Electromagnetism: A Classical Approach

Electromagnetism A Classical Approach

Erik Halliwell

2017 Jasper Place Composite High School

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List of Topics Covered: 𝜀 Methods of Charging Objects 𝜀 Coulomb’s Law 𝜀 Electric Forces and Fields 𝜀 Parallel Plates 𝜀 Electric Potential (Voltage) 𝜀 Current and Charge 𝜀 Magnetic Forces and Fields 𝜀 Hand Rules for Magnetism 𝜀 Particle Motion in Magnetic Fields 𝜀 Motor and Generator Effects

This paper started as a study tool for my Physics 30 AP electromagnetism exam, but ended up morphing into a more lecture style format, which I will deliver some time in the near future. I have enjoyed this process immensely, and this has really created a love for attempting to explain my ideas through writing. Topics I am currently attempting to write about are mechanics, momentum and radioactivity, to name a few. I have always been interested in the field of chemistry, but lately I have become mesmerized with both theoretical and experimental physics. The Feynman Lectures on Physics has become a fixture in my hand, it has provided many explanations for phenomena that I have always thought were interesting, and has also introduced me to concepts that had never crossed my mind. I foresee many more hours spent writing papers like this - there’s so much to know, I wonder if I will ever be able to understand it all. Erik Halliwell February 22nd, 2017

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Methods of Charging Objects There are three ways an object can become charged; note that the ​law of conservation of charge​ is always upheld, which states that the net charge on an isolated system is conserved (physics principle 7). 1. Friction - When two objects are rubbed together, the one that has a looser hold on its electrons will lose electrons and become positively charged, the other object will become negative as it will have an excess of charge. 2. Conduction - When two objects are brought together, touched and then separated, a quantity of charge will be transferred in the form of electrons. If the objects are of similar size and shape, the charge transferred will result in the objects being approximately equally charged. If two spheres are not equal in size, the larger one will receive more charge. When the spheres are separated, the excess charges move to become equidistant from each other. 3. Induction - This process causes migration of charge in a neutral object by allowing polarized charges to escape the neutral object. There is no actual touching in induction. When a negatively charged rod is brought close to a neutral grounded object, the negative charges will move away from the rod and into the ground. When the object is un-grounded, per se, it will have a net positive charge as protons cannot move; they are fixed inside the nucleus of atoms. Coulomb’s Law The formula for electrostatic force, discovered by Charles Coulomb is

| | kq 1q 2 |F e| = r 2 (1) ●

Where |F e| is the electrostatic force ( N )

●

k represents Coulomb’s constant ( 8.99 × 109

●

q 1 is the source charge ( C )

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q 2 is the point charge ( C )

●

r is the separation distance ( m )

N·m 2 C2

)

Figure 1(a)

Figure 1(b)

Figure 1(a) shows the relationship between separation distance r and electrostatic force F e , it is an inverse square shaped hyperbola. Manipulating r by a factor of

1 would r2

produce a linear relationship.

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Figure 1(b) shows the relationship between charge q and electrostatic force F e . It needs no manipulation to produce a straight line, therefore F e is directly proportional to q . Electric Forces and Fields ● Michael Faraday developed the idea of “lines of force” to describe electrical fields ● A field diagram is shows a “sphere of influence” or a visual representation of “action at a distance forces” Electric Fields ● Shown in terms of positive charge only, meaning that the electric field ( E ) is in the direction of force ( F e ) on a positive test charge. ● ● ●

Field lines never cross in drawings, and their strength is shown by the density of arrows in a certain space, not the length. To visualize the field lines in a diagram, imagine placing a massless positive test charge in the figure - the path it then takes shows the force lines. Flux is the change in field strength, commonly used to describe magnetic fields.

The formula for electric field strength is:

E = Fe q (2) 2

Where the following are defined: ● E is the electric field strength ( N /C ) ●

F e is the force that acts on the particle due to the field in which it is placed

●

q 2 is the charge on the test object

Fields in Conductors When a conductor is in the shape of a hollow cylinder or sphere, charge concentrates at the outside surface in order for it to be as far apart as possible (like forces repel). This is why there is no excess charge on the inside of the conductor. We can say that the inside of the conductor is “shielded” from electric fields. Faraday Cages demonstrate this principle, the person inside the cage is not harmed by the strong current, provided he or she does not touch the cage! When a conductor or object is shaped non conventionally, charge tends to concentrate at the most pointed surfaces. Pointed regions in a convexly shaped object will have the density of charge. Conversely, charges will spread out on an irregularly shaped concave conducting object. Multiple Charges ● Field lines always begin at positive charges and end at negative charges. (see figure 2(b). ● The dashed lines shown in the figure below are referred to as equipotential lines, and are always drawn perpendicular to force lines. They show voltage distributions, (the further away from a charge the line is, the lower the force experienced). ● At the centermost point in figure 2(a), no field strength is found as the forces cancel out. 4

Figure 2(a)

Figure 2(b)

After substituting equation 1 into equation 2, we find that the q 2 ’s cancel, and we are left with the equation for field strength experienced by a point charge.

E =

kq 1q 2 r2

q2

→ |E| =

kq 1 (3) r2

Parallel Plates ● Produces a uniform electric field, field arrows should be drawn to show equal density at all points inside the plates. At the edge of the plates there is some warping of field strength, but this is not important to us at the moment. Change in Electric Potential (Voltage) Voltage is described as the change in electric potential per coulomb of charge: V = ΔE q (4) Where: V stands for the voltage ( V or J /C )

ΔE is the energy change ( J ) q is the charge ( C ) Electron volts (eV) can be used to measure energy, 1 eV is known as the energy gained or lost by an electron when it is accelerated across a parallel plate with 1 V of charge provided. Electric Potential Energy ● It takes work to move two like charges together, therefore this increases E p , which is later released as kinetic energy, as the force of repulsion drives the charges apart.

Figure 3.

●

Using the figure above, if a positive test charge was located a small distance d from the positive plate, its Ep initial would be very large, whereas its Ek initial is zero as it is not moving yet. After travelling to the opposite plate as a result of being both attracted and repulsed by the plates, 5

Ek final is now nearly equal to Ep final . It will be exactly equal once the test charge impacts the negative plate. This proves that the conservation of energy is upheld, none is destroyed - just converted from one form to another (physics principle 5). Merging Electric Field Strength and Parallel Plates A formula used to calculate the field strength inside parallel plates can be derived by combining two equations that we were introduced to in physics 20:

W = ΔE and W = F net · d (5,6) Rearranging equation 4 to solve for ΔE and substituting into equation 5 yields:

W = qV (7) Inside parallel plates, there are no other forces active besides the electric forces created by the plates, as particles and test charges have negligible mass, allowing us to ignore the force of gravity in our calculations. Thus, F net = F e , and substituting F e into equation 6 gives us:

W = qE · d (8) Since both equations 7 and 8 give an answer for work, we can set them equal to each other and solve for E , as this will give us the formula for electric field strength in parallel plates.

qV = qE · d (9) E = V (10) d

As we can see, the test charge q cancels, which shows that the field strength is not dependant whatsoever on the particle that is placed in the field. This is similar to the laws of orbital dynamics such that the mass of a satellite does not affect the forces it feels in orbit. Equation 10 has units of V /m or N /C , both can be used interchangeably depending on the question asked. Sometimes it might be advantageous to use one over the other. Current Electric current, measured in amperes, A (named after André-Marie Ampère, a founder of classical electromagnetics) is the amount of charge that flows through a wire in an arbitrarily chosen amount of time. I = qt (11) Quantization of Charge JJ Thompson found the charge to mass ratio for an electron, and discovered the particle. Robert Millikan determined the charge of an electron, giving it a quantized charge; a smallest discrete value in which every other value must be a perfect multiple. He determined this by observing tiny oil droplets moving between parallel plates, fiddling with the voltage until they were perfectly suspended, so the electric force pushing up directly opposed the force of gravity, pulling down. Millikan then found the 6

elementary charge: 1.60 × 10−19C . Every single charge we observe is a perfect multiple of this, no one has been able to find a standalone charge with a lower magnitude than that of the elementary charge. Magnetic Forces and Fields Faraday defined a magnetic field as a three-dimensional region of magnetic influence surrounding a magnet, in which other magnets are affected by magnetic forces. ● Measured in teslas ( T ) ●

Represented by the symbol ( B )

The Earth’s magnetic ends do not line up with the geographic poles, as this is why a compass points north (the Earth’s south magnetic pole is at geographic north). Magnetic fields should be drawn from the north pole into the south pole of a magnet, the density of the lines indicating the magnitude of the magnetic field.

Figure 4(a)

Figure 4(b)

Figure 4(a) shows the pattern of iron filings surrounding a bar magnet. Iron shows this because it has ferromagnetic​ properties, meaning that the iron filings have a permanent high susceptibility to magnetization. Figure 4(b) shows the magnetic field lines, representing direction and magnitude of the magnetic field can replace the iron filings. The number of field lines that exit the magnetic material is always equal to the number of field lines that enter the magnetic material, forming closed loops. Domain Theory and Magnetism There are four elements that are said to be ferromagnetic, as they can become permanent or temporary magnets. ● Iron ● Nickel ● Cobalt ● Gadolinium Magnets are actually made up of tiny regions known as domains, each which behaves like a tiny magnet. Domains which are not generally lined up represent an unmagnetized substance (Figure 5(a)), while a 7

substance with roughly parallel domains is considered magnetized (Figure 5(b)). The strength of the magnet is determined by the parallel-ness of its domains.

Figure 5(a)

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Figure 5(b)

The presence of a magnetic field can cause the domains in a substance to line up and become temporarily magnetized.

Hand Rules for Magnetic Fields Hans Christian Oersted noticed that the needle of a nearby compass deflected every time an electric circuit was switched on. This led him to conclude that there is a relationship between electricity and magnetism, namely being that electricity is the cause of magnetism. It was later shown that if the electric current was directed in a straight line, the magnetic field formed a circular pattern, which follows the first hand rule for magnetic fields.

Figure 6

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Figure 6 shows the direction of magnetic field (direction the fingers are coiled) when the flow of electrons is directed from west to east. For conventional current, use your right hand for all rules.

The first hand rule is applicable in situations involving two current carrying conductors in close proximity to each other. If the current is flowing in the same direction in each conductor, the conductors will attract as a result of opposite field directions, and if the current is flowing in opposite directions, the conductors will repel as a result of identical field directions. The force between the conductors is influenced by: ● the length of conducting wire ● the distance between the two wires ● the amount of current in each wire The second hand rule follows the principle that if an electric current is directed in a coil shape, the magnetic field will form a straight line inside the coil, (as shown below in figure 7). The induced poles are similar to those of a bar magnet. Magnets that use this principle are called electromagnets. Solenoids are an application of electromagnets, which operates mechanical devices. The strength of an electromagnet can be influenced by each of the following: ● increasing the current through the wire 8

● ● ●

increasing the number of loops in the coil increasing the size of the loops in the coil changing the core of the coil

Figure 7

The third hand rule, for motor effect or deflection offers a complete scope of all influences we’ve learned about so far. It states that when a negatively charged particle enters a region of uniform magnetic field strength, it is deflected in a direction perpendicular to both the initial direction of movement and external magnetic field (shown in figure 8).

Figure 8

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The thumb indicates the direction of initial negative charge movement. The extended fingers indicate the direction of the magnetic field, namely being from north to south poles. The palm faces in the direction of the magnetic force, which is the direction the particle will be deflected.

Now that we are working with influences in three different directions, a third dimensional diagram can be used, using the symbols in the figure below.

Figure 9

● ●

The dot symbol indicates a movement of charges or vector force out of the page The X symbol indicates a movement of charges or vector force into the page

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Particle Motion in Magnetic Fields There are three general types of motion induced by magnetic fields, each dependant on the angle in which a particle enters the magnetic field. 1. If the initial motion of a particle is parallel such that θ = 0 to the magnetic field which it travels through, there will be no force felt on the particle. 2. If the initial motion of a particle is directly perpendicular such that θ = π2 to the magnetic field which it travels through, the particle is deflected into a circular arc. In this case of circular motion, the centripetal force or net force is the magnetic force. This case is commonly found in mass spectrometers. 3. If the initial motion of a particle is at some angle which satisfies the inequality 0 < θ < π2 , then the particle will be deflected in a helical path. The phenomenon Aurora Borealis follows this idea, as particles released in solar flares spiral along Earth’s magnetic field lines towards the poles. As they reach the atmosphere, ionizing collisions with air molecules causing photos to be emitted, at wavelengths that are often in the visible light section of the electromagnetic spectrum.

Figure 10

Calculating Magnetic Forces The following ways to calculate magnetic forces follow the third hand rule, always check to make sure you’re calculating the magnetic force with the right direction component. The magnitude of the deflecting force felt on a charge can be calculated using the formula

F m = qv⊥|B| (12) Where the following are defined: ● q is the magnitude of the moving charge ●

v⊥ is the magnitude of the perpendicular velocity component

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|B| is the magnitude of the external magnetic field

When the path of the charge taken into the field is similar to the third type of motion listed above, the perpendicular component can be calculated using the formula v⊥ = v sinθ (13) To calculate the magnitude of the magnetic force for a length of current-carrying wire, use the formula: F m = I l⊥ ||B||

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Where the following are defined: ● I is the current ●

l⊥ is the length of the wire perpendicular to the magnetic field

●

B is the magnitude of the external magnetic field.

Qualitative Measurements of Current The main way to measure electric current is by using a device called a galvanometer, which turns electrical input energy into mechanical energy. In this sense, it works like a motor, although it is not really operating anything besides a little needle on a scale! When there is a current passing through the coil, the induced magnetic forces cause the needle to rotate - the greater the current, the greater the deflection of the needle. One is shown below in figure 11.

Figure 11

The Motor and Generator Effects There are two simple technological devices that can be used to understand the relationship between electromagnetism and energy: motors, which convert electrical or chemical energy into mechanical energy, and generators, which convert mechanical energy into electrical energy. A simple DC motor has three main components: ● the stator - a frame with a coil or permanent magnet to provide a magnetic field ● the armature - a rotating loop of conducting wire ● a commutator - a split metal ring, with two brushes attached to it which complete the electric circuit As electrons pass through the wire, they feel a magnetic force per the third hand rule. In figure 12, this results in a counterclockwise rotation of the armature. The rotation continues by the use of a commutator, which switches the direction of the electrons, keeping the armature moving in the same direction. Basic electric motors are just.. basic! That’s all there is to it.

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Figure 12

Generators operate by electromagnetic induction, meaning that relative motion between a conductor and a magnetic field will induce electrons to move in the conductor. Faraday and Henry were the first to demonstrate this effect. Galvanometers can operate through a generator effect, if the current produced is induced by a change in magnetic flux, or put simply: movement of a magnet in and out of a coil. This is shown below in figures 13(a) and 13(b). Faraday discovered this principle, and Lenz expanded on it, stating that the direction of the induced current in a loop of wire flows in such a way that the induced field opposes the change in magnetic flux. This is why it is harder to pull a magnet out of a coil than pushing into the coil - the induced current creates a magnetic field that tries to keep the magnet in the coil. If it was the other way around, energy would be created out of nothing, and this would violate the law of conservation of energy (physics principle 5). That sure would be a nice way to solve a world energy crisis, but alas - it’s simply not meant to be.

Figure 13(a)

Figure 13(b)

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