential element of surface ds and let makes an angle with as shown in Fig.
The flux crossing is then the product of the normal component of D and ds. The electric flux passing through any closed surface is equal to the total charge enclosed by that surface. Let a positive charge Q is enclosed by a closed surface of any shape. Consider now a positive charge Q situated at the centre of an imaginary sphere of radius r as shown in fig. Three point charges. The total charge enclosed is. How much flux crosses s? Since a point charge is situated at the centre.
Thus Gauss law is proved. But if the charge is not at the centre of the sphere. Equation 1. In equation 2. Awab Sir 76 Example 2: Three point charges are located in air: D due to this charge configuration can be determined. By using some conditions the problem of finding D is simplified. Thus the total charge enclosed is. The sphere of 5 m radius will enclose all three charges in the system. There are two extreme cases of dot product: When and d are normal … When and d are parallel … For the first case the integral goes to zero.
The integral in the expression has dot product term. This product is equal to charge enclosed. For the second case if D. The left hand side of it is electric flux and can be determined by having the knowledge of and d. The same law can also be used to determine D of some known charge distributions. Compute the total flux over a sphere of 5 m radius with centre at 0.
Its not like Gauss law is used to find only flux. The type of surface or surface elements which satisfies which satisfies these conditions are called as Special Gaussian Surfaces. Often the other conductor.
Awab Sir 76 2. While dealing with the problems of positive point charge. V Then equation previous to above equation becomes. D is everywhere either normal or tangential to the closed surface. While solving problems of finding due to given charge configuration. D is essentially constant over that part of the surface where D is normal. The surface is closed. For homogenous medium for which is a constant.
Awab Sir 76 … 3. Laplace Equations in Three Coordinate Systems: Equations for Laplace are derived as follows: In cartesian coordinates: The charge density on the capacitor plates is obtained by. From V. 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. 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. 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: Denoted k e , it is also called the electric force constant or electrostatic constant,  hence the subscript e. The last of these is known as the electrostatic approximation.
When movement takes place, Einstein 's theory of relativity must be taken into consideration, and a result, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the magnetic force, and is described by magnetic fields. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct, but when the charges are moving more quickly in relation to each other, the full electrodynamic rules incorporating the magnetic force must be considered.
The Lagrangian of quantum electrodynamics is normally written in natural units , but in SI units, it is:. The most basic Feynman diagram for a QED interaction between two fermions is the exchange of a single photon, with no loops. Ignoring the momentum on the external legs the fermions , the potential is therefore:. Recognising the integral as just being a Fourier transform enables the equation to be simplified:.
Rearranging these definitions gives:. Continuing in SI units, the potential is therefore. The derivation makes clear that the force law is only an approximation — it ignores the momentum of the input and output fermion lines, and ignores all quantum corrections ie.
The Coulomb potential, and its derivation, can be seen as a special case of the Yukawa potential specifically, the case where the exchanged boson — the photon — has no rest mass. When it is of interest to know the magnitude of the electrostatic force and not its direction , it may be easiest to consider a scalar version of the law. The scalar form of Coulomb's Law relates the magnitude and sign of the electrostatic force F acting simultaneously on two point charges q 1 and q 2 as follows:.
If the product q 1 q 2 is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. Coulomb's law states that the electrostatic force F 1 experienced by a charge, q 1 at position r 1 , in the vicinity of another charge, q 2 at position r 2 , in a vacuum is equal to:.
The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges.
The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. The force F on a small charge q at position r , due to a system of N discrete charges in vacuum is:.
In this case, the principle of linear superposition is also used. For a continuous charge distribution, an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge dq.
The distribution of charge is usually linear, surface or volumetric. It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass m and same-sign charge q , hanging from two ropes of negligible mass of length l. The forces acting on each sphere are three: Let L 1 be the distance between the charged spheres; the repulsion force between them F 1 , assuming Coulomb's law is correct, is equal to.
Coulomb's law. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently great, the angles will be small enough to make the following approximation:. Using this approximation, the relationship 6 becomes the much simpler expression:. In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value. Coulomb's law holds even within atoms , correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons.
This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids. Generally, as the distance between ions increases, the force of attraction, and binding energy, approach zero and ionic bonding is less favorable.
As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable. May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated, that were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?
This series of experiments gave a result which deviated as little as the former or rather less from the inverse duplicate ratio of the distances; but the deviation was in defect as the other was in excess.
We therefore think that it may be concluded, that the action between two spheres is exactly in the inverse duplicate ratio of the distance of their centres, and that this difference between the observed attractions and repulsions is owing to some unperceived cause in the form of the experiment.
Therefore we may conclude, that the law of electric attraction and repulsion is similar to that of gravitation, and that each of those forces diminishes in the same proportion that the square of the distance between the particles increases. It follows therefore from these three tests, that the repulsive force that the two balls — [that were] electrified with the same kind of electricity — exert on each other, follows the inverse proportion of the square of the distance.
From Wikipedia, the free encyclopedia. Fundamental physical law of electromagnetism. Electrical network. Covariant formulation. Electromagnetic tensor stress—energy tensor.
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