Part I

1.1 Electrostatic Phenomena

When we rub glass rods with silk, the glass rod become charged with negative charges. When we rub plastic rods with fur, the rods are charged with positive charges. After being rubbed with fur, plastic rods repel each other. However, if put the plastic rod close to a glass rod, they will attract each other. This kind of interaction between charges is called electrostatic force.

1.2 Electroscope

The electroscope is a tool to measure electrostatic force. It works by electrostatic attraction or repulsion and consists of two metallic leaves suspended from a metal hook. The leaves are inside a glass container to protect it from air currents. The leaves are connected to a metal ball which is on the top of the container. When a charged object is brought near the ball, the charges on the object cause the same charges move away toward the end of the leaves. The leaves are then spread apart due to the repulsion of same charges.

The electroscope can also quantify the amount of charge of an object. A larger charge causes the leaves to spread further apart. Electroscope can show the type of charges of two object. One can first bring an object in touch with the electroscope. This will cause the foil leaves to separate. Then, bringing the second object can be brought next to the electroscope’s post. If the two objects have same type of charges, the second contact will cause a further separation. On the contrary, if the leaves are brought closer than the two objects have different types of charge.

1.3 Benjamin Franklin’s idea

Charge produced on a rubber rod when rubbed by fur and charge produced on a glass rod when rubbed with silk have been found having different properties for a long time. They didn’t gain their names until Benjamin Franklin (1706-1790) introduced the concept “positive” and “negative” around 1750. He noticed stable fluid is present in all objects. During rubbing, they carry similar or different properties as some fluid is transferred from one object to another. Thus, he proposed that the charge on a glass rod when rubbed with silk be called positive. This concept is revolutionary, and Franklin’s model also approaches the reality more, which has electrons being transferred instead of “fluid”.

1.4 Conductors and insulators 

Conductors are materials that permit electrons to move easily from one region of the material to another. For example, electrons can move freely within a copper wire. Most metals are good conductors, while most nonmetals are insulators. Insulators can prevent charge from leaving metal wires. Most metals are conductors, while most nonmetals are insulators. Within a solid metal, one or more electrons in each atom become detached and can move freely throughout the material. In an insulator, electrons cannot move freely through the material. 

1.5 Coulomb force

Coulomb’s law is used to measure the electrostatic force. It is developed by the French scientist Charles Coulomb (1736-1806). 

The force F can be expressed as:

Were q1 and q2 indicates the charges respectively, r is the distance between the charges, and k is a constant. When the charges have the same sign, the forces are repulsive; when they have opposite signs, the forces are attractive. 

In order to measure the weak forces between two charges, Coulomb invented a torsion balance. An insulating rod is suspended at the middle of an enclosure. Two small metal balls are attached at each side of the rod. Both balls are charged. Another ball with the same charges is put close at one ball. It then causes the rod to twist. If the torque needed to create the angle of the twist is measured, the forces produced by two same charges can be measured. Besides, in order to give two balls the same charges, Coulomb placed one charged ball into contact with an uncharged ball, thus divided the charges on one ball into two equal parts. By repeating this process, he can divide charges to one-half, one-quarter and so on.

The electrostatic force has a similar functional form than the gravitational force, we can see they all have inverse-square dependence from the distance:

However, the gravitational force is always attractive. While the electric force depends on charges and can be either attractive or repulsive. For charged particles, the electrostatic force is the only force that is responsible to their interactions.

1.6 Electric field

Two positively charged objects at a distance exert a repulsion force to each other. If we place a negatively charged object instead of one positive object at the same place, the force, although still have the same magnitude, becomes repulsive. Therefore, the electrostatic force between two object depends on each object’s charges. We can define the Electrostatic field as a property generated by one charge only. This can be defined by dividing the Force F generated in between two charges by one of the charges as follows.

The electric field at some point is defined as the electric force F (experienced by a test charge q) divided by the charge q. In other words. The electric field equals to the electric force per unit charge. The object’s electric field is always present and its magnitude can vary by changing the distance from that object. If another charge q enters the field, the force it experiences is exerted by the electric filed of the original object. 

1.7 Electric Field Line

One can define electric field lines to visualize the direction of a given force or a given electric field. The electric field for a positive charge always points away from it, outward, whereas the electric field of a negative charge point towards it, or inwards.

The field direction of a single charge is the direction of the force a positive net charge experienced near the charge. We can find that the test charge is repelled by the original charge when placing near it. The test charge is attracted by the original charge. 

When placing two opposite charges together, they attract each other, so the electric field point toward the negative charge. On the left side of the negative charge, the field line is same to single negative charge’s, and on the right of the positive charge, the field line is also same to that of a single positive charge.

1.8 Electric Potential Energy

The electrostatic force is a conservative force, which means that we can define an electrostatic potential energy. This potential energy leads to the related concept of electric potential. As the change of kinetic energy equals the total work done on the object, if there is only electrostatic force act on the object, the work done by the electrostatic force equals the change in potential energy.

In a uniform electric field as shown above, , If a positive charge moves downward (in the direction of E), the work done by the electric force is positive, so U decreases.

If the charge moves in the opposite direction of E, the work done by the electric field is negative. In other words, an external force is done on the charge, causing U increases. For positive charges, moving along a positive electric field decreases their electric potential energy, while for negative charges, moving along the positive electric field increases U.

Since we usually define U=0 when r is infinite, when moving a particle from infinite far to r. Therefore,

1.9 Electric potential

Electric potential is related to electrostatic potential energy in much the same way as electric field is related to the electrostatic force. The change in electric potential is equal to the change in electrostatic potential energy per unit of positive charge.

Its unit is V or J/C. Electric potential is also called potential energy per unit charge. Like electric field, V exist at a point even if there is no charge.

Part 2

2.1 Electric Circuits 

An electric circuit is defined as a closed conducting path that allows the electric charges in the wire to move from one region to another. Within a conductor, when there is no external electric field, the charges move randomly. Instead, if there is an electric field E is established inside the conductor, charges will experience a force and start to move in one direction. The charges also experience collisions with the metal particles when they pass through conductors. In other words, the kinetic energy supplied by the electric field is dissipated to heat the conductor, which keeps the charges from moving faster.

Electric circuits must be a closed circuit instead of an open circuit or a conductor. In an open circuit, an electric field E₁ will let positive and negative charges accumulate respectively at two sides of the wire. These charges will then create an opposite electric field E₂ , which will offset E₁ and decrease the total electric field to zero.

As a result, we need a device, often called a battery, that act as an external force that can is in opposite direction to that result from E₁ and carry the positive charges from lower potential to higher potential.

Therefore, a complete circuit must consist of a wire, a conductor (or resistance), and a battery.

2.2 Electric current

In a conductor, if the charges experience an electric field, they will start moving in a direction. 

The net charge flowing through the circuit per unit time, or the rate of flow of electric charge is defined as current:

The standard unit of current is ampere, defined as 1 coulomb per second (1A=1C/s).

In different current-carrying materials, either positive or negative charges can be the moving charges. In figure 2.2, positive charges move to right, in the direction of the electric field, while negative charges move to the left, in the opposite direction. For negative charges that moves to the left, they create the same effect as the first case as the net charges on the right increases. In both case, positive charges flow to right, so we define current to be in the direction of the flow of positive charges. Current in this definition is called conventional current.

2.3 An analogy to the flow of water

In a circuit, the charges move through an external conducting path due to the electric field caused by the potential difference at the two sides of the battery. As the positive charges moves from high potential to low potential, the potential of the charges is thus decreasing. In order to maintain the current, as charges pass the battery, the battery provides a force to lift positive charges from low potential to high potential. The work done by the battery increases the potential energy of the charges.

This is analogous to water flowing in a pipe vertically, as shown in Figure 2.3. Water flows down the pipe from high gravitational potential energy to low potential energy. Then, a pump lifts the water back to its original height and increases its potential energy. While batteries provide necessary potential energy for the charges, the pump guarantees the water to flow.

2.4 Electromotive Force 

Electromotive force (emf) is the potential difference, or voltage, of a battery. It is actually not a force. Its equal to the work the battery does on every coulomb of charge that passes through it. It is, in other terms, a potential energy difference, or voltage,  ϵ or V is the symbol for emf. The unit of emf is volt (1V=1J/C)

2.5 Ohm’s Law and Resistance 

Ohm’s Law shows the proportional relationship between the current and voltage. It also shows the current not only depends to the voltage, but also on the resistance. 

The relationship was discovered experimentally by the German physicist Georg Ohm (1789-1854). In symbols:

A quantitative definition of resistance R is:

While the definition of R is:

The unit of resistance is ohm (1Ω=1V/A)

The longer the wire, the greater the resistance; the thicker the wire, the smaller the resistance. Besides, the resistance also depends on the electrical conductivity of the material. Compared with resistance of water’s flow rate in a pipe, a narrow water pipe offers more resistance; the larger cross-sectional area increases flow rate. If the pipe is stuffed with cotton, the resistance also increases. 

2.6 Series and parallel circuits

In a series circuit, the circuit elements are connected in sequence, lining up on a one after the other. In series, the current I remains the same through the path. 

The total resistance of the combination R_s is:

For each resistor, since they all have the same current, we can know their voltage respectively:

The resistance across two sides of the battery is:

Since the total resistance of a series combination is greater than any individual resistance, so connecting a bulb is series will make it glow less brightly.

2.6.2 Parallel circuits

In parallel circuits, each resistors provides an alternative path for the current. The potential difference is the same across each element.

The total current equals to the sum of the three currents in the resistors:

The current in each resistor is:

So the total resistance R is:

The total resistance of a parallel combination is less than any individual resistance. As the current in a bulb connected in parallel is independent to the adding of other resistors, connecting a bulb in parallel doesn’t change its brightness.

2.7 Electric Energy and Power

The negative and positive charges in a battery are kept at the opposite poles of the battery thanks to the chemical reactions happening inside the battery. This chemical reaction are generate by a chemical energy which keeps the positive and negative charges separated on the pole. This latter chemical energy corresponds to the electrical energy which is created within the battery and can be used through the electric circuit to move the charges from one point to another. As the charges pass resistors, the potential energy is dissipated as the charges collide with atoms in the resistors. 

The rate of transferring energy from the battery is called power:

The unit of power is watt: 1W=1J/C. The power delivered to any resistor is:

Where V_r  is the voltage across the resistor and I is the current in the resistor.

Part 3

3.1 What are magnetic poles?

Of a bar magnet, one end is called a north pole or N pole, and the other end is called a south pole or S pole. Opposite poles attract each other, and like poles repel each other. This is similar to electric interactions. We can also describe the interactions between two magnets as one sets up a magnetic field around it, and the other magnet moves in response to the magnetic field.

3.2 Is the Earth a magnet?

In early years, people sail with a compass which always points to the north. The Earth’s north is defined by the direction of the compass’s north points. As a result, Earth’s geographic north pole is like a magnet’s south pole, which attracts the compass’s north pole. The earth’s magnetic axis, though, is not parallel to its rotation axis, so the compass’s reading derivates a little from the true north. 

3.3 Magnetic field lines

While magnetic poles may be similar to electric charges, unlike isolated positive and negative charges, magnetic poles always exist in pairs. When a magnet is broken to two, each end becomes a new pole. The magnetic poles in pair are called a magnetic dipole, and an isolated magnetic pole is called magnetic monopole, which is not yet discovered.

Magnetic field lines emerge from the north pole and go into the south pole. Different from electric field lines, magnetic field lines do not end at the south pole, instead, they form a circle back to the north pole.

If we put a magnetic dipole in a magnetic field, the magnetic dipole tends to align with the magnetic field at its position due to magnetic attraction forces and repulsion forces.

3.4 An unexpected effect

The first evidence of the relationship of magnetism and electric current was discovered by Danish scientist Hans Christian Oersted. When he was giving a lecture, he surprisingly found a compass needle was deflected by a current-carrying wire. When the wire carries current, a compass placed directly over the horizontal wire deflects. As shown in figure 4, when the current is towards north, the needle deflects towards northeast. 

The electric and magnetic interactions in this case around a straight, current-carrying wire forms circular magnetic fields. The magnetic field’s direction is determined by a right-hand rule. Point the thumb of your right hand in the direction of the current. Your fingers now curling around the wire in the direction of magnetic field lines.

3.5 The magnetic force on a current-carrying wire

Since there are magnetic fields near a current, will this magnetic field exert a force to another current-carrying wire?

Ampere discovered that there is a force exerted to another current-carrying wire due to the magnetic field around the original wire. Ampere first ensured the force are not mainly caused by electrostatic effects. Then, he found out the magnetic force between two wires is proportional to the currents and inversely proportional to the distance r between the two wires.

This relationship can be expressed as:

where constant k’  is equal to   , I1 and and I2 are the current flowing through each of the wire, respectively, and l is the length of the wire.   is the force per unit length of the wire. The longer the wires, the greater the force. The force between the wires is attractive when the currents are flowing in the same direction and repulsive when the currents are flowing in the opposite directions.

3.6 The magnetic force on moving charges

By experiments, we know that the magnetic force on a moving charge is proportional to the magnitudes of the charge. When its charge doubles, the magnetic force doubles. The force is also proportional to the magnetic field strength. If the magnetic field is stronger while the charge remains the same, the force is stronger. If we have a moving charge q which moves through a magnetic field B then the magnetic force also depends on the velocity of the charge. If the charge is at rest, it experiences no magnetic forces. At last, the force is only related to the component of velocity   that is perpendicular to the field. When is parallel to the magnetic field, the force is zero.

The magnitude of magnetic force is given by: 

Sinceand are vectors, the direction of their product can be determined by a right-hand rule. First, point the index finger of your right hand in the direction of the velocity of the positive charge. Then, point the middle finger in the direction of the magnetic field. The direction of the thumb is then the direction of the magnetic force. The force on a negative charge is in the opposite direction of a positive charge moving in the same direction.

About the author

Zixuan Wang

Zixuan is currently an 11th grade student at the Shanghai Jianping High School.

The post The Differentiating Impacts of Electricity and Electromagnetism on Force and Magnetic Fields. appeared first on Exploratio Journal.

1.Introduction

Lightning rods, mostly made of copper, is a structure that protects buildings from being damaged by attracting flashes through electric-magnetic force and guide the current to the ground. After learning a bit about electricity and experiencing a night of thunder and lightning, I intend to explore how lightning rods work. Therefore, in this presentation, I will first introduce the historical research on lightning rods, and then explain how lightning rods work in general using electrostatic principles and some easy-to-understand analogies. I will then write a program to calculate the effective range of the lightning rod based on the Monte Carlo technique and finally propose a lightning protection solution in conjunction with the 3D street view.

2. The History of Lightning Rod

2.1 Franklin Kite Experiment

In 1746, Franklin turned his home into an electrical laboratory after occasionally discovering the electrical experiments of other scientists, and in a letter he described receiving an electric shock as “a numbing sensation from the beginning to the end”.

In 1747, thanks to Franklin’s discoveries, people stopped using glassy and resinous to describe electricity. They began to use positive and negative electricity.

In 1749, Franklin began to make analogies between lightning and batteries, and from then on lightning became palpable. He explained by analogy the bifurcation in lightning, the color of the lightning, and the deafening sound, and was determined to prove that lightning and electricity were directly related. in 1750, he began to focus his research on the protective devices for lightning. This was man’s first step toward the lightning rod

Fifteen years later, Franklin’s close friend recorded in his diary Franklin’s famous kite experiment. He took the risk of using a kite to try to get up close and personal with lightning. He even tied a key to the kite in order to attract an electrical charge. Even though the string of the kite was already made of insulating silk, this was still a very risky act considering the strength of the lightning bolt could even make the insulator relatively conductive. The conclusion of this experiment was that the key was seen to receive the electric charge brought by the lightning, and Franklin thus proved that lightning is electricity.

2.2 Tip Lightning Rod or Round-end Ones?

Almost simultaneously with the kite experiment, Franklin realized the fact that iron needles can conduct electricity, and tried to integrate this into the “lightning rod” invention. In his diary, he envisioned, “Could there be a way to protect people from sudden lightning strikes by inserting thin needles directly into clouds and pulling the electricity out of them before the lightning strikes the ground?”

Franklin focused on elevating the tip of the lightning rod, while Benjamin Wilson, a member of the Royal Court circle of George III, believed that the pointed lightning rod would attract lightning (and this property remained unchanged and became the main principle of the modern lightning rod) and was not as safe as the round-headed lightning rod. Most scholars at the time also supported Benjamin Wilson’s view, so much so that this eventually turned into a political showdown, with proponents of Franklin’s lightning rod being falsely accused of “trying to establish their own political group in England. The war between science and politics officially ended when an East India Company was struck by lightning, and Franklin’s spiked design is still used today.

2.3 Three primary Modern lightning rods

2.3.1 Early Streamline Emmision (ESE)

ESE systems are more similar to conventional lightning rods. They are designed to trigger early initiation of upward flow, which increases the effective protection range. This discharge trigger increases the probability of triggering a “streamline” discharge at or near the tip of the rod as the ionized “leader” approaches. 

2.3.2 Charge Transfer System (CTS)

The CTS is characterized by its designated protection zone. It is the only system that deters lightning strikes, rather than encouraging them. CTS technology is based on existing physical and mathematical principles. The CTS collects the induced charge from the thunderstorm clouds in the area and transfers it to the surrounding air via an ionizer, thereby reducing the electric field strength in the protected area.  The resulting reduction in the potential difference between the site and the clouds inhibits the formation of upward currents and thus reduces electric shocks.

2.3.3 Dissipation Array System (DAS)

DAS is a special type of CTS. Based on the “protected area” of CTS, DAS can completely isolate a facility from direct lightning strikes during a thunderstorm by releasing the induced charge within the protected area to 55% of its pre-installation level in relation to its surroundings. When the electric field is reduced, the upward current does not get enough energy, and it is the connection of the upward and downward currents that is required for lightning to occur. Without energy, the connection cannot be made, so lightning cannot be generated.

3. The working principle of the lightning rod

3.1 Electron distribution

3.1.1 Electrons in the earth’s crust

Before asking how lightning rods come into use, let’s first examine the function of electrons that makes lightning occurs. Before we go into how lightning rods work, let’s take a look at how electrons work to cause lightning. To begin with, the ground’s surface is made up of positive charges because the dipole cloud produces an electric field that forces electrons to flow to the earth’s core. The earth’s crust is plainly devoid of negatively charged electrons, resulting in a positively charged ground. Colors have been employed to depict the phase cancellation process, with yellow indicating negative charges and blue representing positive charges. The green hue created by combining blue and yellow is neutral, but the absence of either color gives it a bluish/yellowish appearance.

The positive charge upon the ground produces an electric field between the earth and the clouds, resulting in a negative charge covering the bottom of the clouds. And this electric field can reach 400,000 volts, creating a powerful electric field that lingers in the atmosphere. The procedure of positive and negative charge exchange in the clouds is essentially like an ion engine that repels the negative charge of the entire planet to the opposite side, according to the principle that different charges attract and the same charges repel.

We all know that when the electric field’s dipole reaches a particular level, clouds unleash lightning, which neutralizes the charge at the cloud’s bottom compared to the ground, and then the clouds repeat the process to rebuild the potential difference in the form of an exponential equation. Here’s a visual representation of this.

The time it takes to recharge is known as the resetting time, and we use it to determine the power of the ion pump described before, for which we have data of around 5 seconds. I=Q/T. When the experiment is replaced, the result is a cloud with a charge of -20C and a resetting time of 5 seconds.

Despite the fact that 4 amps may appear to be a little quantity, comparable to double the current of a mobile phone charger (2A) or the current used in street lights (4A), it will inflict a great deal of damage due to the fact that it is released from a very small hole.

3.1.2 Electrons in clouds

Clouds that carries lightning consists soft hail particles and ice particles. Soft hails has more weight than ice particles, therefore they fall to the bottom during a thunderstorm while the small crystals were uplifted to the top. This falling process allowed negatively charged hails stay at the bottom(6-8km) and positively charged ice floats to the upper part of the cloud to 10km.

3.1.3 Generation of lightning

There are three major hypothesis about how lightning comes into place. 

• The electric field inside a stormy cloud is far higher that what has been calibrated. 

• Lightning is created via hydrometeors, which means water particles in the cloud
• Energetic runaway electrons initiate the lightning.

 Because the overactive electrons (hypothesis 3) in the above figure are hydrometeors (hypothesis 2), ice crystals and water droplets traveling through the cloud, and the situation presented by hypothesis 2 usually boosts the electric field strength by a large margin, due to the equation of Coulomb’s law, these three points are actually interconnected.

It can be found that the electric field strength is inversely proportional to the square of the distance. Thus overactive electrons are in between many electrons of different charges, causing a huge electric field and thus the birth of lightning.

Most lightning is intra-cloud lightning, while lightning that occurs outside of clouds is divided into four main types, two from the ground to the thunderclouds, which are beyond the scope of this report, and two from the thunderclouds to the ground. One of them is downward lightning negatively-charged leader caused by a negative charge at the bottom of the cloud and a positive charge for activation, and the other is downward lightning positively-charged leader caused by a negative charge leading from the positive charge at the top of the cloud connected to the ground charged leader.

3.2 Lightning propagation methods

The negative step leader, as its name suggests, will extend the length of the leader channel by step propagation. In the study of lightning pathways, early studies based on photography were skewed because some of the steps were too tiny to be seen with the human eye. The multiple-station dE/dt technique was utilized by J. Howard, M.A. Uman, C. Biagi, D. Hill, V.A. Rakov, and D.M. Jordan in 2011 to localize each step. When Step brings the lightning to the ground, the length charge is around 10^-3C/m, and the earth sends a return stroke to contact with it, resulting in lightning. TThis generally happens in the conductor nearest to the elevation, since lightning, like an item in an automated pathfinding system, will want to walk on the side with the least “resistance,” that is, the side with the quickest potential reduction movement. Because the charges at the ground and at the bottom of the cloud are generally different, the ground attracts the leader in this scenario.

3.3 How do conductors work?

 A conductor is a substance that electrons are relatively free to move compared to the insulators, which is the property needed to create the lightning rod–any net charge resides on the surface because ρ=0 inside a conductor according to Gauss’s law. So that negative charges are attaching on the surface of the lightning rod, making it easy to be strike. The reason is, usually a ground is conductive and there are negative charges throughout the ground. When you put a conductor such as a metal on the ground, the electrons of the ground moved to the metal, and the protons in the metal moved to the ground until the metal and the ground are equipotential and the metal and the ground can be regard as a system because of the formula E = -⊽ V , Where E represents the electric field and it equals the negative product of divergence of electric potential. When the system is a closed loop the divergence is 0 so that there is no electric field and electrons are static again. The metal can then be seen as a whole with the ground. This reasoning is also valid for conductive buildings and lightning rods, which become more vulnerable to lightning strikes as if they were a mountain range raised on the natural landscape.

What will the lightning rods do to the lightning that it had intercepted? When lightning occurs, the lightning rod can attract the discharge channel of lightning, so that the lightning current flows from the lightning rod into the earth’s land, avoiding huge currents to cause damage to buildings, equipment, trees or injury to people or animals that happen to walk above the ground.

4. The effective area of lightning rods.

4.1 Monte Carlo Technique

Abhay Srivastava calculated the protection of the lightning rods by applying a mathematic model conducting rod using Monte Carlo technique. It is a computer simulated model that randomized the distribution of lightning strokes. It assumes a concave lateral surface of the cone, using the concept of striking distance  in Golde’s formula, where d_s means the striking distance, A=10 and σ = .65  are constants.

 As the graph suggests, it initialized the sky with height k, the starting point of the lightning as h0, which will randomly stepping down towards the ground until it sees a postive charged object in its detecting sphere, which is assigned H_n. H_n-1 and H_n-2 are the last two steps from the striking point, and each point in this graph is given a three dimension coordinate.

The author assumes that the cube is 100*100*1000, and that a Cumulonimbus Cloud capable of storing lightning has a height of 500-16,000 meters.

The lightning will begin at a random location at the maximum height,1000m.

int[][] origin=new int[Math.random()*101][Math.random()*101];

In 5% to 80% range of the striking distance it generate some random variables to select the step length of leader.。Then it designate two angels  of the spherical polar coordinates: the inclination angle α lying between π/2  and 3π/2 and the azimuthal angle β lying between 0 and 2π. 

The mathematic formula is as follow. Iteration in Java should be used to infer where the lightning will strike. The loop terminates when the program determines that the lightning leader has reached the monitoring range, which is simulated as a sphere of radius 20m.

/*

*Precondition: it checks every step of the lightening

*Postcondition: true means that the lightning has successfully been 
*intercepted, and therefore will not be accounted as lightenings that have 

*caused damage.

*/

Public boolean inRegion(Object leader,int r){

if(Math.sqrt(Math.pow(leader.getx()-rod.getx(),2)+Math.pow(leader.gety()-rod.gety(),2)+Math.pow(leader.getz()-rod.getz(),2)<=r)}

//Euclidean distance

return true;

return false;

}

Then comes the main program for generating the path, written according to the following mathematical equation.

public int[][] stepProcess(int[][] before,Object leader,int r){

int[][] after=int[][] before;

I=leader.getx();

j=leader.getz();

K=leader.gety();

while(k!=0&&!inRegion(leader,r)){

i=i+r*Math.sin( )*Math.cos(β);

j=j+r*Math.sin( )*Math.sin(β);

k=k+r*Math.sin( );

}

if(inRegion(leader,r)){

return before;

}

else{

after[i][j]=after[i][j]+1;

return after;

}

//the higher the number in the array is, the more dangerous is the area

}

After running the program, I tested 1000 times, assuming 10 strikes per year in this area, which would be all the lightning this virtual area has suffered in 100 years, and drew a graph of the conclusions drawn, where the yellow area represents relative safety(have been struck once), green represents absolute safety, and red represents danger(more than once).

 4.2 Real Life Application

In the 3D street view of Gaudet Map, I intercepted a dense map of high-rise buildings of about 1000*1000 and used the model for the simulation of lightning rod placement. It is assumed that all the buildings need protection, but we can ignore the open space. Here are the before-after graph of the map. When lightning rods were applied in that area, it meant to make sure every building to stay in the green or yellow circle of fig 12/13, which take the height of the lightning rods, r, as a variable and execute the program.

5. Reference List

1.“Modern Lightning Protection Lightning Rods with Lightning Eliminators.” Edited by LEC By admin, LEC, 19 Sept. 2018, www.lightningprotection.com/lightning-rods-are-old-new-lightning-protection-part-3/. 

2. J. Howard, M.A. Uman, C. Biagi, D. Hill, V.A. Rakov, D.M. Jordan, Measured close lightning leader-step electric-field-derivative waveforms, J. Geophys. 

Res. 116 (2011) http://dx.doi.org/10.1029//2010JD015249. 

3. E.P. Krider, C.D. Weidman, R.C. Noggle, The electric field produced by lightning stepped leaders, J. Geophys. Res. 82 (1977) 951–960.

4. Srivastava, Abhay, and Mrinal Mishra. “Lightning Modeling And Protection Zone Of Conducting Rod Using Monte Carlo Technique – ScienceDirect.” Lightning Modeling And Protection Zone Of Conducting Rod Using Monte Carlo Technique – ScienceDirect, Www.sciencedirect.com, 13 June. 2013, https://www.sciencedirect.com/science/article/pii/S0307904X13003478?via%3Dihub.

“Franklin’s Lightning Rod | The Franklin Institute.” The Franklin Institute, Www.fi.edu, 8 March. 2014, https://www.fi.edu/history-resources/franklins-lightning-rod.

5. Godwin, Ian . “Franklin Letter To King Fans Flames Of Lightning Debate › News In Science (ABC Science).” Franklin Letter To King Fans Flames Of Lightning Debate › News In Science (ABC Science), Www.abc.net.au, 26 March. 2003, https://www.abc.net.au/science/articles/2003/03/26/816484.htm.

6. M. Vargas, H. Torres. On the development of a lightning leader model for tortuous or branched channels – Part II: model description

7. J. Electrostat., 66 (2008), pp. 489-495

8. M.A. Uman, The Lightning Discharge, Academic Press, London, 1987, 376 pages, revised paperback edition, Dover, New York, 2001. 

9. K. Berger, Blitzstrom-Parameter von Aufwärtsblitzen, Bull. Schweiz. Elektrotech. Ver. 69 (1978) 353–360.

About the author

Douyun (Winnie) Shi

Winnie is a Physics learner at the Starriver Bilingual School in Shanghai, China.

The post How do lightning rods work? appeared first on Exploratio Journal.