fields

Physics 40S

Unit 2 - Fields

Kepler’s Laws

• Johannes Kepler (1571-1630) was born in southwest Germany. He had been greatly influenced by the work of the Greeks and spent a great deal of time trying to devise systems with which to explain the motion of the planets about the sun.

Topic 2.1 – Exploration of Space

Kepler’s Laws


Kepler’s 1st Law

• Using Tycho's observations of the planet Mars, Kepler found that its motion fitted an elliptical orbit with a high degree of accuracy. He later found that the orbits of the other known planets could be drawn as ellipses, with the sun at one of the foci.

• The motion of the planets in ellipses became known as Kepler's First Law.


Kepler’s 1st Law

• An exaggerated diagram of an ellipse and two foci is shown below. The sun is at one of the foci, F1.


Kepler’s 1st Law• An ellipse can be defined as a closed curve such that

the sum of the distances from any point P on the curve to two fixed points (called the foci F1 and F2) remains constant. That is, the sum of the distances F1P + F2P is the same for all points on the curve. A circle is a special case of an ellipse in which the two foci coincide at the centre of the circle.

• We can state Kepler's First Law as follows. The planets move about the sun in elliptical orbits, with the sun at one focus of the ellipses.


Kepler’s 2nd Law• Kepler's Second Law further describes the motion of a planet

around the sun. This law examined the areas swept out by the planets as they moved around the sun.

• The diagram shows two equal time intervals Δt1,2 and Δt3,4. For these time intervals, the areas swept out by the line joining the sun and the planet are equal, that is area A = area B. This means that the closer a planet is to the sun, the faster it will move, and it will move in a predictable way.


Kepler’s 2nd Law

• Kepler's ellipses put an end to Greek astronomy. The long established concept of celestial spheres and perfectly circular motion had been destroyed. Kepler's scheme of the solar system has been followed by astronomers and scientists ever since, without significant change.

• Kepler's Second Law can be stated as follows.

The straight line joining the sun and a given planet sweeps out equal areas in equal intervals of time.


Kepler’s 3rd Law

• In 1619, Kepler published a second book. It contained what is now known as Kepler's Third Law. It stated that the square of the orbital period of a planet is proportional to the cube of its mean distance from the sun. It can be expressed mathematically as

• where R is the mean radius of the orbit of the planet, T is the period of revolution about the sun, and K is a constant, usually referred to as Kepler's constant.


Kepler’s 3rd Law• Kepler's constant is K = 3.35 x 1018 m3/s2. By using

Kepler's Third Law, it is possible to determine the period of revolution, or the mean radius of orbit for any object in orbit around the sun.

• Kepler's constant can also be determined for objects in orbit around other central bodies. So for example, the motion of the moon around Earth could be used to determine K for our earth. Once we know what the value of K is, then we can also determine the period of revolution or the mean radius of orbit for that object.


Kepler’s Law Experiment

Universal Law of Gravitation• Newton graduated from Cambridge University in 1665 after

majoring in mathematics. In the few years after this, Newton did some of his best thinking. Legend has it that while watching an apple fall, Newton began to wonder if the force that accelerated the fall of the apple might also be the one responsible for maintaining the path of the moon's orbit. The moon is moving around Earth in a circular orbit and thus has centripetal acceleration. This acceleration is caused by the gravitational pull between the moon and Earth. The amount of this acceleration due to gravity was proportional to the gravitational force between Earth and the moon. By using his knowledge of circular motion and Kepler's Third Law, Newton deduced that the gravitational force was inversely proportional to the square of the distance from Earth to the moon.


Universal Law of Gravitation



• Newton's reasoning was similar to the following. From our work on circular motion, we know that the centripetal force required to keep an object moving in a circle is

• From Kepler's Law and



• By substituting this value for T2 into the equation for force, we obtain

• Since the expression (4π2K) is a constant, the force of gravitational attraction on a planet is directly proportional to the mass of the planet and inversely proportional to the square of its distance from the sun.



• We saw in the previous lesson that Kepler's constant is the same for any object circling the sun, but it would have a different and smaller value for a satellite of Earth. Newton reasoned that this is because the mass of Earth is smaller than the mass of the sun. In other words, Kepler's constant is proportional to the mass of the object exerting the force of attraction at the centre of the orbit. So the expression 4π2K is proportional to mc where mc refers

to the mass of the central body (like the sun or Earth).


Universal Law of Gravitation• We can thus rewrite this expression as 4π2K = Gmc. The value

G is referred to as a universal gravitational constant. Henry Cavendish first experimentally measured it in 1798. Newton's remarkable extension was to say that if this equation for force applies to the sun and its planets, and to Earth and its moon, then it should apply to any body in the universe that has mass. We can thus summarize Newton's Law of Universal Gravitation as

• Any two bodies attract each other with forces proportional to the mass of each and inversely proportional to the square of the distance between them. where G is the Universal Gravitational Constant: G = 6.67 x 10-11 N·m2/kg2


Gravitational Potential Energy

• How do you calculate the gravitational potential energy between 2 masses?

• What is a potential well?

• What is the escape velocity? How is it calculated?

• What is binding energy? How does the binding energy compare in magnitude to the PEg?



• How do you calculate the gravitational potential energy between 2 masses?

• To increase the separation of the two masses from R1

to R2 requires work to be done to overcome the force

of attraction, similar to stretching a spring. When this work is done, the gravitational potential energy increases. To derive a mathematical equation for gravitational potential energy, we need to remember the relationship between gravitational force and separation.



• A force versus separation graph

for this situation is like the one shown.

• From our work on work and energy, we learned that the work done by a varying force is simply the area under a force-displacement graph for the interval. To find the area under an inverse square curve requires doing an integration using calculus.



• The resulting change in

potential energy is• When written in this way, the first term in the

expression depends on R2 and the second term on R1.

Thus, each term is an expression for the gravitational potential energy at that separation. Therefore, at any separation distance R, the gravitational potential energy, PEg, between two masses M1 and M2 is given

by




• The equation always produces a negative value. As R increases, that is, as the masses get farther apart, PEg increases by becoming less

negative. As R increases to the point of approaching infinity, R→ ∞, the potential energy approaches zero, PEg→0. In other words, the zero value of

gravitational potential energy between two masses occurs when they are infinitely apart.



• This type of potential energy relationship, where two objects have a force of attraction between them, resulting in their potential energy being negative, is called a potential well.

• An object must rise out of the potential well to be free of the gravitational force.



• For example, one of the masses could be Earth. The other mass could be a rocket. For the diagram above, R1 could represent the

radius of Earth. If the rocket is at rest on Earth, then it does not have any kinetic energy, KE, but only gravitational potential energy, PEg

which is equivalent to .



• The total mechanical energy of the rocket, ET

is the sum of the gravitational potential energy and the kinetic energy: ET = PEg + KE but at

the surface of Earth, the total energy is just the gravitational potential energy:

• ET = PEg = .



• Now assume that the rocket rises above Earth's surface to a height of R2. At this height, it has

less gravitational potential energy and some kinetic energy. The total mechanical energy of the rocket remains at the same constant value it had before.

ET = KE + PEg = KE + .



• What is the escape velocity? How is it calculated? • We can calculate the minimum velocity the rocket

must have to escape from Earth's potential well. To do this, the rocket's initial kinetic energy must exceed the depth of the potential well at Earth's surface, thereby making its total energy positive. In the ideal case, this would mean that the rocket must reach an infinite distance where PEg = 0 before coming to rest.

This minimum velocity is called the escape velocity.


Gravitational Potential Energy• At the surface of Earth, the total energy is the

gravitational potential energy: ET = .

• This energy is changed to kinetic energy: KEr = .

• Therefore,

• We take the absolute value of the gravitational potential energy so as to have a positive value for the velocity. For Earth, the escape velocity becomes


Gravitational Potential Energy• For Earth, the escape velocity becomes

• Thus a rocket launched from Earth with a velocity greater than this will move away from Earth, losing KE and gaining PEg as it does so. However, since its KE is greater than the depth of its PEg well at any point, its total energy will always remain positive. Theoretically, it will reach an infinite separation distance from Earth with some KE left, and escape. For launch velocity less than the escape velocity, the rocket might come to rest at some finite distance, then fall back to Earth.



• What is binding energy? How does the binding energy compare in magnitude to the PEg?

• In order for a rocket to escape from Earth's potential well, its kinetic energy must exceed the gravitational potential energy. If the KE does not exceed the PEg,

then the rocket cannot escape from the potential well, and it is said to be “bound” to Earth. The binding energy is the amount of additional kinetic energy it needs to escape.



• For a rocket at rest on the surface of Earth, the binding energy is identical in magnitude to the gravitational potential energy at the surface of Earth, since at this point, the rocket is not moving.

• Ebinding =



• If a rocket is in orbit at any radius Ro in the

potential well of Earth, then the centripetal force necessary to keep the rocket is the circular orbit is provided by the force of gravitational attraction between the rocket and Earth. If the rocket has a mass mR and an

orbital velocity of vR, then

• Fc = Fg =



• The total mechanical energy of the orbiting rocket is thus

• This is a very significant result. The total energy of a satellite in a circular orbit at any radius of orbit Ro is negative, and is equal to one-half the

value of the gravitational potential energy at this radius. The satellite is bound to Earth, and its binding energy is


Newton's Thought Experiment and Satellite Motion . . .

Topic 2.2 – Low Earth Orbit

. . . leads to Microgravity

Micro-Gravity and Apparent Weight


Micro-Gravity and Apparent Weight • We define the weight of a body as the gravitational force,

Fg, on the body. But what our bodies feel is really the reaction force of whatever surface we are in contact with (i.e. a chair, the ground, etc.) This upward force that counteracts the force of gravity is the normal force, FN, and it is the apparent weight.

• Micro-gravity can be defined as an environment in which the apparent weight of a system is smaller than its actual weight. Some of the possible environments in which this can occur are an elevator falling freely in a vacuum, a space capsule orbiting the earth, a spacecraft drifting in outer space with its engines off, a roller coaster on a downward path, a parabolic flight on a downward path, and so on.



• Imagine standing in an elevator that is at rest. The floor pushes upwards on your feet with the same force that gravity is pulling you down. Your apparent weight is the same as your real weight.

• FN - Fg = 0

FN = Fg



• If the elevator accelerates downwards, the normal force is less and your apparent weight is less. If the elevator accelerates downward at ½ g , then the normal force is reduced to ½ Fg and your apparent weight is one-half the force of gravity.



• If the elevator is in free fall, then the acceleration is equal to g and the person feels weightless. There is no normal force.

• FN - Fg = -ma

FN - mg = -mg

FN = 0


Micro-Gravity and Apparent WeightExperimentation • Your assignment is to research

how these 2 concepts allow roller coaster engineers to make the thrill of a roller coaster more exciting.


Gravitational Slingshot

• Interplanetary space probes often make use of then “gravitational slingshot” effect to propel them to high velocities.

• Your assignment is to research this maneuver and explain how a higher velocity is achieved.

• Be prepared to explain how this happens to your classmates.


Introduction to Electric Fields

• Electric fields are created by electric charges, and the fields created by these charges cause other charges to be influenced by them.

• We can define the electric field as the region of space around a charge where a positive test charge experiences a force caused by one or more other charges.

Topic 2.3 – Electric & Magnetic Fields


• There are two kinds of charge, positive and negative. Both charges can create an electric field around them .

• Electric field strength, E, can be defined to be the ratio of the electric force, F, to the charge, q, experiencing the force.



• Electric field lines are lines representing the direction of the electric field. Since electric field is defined to be the electric force per unit charge, the electric field lines are sometimes called lines of force.


Electric Field Lines for a Positive Point Charge


Electric Field Lines for a Negative Point Charge


Electric Field Lines for Two Oppositely Charged Point Charges


Electric Field Lines for Two Same Sign Point Charges


Electric Field Lines for Two Plates


Electric Field Lines for Two Plates & a Positive Charge


Electric Field & Electric Force


• As we have seen, there are two kinds of charge, positive and negative. Now we need to be able to say how much negative or positive charge there is on an object. The internationally accepted SI unit of charge, q, is the coulomb (C).



• The electric force on the charge q isFe = qE = (+5.00C)(10.0N/C) = +50.0N

• In general, we can say that the electric field vector and the electric force vector are in the same direction if the charge brought into the field is positive.

• Repulsion



• The electric force on the charge q isFe = qE = (+5.00C)(10.0N/C) = +50.0N

• In general, we can say that the electric field vector and the electric force vector are in the opposite direction if the charge brought into the field is negative.

• Attraction

Electric Field & Electric Force in 1 Dimension


• If there exists at a place in space, electric fields caused by more than one source, then the fields will add together like vectors since electric field is a vector.

• The total electric field at position P is ET = E1 + E2 = 10.0 N/C + (-15.0 N/C) = -5.0 N/C or 5.0 N/C west.

If a charge of 2.0 C is placed at position P, then the force on this charge is F = qET = (2.0 C)(5.0 N/C) = 10. N. The direction of this force is in the same direction as the electric field vector, that is west.

Electric Field & Electric Force in 2 Dimension


• Two electric fields are at right angles relative to each other. The positive charge, q1, creates and

electric field of 10.0 N/C north while the charge, q2, creates and

electric field of 15.0 N/C east.• In adding together the two electric

field vectors. We attach them head to tail.

Parallel Plates & Millikan


Millikan’s Experiment


Millikan and the Elementary Charge


• The currently accepted value for the elementary charge is 1.602 177 33 x 10-19 C. The value we will use in our calculations is 1.60 x 10-19 C Thus the total charge, q, on any object is a whole number multiple, N, of this elementary charge, e.

q = Ne

Gravitational PE vs Electric PE


Calculating Electric PE


• Assume there exists a charge between the plates, (+10.0 C) at a 2.00 cm above the negative plate (h1 = 0.02 m).

• The positive charge is attracted towards the negative plate and repelled from the positive plate.

• The force the charge experiences is a constant force of FE = qE towards the plate. Assume that this force is F = 5.00 N. At the negative plate, the charge does not have any potential energy.

• At a distance of 2.00 cm away from the plate, the potential energy at that location relative to the negative plate is

• PE1 = Fh1 = (5.00 N)(0.0200 m) = 0.100 J

Calculating Electric PE


• Now the +10.0 C charge is pulled away from the negative plate towards the positive plate. The force of 5.00 N is used to do this. Suppose that the new distance away from the negative plate is 8.00 cm. The new potential energy of the charge at this location is

• PE2 = Fh2 = (5.00 N)(0.0800 m) = 0.400 J• The charge has a greater potential energy at location

B than it did at location A.• The change in potential energy is • ΔPE = PE2 - PE1 = 0.400 J - 0.100 J = 0.300 J • This same amount of energy can be derived from the

force-distance graph. The area of the rectangle would be F(Δh) = 5.00 N (0.0800 m - 0.0200 m) = 0.300 J.

Work & KE


• To summarize, the change in electric potential energy of a charge q in moving from A to B is equal to the work required to move q from A to B against the electric field.

• W = ΔPE = PEB – PEA

• If work had to be done to move a charged particle to a higher potential, then when the particle is released and allowed to move freely under the influence of the electric field, the potential energy of the particle will decrease and it will gain in kinetic energy.

• We know that the kinetic energy gained is, KE = ½ mv2. If the mass of the charged object is known, then the speed of the object can be found.

Electric Potential


• The electrical potential energy is dependent on the quantity of the charge at a given location. But what if we wanted to measure the amount of energy that is independent of the charge placed at a location? In such a situation, we would have to have a measure of the electric potential energy per unit charge. That quantity is called the electric potential. The symbol “V” is used for electric potential.

Electric Potential


• We can formally define electric potential as follows.

• The electric potential V at a given point is the electrical potential energy, PE, of a test charge qo situated at that point divided by the charge itself.

• The SI unit of electric potential is the joule/coulomb also called the volt (V).

Electric Potential


• The terms electric potential energy, PE, and electric potential, V, have similar names, but they are not the same. Electric potential energy is, as the name implies, a measure of energy, and is therefore measured in joules. In contrast, the electric potential is the quantity of energy per unit charge, and is measured in joules per coulomb, or volts.

• The term “potential difference” is simply the difference in potential between two points. It also has the units of volts.

Calculating the Electric Potential & Potential Difference


• To calculate the electric potential we use,

• The potential difference between these two points is

• ΔV = V2 - V1

• The change in electric potential energy of a charge q in moving from A to B is equal to the work required to move q from A to B against the electric field. This can now be expressed as follows.

• ΔPE = qVB - qVA = q(VB - VA) = qΔV

The Electron Volt


• The change in the potential energy of a charge q in moving through electric field can be calculated using ΔPE = qΔV. The traditional unit for energy is the joule (J). However, the joule is not a convenient unit for small charges. Therefore another unit called the “electron volt” is sometimes used.

• We can define the electron volt as the energy acquired by an electron as a result of moving through a potential difference of 1V.

• Since the magnitude of the charge of an electron is 1.602 x 10-19 C, and the change in potential energy is qΔV we can say that 1 eV is equal to:

(1.602 x 10-19 C)(1V) = 1.602 x 10-19 J

The Relation Between Electric Potential and Electric Field


• Since the work involved in moving a charge against the electric field

between two plates is W = qΔV and , another way of determining the work is

W = qEd.

Equipotential Lines


• Electric potential can be represented in a diagram by drawing equipotential lines. An equipotential line is one where all points on that line are at the same potential. That is, the potential difference between any two points on the line is zero. Because of this, no work is required to move a charge from one point to another.

Electrical Potential Video


Electric Potential & Potential Energy for a Point Charge


• Gravitational Potential Energy for Two Masses

• Electric Potential Energy for Point Charges

Electric Potential Energy for Point Charges


• It is important to realize that the “zero” of potential energy occurs when the point charges are very far apart, in fact when they are at an infinite separation distance.

Electric Potential for Point Charges


Electric Potential for a Positive Point Charge


Electric Potential for a Negative Point Charge


Total Electric Potential


• We have seen that a single point charge can raise or lower the electric potential at a given location depending on whether the charge is positive or negative. If there are two or more charges, then we should follow the following rule to determine the total potential.

• When two or more charges are present, the potential due to all the charges is obtained by adding together the individual potentials.

Total Electric Potential


Lines of Equipotential


• Equipotential lines are perpendicular to electric field lines. We can also apply these ideas to point charges.

Potential Energy, Kinetic Energy, and Electric Field: Example


• The diagram shows a negatively charged drop moving between two deflecting plates between which a uniform, downward-pointing electric field, E, has been set up. The drop is deflected upward and then strikes the paper at a position that is determined by the strength of the electric field and the charge on the drop.

A Charged Particle Moving at Constant Acceleration: Example


Magnetic Fields


Ferromagnetism


• Some materials, such as iron, nickel, cobalt, and gadolinium, are not normally magnetized, but under certain circumstances, they can become magnetized. Such materials are called ferromagnetic.

• Ferromagnetic materials are strongly attracted by magnets. Ferromagnetic substances are commonly referred to as “magnetic substances.”

Domain Theory of Magnetism


• How ferromgnetic substances are able to acquire magnetic properties may be explained using the Domain Theory of Magnetism.

• Ferromagnetic substances are composed of a large number of tiny regions (less than 1 micrometer long or wide) called “magnetic domains.”

• Each domain behaves like a tiny bar magnet, with its own north and south poles. When the material is in an unmagnetized state, these millions of domains are oriented at random, so that their magnetic effects cancel out.



• In the diagram below, the arrows represent the domains in a small piece of unmagnetized iron.



• However, if a piece of ferromagnetic material such as iron is placed in a strong enough magnetic field, a strange phenomenon occurs. Some domains actually rotate slightly to align with the external field while others that are already aligned tend to grow at the expense of other non-aligned domains. The net result is that there is a preferred orientation of the domains in the same direction as the external field, causing the material to behave like a magnet.

The Magnetic Field of the Earth




• We will define the north magnetic pole as the pole that is found in the northern hemisphere. This pole acts as though it is a south pole on a bar magnet. The north magnetic pole is so named because it is the location toward which the north end of a compass needle points.

• The south magnetic pole is the pole that is in the southern hemisphere. This pole acts like the north pole of a bar magnet. The north end of a compass points away from this pole. At the poles, a magnetic compass would just rotate freely.



• The north geographic pole is that point where the earth’s axis of rotation crosses the surface in the northern hemisphere. The north magnetic pole does not coincide with the north geographic pole, but instead lies in Hudson Bay some 1300 km to the south. The south magnetic pole is located in Antarctica near the Ross Sea. The axis of the fictitious bar magnet and the earth’s rotational axis is about 11.5o.

Magnetic Declination and Magnetic Inclination


• The north magnetic pole does not coincide with the north geographic pole, but instead lies in Hudson Bay some 1300 km to the south. The south magnetic pole is located in Antarctica near the Ross Sea.

Magnetic declination is the angle between magnetic north and geographic north and varies from position to position on the earth’s surface.

Magnetic inclination or angle of dip is the angle between the earth’s magnetic field at any point and the horizontal, is called the magnetic inclination and it is measured with a magnetic dipping needle.

The Magnetosphere and the Auroras


• The magnetosphere is a region of the upper atmosphere beyond approximately 200 km in which the motion of charged particles from space is governed by the magnetic field of the earth. The magnetosphere on the side facing the sun extends beyond the earth’s surface approximately 57 000 km or about 10 earth radii. On the side away from the sun, the magnetosphere probably extends outward for hundreds of earth radii. The elongated shape results from the influence of the onrushing solar wind. The solar wind consists mainly of protons and electrons emitted by the sun, and this compresses the magnetosphere on the side nearest the sun.



• Auroras, commonly called the northern or southern lights, are caused by high energy particles from the solar wind that are trapped in the Van Allen belts of the earth’s magnetic field. As these particles oscillate along the magnetic field lines, they enter the atmosphere near the north and south magnetic poles. Energetic electrons collide with the oxygen and nitrogen molecules in the atmosphere. These collisions excite the molecules. When they escape from their excited states, they emit the light we see in the auroras. This forms a curtain of light that may hang down to an altitude of 100 km. Green light is emitted by oxygen, and pink light by nitrogen, but often the light is so dim that only white light can be seen.

fields

Education

keplers laws topic

keplers lawand topic

keplers scheme

ndlaw keplers ellipses

rdlaw keplers constant

keplers laws johannes

circular orbit

planet mars