Electromagnetism: Difference between revisions
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== Electric Fields == | == Electric Fields == | ||
An electric field is produced by any electric charge. If there is no charge, there is no field. In the real world, electric charges are all around us so there is always an electric field around us. If two electric charges would collide together or push away from each other with some force, the field tells us what force would happen if you placed a charge in a certain spot in space and time. That is why if there is a charge, there will be a field because any second charge you add will be attracted or pushed away by the first charge. The force experienced by any two charges can be expressed by k<sub>e</sub>*(q*Q)/(r<sup>2</sup>). We label charge as q or Q depending on which is bigger. This essentially says that the force between these two charges is proportional to both of the charges thus q*Q and the force gets smaller the farther away they get thus you divide by the distance r between them and because of the Newton’s third law pair, both charges feel an equal force but have to factor in both their distances which both equal r in this notation. Therefore, you divide by r<sup>2</sup>. This is only a rough approximation of the force experienced and will later be expressed in an integral to take care of what happens when charge is not uniform. | An electric field is produced by any electric charge. If there is no charge, there is no field. In the real world, electric charges are all around us so there is always an electric field around us. If two electric charges would collide together or push away from each other with some force, the field tells us what force would happen if you placed a charge in a certain spot in space and time. That is why if there is a charge, there will be a field because any second charge you add will be attracted or pushed away by the first charge. | ||
The force experienced by any two charges can be expressed by k<sub>e</sub>*(q*Q)/(r<sup>2</sup>). We label charge as q or Q depending on which is bigger. This essentially says that the force between these two charges is proportional to both of the charges thus q*Q and the force gets smaller the farther away they get thus you divide by the distance r between them and because of the Newton’s third law pair, both charges feel an equal force but have to factor in both their distances which both equal r in this notation. Therefore, you divide by r<sup>2</sup>. This is only a rough approximation of the force experienced and will later be expressed in an integral to take care of what happens when charge is not uniform. The relation of electric field to electric force is very tight. In fact, the electric field notated '''E''' is a way to measure what electric force would happen if there was a charge in that field. This means you can express the electric force as '''F<sub>e</sub>''' = q*'''E''' where q is the charge you place in the field '''E'''. This means you can also express the electric field '''E''' as '''E''' = (k<sub>e</sub>*Q)/r<sup>2</sup>. | |||
Since Q is not usually uniform, you can express q as a density function ρ of the position vector '''r'''. This can lead to the integral for '''E''' is '''E''' = ∫d'''E''' = ∫(dQ)/(r<sup>2</sup>)*'''r̂''' = ∫(ρ('''r'''))/(r<sup>2</sup>)*'''r̂'''dV, or using derrivitives, ∇ • E = ρ/ε<sub>0</sub>. Using these equations, you can calculate the entire vector field representing '''E'''. This equation will give you the '''E''' field centered at some point (x,y,z) that is some radius r away from dQ and has an '''r̂''' unit vector pointing in the direction of the radius r. You can use this to calculate various different points in space and get a 3-dimensional vector field. | |||
The electric field, mathematically, changes in time as described by dE/dt = ((- ∇ × B)/μ<sub>o</sub> - J) / ε<sub>0</sub>. | |||
== Magnetic Fields == | |||
A magnetic field is similar to an electric field with a few key differences. Just like charges produce electric fields, magnets produce magnetic fields. All magnets produce a magnetic field. This means that in the real world, we are living in a magnetic field. This field is notated '''B'''. Since it is impossible to have a north or south pole along, the equation ∇ • B = 0 holds, and can also be written as B = ∇ × A, as the divergence of the curl is 0. | |||
The magnetic field, mathematically, changes in time as described by dB/dt = - ∇ × E. | |||
[[Category:Physics]] | |||
Latest revision as of 21:59, 20 June 2025
Electromagnetism is the study of electric fields and magnetic fields and how they interact.
Electric Fields
An electric field is produced by any electric charge. If there is no charge, there is no field. In the real world, electric charges are all around us so there is always an electric field around us. If two electric charges would collide together or push away from each other with some force, the field tells us what force would happen if you placed a charge in a certain spot in space and time. That is why if there is a charge, there will be a field because any second charge you add will be attracted or pushed away by the first charge.
The force experienced by any two charges can be expressed by ke*(q*Q)/(r2). We label charge as q or Q depending on which is bigger. This essentially says that the force between these two charges is proportional to both of the charges thus q*Q and the force gets smaller the farther away they get thus you divide by the distance r between them and because of the Newton’s third law pair, both charges feel an equal force but have to factor in both their distances which both equal r in this notation. Therefore, you divide by r2. This is only a rough approximation of the force experienced and will later be expressed in an integral to take care of what happens when charge is not uniform. The relation of electric field to electric force is very tight. In fact, the electric field notated E is a way to measure what electric force would happen if there was a charge in that field. This means you can express the electric force as Fe = q*E where q is the charge you place in the field E. This means you can also express the electric field E as E = (ke*Q)/r2.
Since Q is not usually uniform, you can express q as a density function ρ of the position vector r. This can lead to the integral for E is E = ∫dE = ∫(dQ)/(r2)*r̂ = ∫(ρ(r))/(r2)*r̂dV, or using derrivitives, ∇ • E = ρ/ε0. Using these equations, you can calculate the entire vector field representing E. This equation will give you the E field centered at some point (x,y,z) that is some radius r away from dQ and has an r̂ unit vector pointing in the direction of the radius r. You can use this to calculate various different points in space and get a 3-dimensional vector field.
The electric field, mathematically, changes in time as described by dE/dt = ((- ∇ × B)/μo - J) / ε0.
Magnetic Fields
A magnetic field is similar to an electric field with a few key differences. Just like charges produce electric fields, magnets produce magnetic fields. All magnets produce a magnetic field. This means that in the real world, we are living in a magnetic field. This field is notated B. Since it is impossible to have a north or south pole along, the equation ∇ • B = 0 holds, and can also be written as B = ∇ × A, as the divergence of the curl is 0.
The magnetic field, mathematically, changes in time as described by dB/dt = - ∇ × E.