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Transcript
12/1/2016 AP PHYSICS 2 UNIT 3 Electrostatics: electric force, electric field, and electric potential. CHAPTER 14 ELECTRIC CHARGE, FORCE, AND ENERGY EXPERIMENTS The earliest account of electric phenomena comes from the Ancient Greeks. In the 1700’s, an American scientist conducted extensive research in these phenomena. He sold many of his possessions to fund his work. They observed that after amber tools were cleaned with animal fur, they would attract The word electricity comes from the Greek word for small bits of dust and amber – electron. feathers. EXPERIMENT 1 – COMB AND BITS OF PAPER • Everyday observations show that uncharged lightweight objects, such as small bits of paper that have not been rubbed against anything, are attracted to charged objects. EXPERIMENT 2 – TRANSPARENT TAPE • Take two 9-inch-long pieces of transparent tape and place them sticky side down on a plastic, glass, or wooden tabletop. Now pull on one end of each tape to remove them from the table. Bring the pieces of tape near each other. They repel each other, as shown in the figure. Can we explain the repulsion of the tapes as an electric interaction? 1 12/1/2016 EXPERIMENT 3 – BALLOONS If you rub two balloons the same way with a wool cloth, Just like the experiments in class, Benjamin Franklin recorded that when an amber rod was rubbed with fur, the amber would attract the fur that it was rubbed with. Is the electrical interaction just a magnetic interaction? However, he also discovered that two amber rods rubbed with fur would repel one another. • Perhaps the electric interaction is He was able to achieve similar results with a glass rod rubbed with silk, but with one major difference: actually the magnetic interaction, just they will attract the The rubbed glass rod and the rubbed amber rod repel one another too ! wool but repel each other. EXPERIMENTS WITH MAGNETS Ben Franklin’s Conclusions described using different terminology. Ben Franklin’s Conclusions • The property acquired by materials due to rubbing is called electric charge. • Objects acquire positive or negative electric charge (He could have named them anything else – the names are arbitrary) • Electrostatics is not the same phenomenon as magnetism! • These objects did not have two poles! Charged rod attracted south pole Charged rod attracted north pole • Objects that interact with each other because they are charged exert electric forces on each other. • When charged objects are at rest, the force is an electrostatic force. 2 12/1/2016 Ben Franklin’s Conclusions • Materials rubbed against each other acquire electric charge. • Two objects with the same type of charge repel each other. • Two objects with opposite types of charge attract each other. Charged objects attract uncharged objects • Uncharged objects are attracted to WHITEBOARD • You pull your sweater and shirt off together, and then pull them apart. You notice that they attract each other— a phenomenon called "static" in everyday life. Explain the mechanism behind this attraction and suggest an experiment to test your explanation. both positively and negatively charged objects; uncharged metal objects are attracted more strongly than nonmetal • Two objects made of different materials rubbed against each other acquire opposite charges. HISTORICAL MODELS FOR ELECTRIC CHARGE • Fluid models of electric charge • Too much electric fluid in an object makes it positively charged and too little makes it negatively charged. • Particle model of electric charge • Electric charge is carried by microscopic particles. Removing or adding these particles to an object changes the charge of the object. objects. CONTEMPORARY MODEL FOR ELECTRIC CHARGE • Two objects start as neutral—the total electric charge of each is zero. • During rubbing, one object gains electrons and becomes negatively charged. • The other object loses an equal number of electrons and with this deficiency of electrons becomes positively charged. What exactly is electric charge, anyway?? 3 12/1/2016 Early Theory of Electric Charge Physicists like Benjamin Franklin and Charles Coulomb believed that electric charge was a fluid. They theorized that when two objects are rubbed together, the fluid travels. Their theory also said that objects that have lots of the fluid are positively charged, and objects that have a lack of the fluid are negatively charged. Coulomb trying to capture electric fluid in a jar (unsuccessfully) Elementary particles have two key components WHAT ACTUALLY IS ELECTRIC CHARGE? Electric charge is a property of subatomic particles (protons and electrons) that cause them to feel a force when other protons or electrons are present. Since all matter is composed of protons, electrons and neutrons, it is possible for any object to become positively or negatively charged! (but very difficult to accomplish for certain materials) 2) Electric Charge Mass gives particles two fundamental properties. Electric charge gives particles one fundamental property. b) Mass causes particles to attract all other particles that have mass This is the gravitational property of matter. Protons and neutrons have roughly the same mass as one another, and they have about 10,000 times as much mass as electrons. If an object is negatively charged, it means that it has excess electrons. If an object is positively charged, it means that it has a lack of electrons. If an object has an equal number of protons as electrons, then it is electrically neutral. PHYSICAL QUANTITY: ELECTRIC CHARGE Elementary particles have two key components 1) Mass a) Mass causes particles to resist being accelerated This is called the property of inertia. CHARGED OBJECTS a) Electric charge causes protons and electrons to create an electric field This means that all other charged particles in their vicinity will feel an electric force Protons and electrons exactly opposite charges to one another. (This shows the complete independence of mass and charge) Neutrons have an electric charge of 0. • Symbol • Units q Coulomb • Units symbol C Electric charges are small and they are usually written using prefixes of the metric system q = +3C q = +3x10-6 C 4 12/1/2016 Protons and electrons are the fundamental building block of all charged objects. FUNDAMENTAL CHARGE • Electron: • Proton: q = 1.6×10−19 C q= 1.6×10−19 This is called the elementary charge, e (e = 1.6 x 10-19 C) C • Ohhhhh yesss !!! electrons and protons have the same magnitude of electric charge !! • To differentiate the charge of an electron from the charge of a proton we use the NEGATIVE SIGN for the CHARGE OF AN ELECTRON. q= -1.6×10−19 No object could ever exist in nature that has a charge of 1 x 10-20 C The most physics-y way of stating this concept is that electric charge is quantized. Electric charge comes in indivisible “packages” of magnitude e. Any charged object in nature must have a charge that is a direct multiple of e. C + + + -- • Metals have free electrons that are free to move inside. + + + + + + + + + + +- +- ++- +- ++- +- +- • Plastic, glass, and other nonmetal materials do not have free electrons or any other charged particles that are free to move inside. • Both materials are composed of oppositely charged particles with a total charge of zero. • In conductors, some of can move freely. - +- +- ++- +- ++- +- +- Neutral atoms with charge uniformly distributed. Polarization of dielectrics - + -+ - +- +- + - When the positively charged object is brought near the dielectric material, its atoms and molecules become polarized. A similar thing happens if we bring a negatively charged object near a nonconducting material. the charged particles • In dielectrics, charges can only be redistributed slightly (set off), a process called polarization.. POLARIZATION OF A DIELECTRIC Copper wire - - - - - • Materials can be divided into two groups: conductors and insulators (dielectrics). Why? DIELECTRICS vs. CONDUCTORS Latex Balloon Electric properties of materials - ++ ++ ++ + + - + - ++ - ++ - + - ++ - ++ - ++ - + - ++ - ++ - ++ - + - + - Charges can only be redistributed slightly (set off) 5 12/1/2016 Polarization of dielectrics - + -+ Conductors: Separation of charges - - + - ++ ++ ++ + +- +- + - - + - + - ++ - ++ - + ++ ++ ++ + - - ++ - ++ - ++ - + - + - - + - +- +- + - +- - + +- + +- + +- ++ + +- + +- + +- + +- + + - ++ + +- - Charged objects + + + + + + + + + + Positively charged object: Lack of negative charges + + - - - + +- - + +- - - ++ - - - - Negatively charged object: Excess of negative charges - +- +- + - + + + + + + + + + + + + + + + + + - - - + +- - + +- - - ++ - - - - + - Negative charges on metal cups are free to move, and therefore repelled by the negatively charged rod When separated, the metal cups are charged CONDUCTORS + + + + -+ - - + - When separated, the plastic cups are uncharged + + -+ - Conductors: Separation of charges FREE ELECTRONS ? • In metals, some electrons can move freely throughout the metal. – When we bring a positively charged rod next to a metal bar, the free electrons move closer to the positively charged rod, leaving the other side with a deficiency of electrons. © 2014 Pearson Education, Inc. 6 12/1/2016 CONDUCTORS Conductors Is the human body a conductor or a dielectric? As your hand approaches a positively charged cup, the free electrons on the surface of your hand and arm move toward it until the net charge on the cup is zero. ELECTROPHORUS Grounding • If the cup has a positive charge, negative electrons in the ground are attracted toward the cup and travel from the ground to the cup, causing it to become neutral. Discharging by moisture in the air • If you leave a negatively or positively charged object alone for a long time interval, the object can become neutral in part because of the presence of water vapor in the air. The electroscope • An electroscope consists of a metal ball attached to a metal rod. A very lightweight needle-like metal rod is connected on a pivot near the bottom of the larger rod. 7 12/1/2016 The electroscope • The angle of deflection is related to the magnitude of the electric charge of the electroscope. Several years later, while Benjamin Franklin was helping to build our nation, a French scientist named Charles-Augustin de Coulomb ventured a step further. “The attractive force between two oppositely charged spheres is proportional to the product of the quantities of charge on the spheres and is inversely proportional to the square of the distance between the spheres” Coulomb's force law • In 1785, Charles Coulomb determined the relationship between distance and magnitude of charges, and the force between charges. • The experimental apparatus Coulomb used is called a torsion balance. (Originally in French) Coulomb's data Coulomb’s Law Electric Force (FE ) Electrostatic interactions • Sometimes more vigorous rubbing leads to a greater force exerted by the rubbed objects on each other. q1 q2 d • The magnitude of the force that the charged objects exert on each other increases when the distance between the objects decreases. 𝐹𝐸 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑑2 𝐹𝑞1𝑜𝑛𝑞2 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑑2 AP Equation sheet AP Book 8 12/1/2016 Coulomb’s Law 𝐹𝑞1𝑜𝑛𝑞2 Coulomb’s Law Constant 𝑘 ∙ 𝑞1 ∙ 𝑞2 = 𝑑2 • q1 is the magnitude of the charge of object 1. • q2 is the magnitude of the charge of object 2. 𝑘= 1 4 ∙ 𝜋 ∙ 𝜀0 0 is the vacuum permittivity (constant number) −12 2 𝜀0 = 𝑘= • d is the distance between the charged objects. 8.85𝑥10 𝐶 𝑁 ∙ 𝑚2 1 8.85𝑥10−12 𝐶 2 4∙𝜋∙ 𝑁 ∙ 𝑚2 • k is Coulomb’s Law Constant 𝑘= What charge exerts a bigger force? 𝐹𝐸 q2 q1 d 𝐹𝑞1𝑜𝑛𝑞2 = 𝐹𝑞2𝑜𝑛𝑞1 Newton’s third Law !!! Coulomb’s Law 𝑘 ∙ 𝑞1 ∙ 𝑞2 = 𝑑2 If the two objects have the same type of charge, they will repel one another. + + _ _ 𝐹𝑞1𝑜𝑛𝑞2 = 9𝑥109 𝑁 ∙ 𝑚2 𝐶2 With the math model you can get you the magnitude of the electric forces exerted on each object, and you can use some reasoning to find the direction. If the two objects have different types of charge, they will attract one another. + 𝐹𝑞1𝑜𝑛𝑞2 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑑2 1 𝑞1 ∙ 𝑞2 4 ∙ 𝜋 ∙ 𝜀0 𝑑2 WHITEBOARD Find magnitude and direction of • FqA on qB • FqB on qA qB = -9C qA = +3C + d = 0.05 m _ The unit of electric charge (q) is the Coulomb [C], in honor of Charles Coulomb. • FqA on qB = - 97.2 N • FqB on qA = + 97.2 N 9 12/1/2016 WHITEBOARD Four electric charges are fixed on a grid. Find the 4 missing quantities. qA = -4C - FqAonqB = ? d = 0.04 m + • d = 0.08 m d=? FqBonqD = +14.0625 N • qD = -2C - d = 0.04 m qD = ? FqConqD = -90 N WHITEBOARD • Three identical metal spheres A, B, and C are on separate insulating stands. You charge sphere A with charge +q. Then you touch sphere B to sphere A, and separate them by distance d. 1. Write an expression for the electric force that the spheres exert on each other. • You now take sphere C, touch it to sphere A, and then remove sphere C. You then separate spheres A and B by a distance d/2. 1. Write an expression for the magnitude of the force that spheres A and B exert on each other. • FqA on qB = -112.5 N qB = +5C + d=? FqAonqC = +45 N qC = +8C WHITEBOARD - SOLUTION • FqA on qD = 9 N FqAonqD = ? 𝐹𝑞𝐴 𝑜𝑛 𝑞𝐵 𝑘 ∙ 𝑞2 = 4𝑑2 10 12/1/2016 Whiteboard Reasoning with Coulomb’s Law F +q F +q 𝐹𝑞 = d 𝐹𝑞𝐴 𝑜𝑛 𝑞𝐵 𝑘 ∙ 𝑞2 = 2𝑑2 If one of the spheres is replaced with a sphere of charge +2q, and they are brought to half the distance apart, what will be the resulting force exerted on each of the spheres? a +2q b +2q +3q d c +q 𝐹𝑞 = 𝟖 ∙ 𝑘 ∙ 𝑞2 𝑑2 a>b>c WHITEBOARD (RANKINGS) (RANKINGS) (RANKINGS) +q c +q d a=b=c +q a -q d +q b +q d +q d WHITEBOARD b +q d WHITEBOARD a d (RANKINGS) +2q d/2 -q 𝑘∙ 𝑑2 Two charged spheres are held near one another, as shown above. +q -q WHITEBOARD 𝑞2 +q c +q 2d a>b>c +q a +q 3d +q b +q d +q c +q 2d +q +q ½d c>a>b 11 12/1/2016 WHITEBOARD WHITEBOARD Both spheres are positive (RANKINGS) a b +q +q +2q d c +q 2d +3q +q 1/3 d c>a>b WHITEBOARD Electrostatic Conceptual Question Two isolated charges, +q and -2q, are 2 centimeters apart. If F is the magnitude of the force exerted on charge -2q, what are the magnitude and direction of the force exerted on charge +q? Magnitude (A) (B) (C) (D) (E) F/2 2F F F 2F Direction Toward charge -2q Away from charge -2q Toward charge -2q Away from charge -2q Toward charge -2q Newton’s Third Law! A positively charged sphere with charge q and mass m hangs at the end of a string. Another positively charged sphere of a different charge 5q is secured at the top end of the string to a wooden support. + Draw a force diagram for the hanging sphere (qh) Represent mathematically using Newton’s 2nd Law + Draw Newton’s 3rd Law forces that the spheres exert on each other. FT on qh Fq2 on qh + qh FE on qh FE on qh - q2 + q2 + qh Fq2 on qh Fqh on q2 Calculate the ratio of the strength of the electric force that each particle exerts on the other to the strength of the gravitational force that particle exerts on the other. q1q2 r2 Fgravitational = G Fgravitational = kq1q2 Gm1m2 Fq2 on qh qh + Felectric = k Felectric FT on qh + qh Fqh on q2 A hydrogen atom is composed of a proton and an electron. mproton = 1.67 x 10-27 kg melectron = 9.1 x 10-31 kg G = 6.67 x 10-11 Nm2/kg2 Fq2 on qh FT on qh = Fq2 on qh FT on qh + Fq2 on qh + FE on qh = FE on qh Whiteboard: The Hydrogen Atom qelectron = -1.6 x 10-19 C qproton = 1.6 x 10-19 C k = 9 x 109 Nm2/C2 Hanging sphere is positive, top sphere is negative m1m2 r2 = 2.27 x 1039 times as strong!!! The electrical force between the particles is much, much stronger than the gravitational force! 12 12/1/2016 The Principle of Superposition The net force exerted on an object is the vector sum of the individual forces exerted on it. Treat each force separately using Coulomb’s Law, and then add them as vectors! Whiteboard: Superimpose! Whiteboard: Solution Determine the magnitude and direction of the net force exerted on q1 in the configuration shown below. Fnet = 3.6 N 60° + 60° q1 = 6 x 10-7 C 60° + 3 cm + _ q2 = -6 x 10-7 C + _ 3 cm 3 cm + q3 = 6 x 10-7 C + Whiteboard 60° + The y-components of the forces negate one another, and the xcomponents add to one another! Whiteboard Whiteboard: solution Fq1onq3 60° X-component Y-component -0.00102 0.001357 L +q qq + L + +q d • Determine the signs of the charges q1 and q2. • Calculate the magnitude of the net electrostatic force exerted on particle 3. • Where should another positively charged particle could be fixed in place so that the electrostatic force on particle 3 is zero? Fq2onq3 -0.0018 -0.001354 F -0.00282 0 Two objects of mass m and charge +q hang at the bottom of a string of length L. • Draw a Force diagram for any +q • Write an expression for any known force (or components) • Write an expression for the electric charge that keeps the system in equilibrium. 13 12/1/2016 FT q +q Fq 𝐹𝑞 = 𝑘 ∙ 𝑞2 𝑑2 + FTx FTy 𝐹𝑇𝑦 = 𝐹𝑇 ∙ 𝑐𝑜𝑠 𝜃 𝐹𝑇𝑦 = 𝑔𝑚 𝐹𝑇𝑥 = 𝐹𝑇 ∙ 𝑠𝑖𝑛 𝜃 FG Σ𝐹𝑦 = 0 Σ𝐹𝑥 = 0 𝐹𝐺 = 𝑔𝑚 Whiteboard 𝐹𝐺 + 𝐹𝑇𝑦 = 0 𝐹𝑞 + 𝐹𝑇𝑥 = 0 𝑔𝑚 + 𝐹𝑇 ∙ 𝑐𝑜𝑠 𝜃 = 0 𝑘𝑞 2 + 𝐹𝑇 ∙ 𝑠𝑖𝑛 𝜃 = 0 2 𝑑 𝑔𝑚 𝑘𝑞2 𝐹𝑇 = − 𝐹𝑇 = − 2 𝑐𝑜𝑠 𝜃 𝑑 ∙ 𝑠𝑖𝑛 𝜃 𝑑2 𝑘𝑞2 𝑔𝑚 = ∙ 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑞=𝑑 𝑔𝑚 ∙ 𝑡𝑎𝑛 𝜃 𝑘 Suppose in the scenario with the hanging charged spheres, one of the spheres had a charge +q, and the other had a charge +2q. The spheres have the same mass. Which of the following shows the equilibrium position that will be reached by the spheres? (A) (B) (C) (D) None of the above are correct. WHITEBOARD Metal spheres on insulating stands have the F2q on q Fq on 2q following electric charges: qA = +2.0 nC, qB = +2.0 nC, and qC = –4.0 nC. The spheres are 𝐹𝑁𝐸𝑇 = 124.7 𝑛𝑁 placed at the corners of an equilateral triangle The magnitude of the electric force exerted on each particle is the same. Newton’s 3rd Law is a beautiful thing! whose sides have length d = 1.0 m, with qC at the top of the triangle. What is the magnitude of the total (net) electric force that spheres A and B exert on sphere C? 14 12/1/2016 Whiteboard Whiteboard: solution X-component • Calculate the magnitude and direction of the net electric force exerted on the object of charge q1. • If the mass of each of the particles is 30 grams, and the particles are released from rest in the configuration shown above, what will be the magnitude and direction of the instantaneous acceleration of the object of charge q1 at the moment of release? Whiteboard: solution Y-component Fq2onq1 0 0.02025 Fq3onq1 0.011691 -0.00675 F Whiteboard 0.011691 Fq2onq1 𝐹𝑁𝐸𝑇 = 0.0179 𝑁 𝜃 = 49.1 Fq3onq1 0.0135 Whiteboard 𝑎= Σ𝐹 𝑚 𝑎= 0.0179 𝑁 0.03 𝑘𝑔 𝑎= 0.595 𝑚 𝑠2 Whiteboard • On the axes below, qualitatively sketch a graph of the acceleration a of the object of mass m2 versus the distance d between the objects after the string has been cut. • Two small objects, each with a charge of -4.0 nC, are held together by a 0.020 m length of insulating string as shown in the diagram above. The objects are initially at rest on a horizontal, nonconducting frictionless surface. The gravitational effect on each object due to the other is negligible. 𝐹𝑞1 𝑜𝑛 𝑞2 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑑2 𝑎= 𝐹𝑞1 𝑜𝑛 𝑞2 = +0.00036 𝑁 𝑎= +0.00036 +0.006 𝑚 𝑎= 0.06 𝑘𝑔 𝑠2 𝐹𝑞2 𝑜𝑛 𝑞1 = −0.00036 𝑁 𝑎= −0.012 𝑚 −0.00036 𝑎= 𝑠2 0.03 𝑘𝑔 • The masses of the objects are m1 = 0.030 kg and m2 = 0.060 kg. The string is now cut. • Calculate the magnitude of the initial acceleration of each object. Σ𝐹 𝑚 • Describe qualitatively what happens to the speeds of the objects as time increases, assuming that the objects remain on the horizontal, nonconducting frictionless surface. 15 12/1/2016 1 Felectric a 2 d & a a Fnet 1 aa 2 d Whiteboard Bonus Follow-Up Question! What would the velocity vs time graph for m2 look like after the string was cut? v A positively charged cannonball is held near another fixed positively charged object in the barrel of the cannon. Build an Energy bar chart for the initial and final states. After the string is cut, m2 will continue to speed up forever, but with decreasing acceleration. m2 will never slow down, because it is always being repelled by m1! 0.006 m/s2 However, it will speed up by less and less each second as it gets further away. Electric potential energy: A qualitative analysis • Some type of energy must decrease if gravitational and kinetic energies increase in this process. 0.07 0.06 This shows a decreasing acceleration (slope) t m2 is always being repelled by m1, so it will continue to speed up forever. However, as they get further apart, Felectric decreases. This causes m2 to speed up with a decreasing acceleration. 0.02 m 0.08 Velocity keeps increasing, but by less and less each second. Cannonball at a some elevation Moving Cannonball Whiteboard Consider two oppositely charged blocks, one of which can slide without friction.. Build an Energy bar chart for the initial and final states. Electric potential energy: A qualitative analysis • When the negatively charged block is released and moves nearer the nut, the kinetic energy of the system increases. 8.00E+00 6.00E+00 0.05 4.00E+00 0.04 2.00E+00 KE cant be negative 0.00E+00 0.03 -2.00E+00 0.02 -4.00E+00 0.01 Uqi Ugi Kei UqF UgF KeF -6.00E+00 0 Uqi Ugi Kei UqF UgF KeF -8.00E+00 -1.00E+01 -1.20E+01 Cannonball at a some elevation Cannonball at a higher elevation The electric potential energy of oppositely charged objects is negative Uq becomes more negative as the charged object gets closer 16 12/1/2016 analyze a situation where only the electric potential energy changes when work is done. Distance decreases, electric force increases. It is harder to push ! Electric potential energy 𝑊 = 𝐹𝑞 ∙ ∆𝑟 𝑊= 𝑘 ∙ 𝑞1 ∙ 𝑞2 ∙ ∆𝑟 𝑟2 𝑈𝑞 = Calculus ! 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑊= − 𝑟𝐹 𝑟𝑖 𝑊 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑟 • Sign of the charge does matter ! 1 1 − 𝑟𝐹 𝑟𝑖 17 12/1/2016 Tip Skills for analyzing processes involving electric force and electric potential energy ELECTRIC POTENTIAL ENERGY In conjunction with your problem-solving strategy: • Decide whether you can consider the charged objects to be point-like. PRACTICE PROBLEMS • If you are using the work-energy principle, construct an energy bar chart. Decide where the zeros for potential energies are. WHITEBOARD: Graph electric potential energy versus distance • Because of the 1/r dependence, the electric potential energy approaches positive infinity when the separation approaches zero, and it becomes less positive and approaches zero as like charges are moved far apart. WHITEBOARD 1 / 6 • Two oppositely charged objects (with positive charge +q and negative charge –q) are separated by distance ri. Will the electric potential energy of the system decrease or increase if you pull the objects farther apart? • Explain (Use a bar chart) • Graph Uq vs r WHITEBOARD SOLUTION 4 3 2 1 0 -1 Uqi w UqF -2 -3 -4 -5 -6 18 12/1/2016 WHITEBOARD 2 / 6 • Suppose that a radon atom in the air in a home is inhaled into the lungs. While in the lungs, the nucleus of the radon atom undergoes radioactive decay, emitting an α (alpha) particle, which is composed of two protons and two neutrons. During this process, the radon nucleus turns into a polonium nucleus with charge +84e and mass 3.6 x 10–25 kg. The α particle has charge +2e and mass 6.6 x 10–27 kg. Suppose the two particles are initially separated by 1.0 x 10–15 m and are at rest. How fast is the α particle moving when it is very far from the polonium nucleus? WHITEBOARD 3/6 (PROBLEM 24) 𝑈𝑞𝑖 = 𝑈𝑞𝑓 + 𝐾𝐸𝑓 𝑈𝑞𝑖 = 𝐾𝐸𝑓 Uq = 0 R= 𝑣= 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑚 ∙ 𝑣 = 𝑑 2 2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑚∙𝑑 𝑣= 𝑣= 108,302,606 m 𝑠 2 2 ∙ 𝑘 ∙ 84𝑒 ∙ 2𝑒 6.6𝑥10−27 ∙ 1𝑥10−15 • Determine the change in electric potential energy of a system of two charged objects when a -1.5 C charged object and a -4.0 C charged object move from an initial separation of 500 km to a final separation of 100 km. • Sketch • Energy bar chart • Find Uq 𝑈𝑞𝑖 + 𝑊 = 𝑈𝑞𝑓 𝑊 = ∆𝑈𝑞 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑘 ∙ 𝑞1 ∙ 𝑞2 − 𝑟𝐹 𝑟𝑖 ∆𝑈𝑞 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 ∆𝑈𝑞 = 𝑘 ∙ −1.5 𝐶 ∙ −4 𝐶 ∙ 1 1 − 𝑟𝐹 𝑟𝑖 1 1 − 100000 500000 ∆𝑈𝑞 = 432,000 𝐽 19 12/1/2016 WHITEBOARD SOLUTION WHITEBOARD 4/6 (PROBLEM 33) ∆𝑈𝑞 = 𝑘 ∙ 𝑞1 ∙ 𝑞2 600000 540000 500000 432000 400000 300000 200000 108000 100000 0 Uqi W UqF WHITEBOARD SOLUTION 600 • A stationary block has a charge of +6.0x10-4 C. A 0.80 kg cart with a charge of 4.0x10-4 C is initially at rest and separated 4.0 m from the block. The cart is released and moves along a frictionless surface to a distance of 10.0 m from the block. • • • • Sketch Find the change in electric potential energy. Find the speed of the cart. Energy bar chart ∆𝑈𝑞 = 𝑘 ∙ 6𝑥10−4 ∙ 4𝑥10−4 ∙ 540 ∆𝑈𝑞 = −∆𝐾𝐸 324 ∆𝑈𝑞 = − 200 100 0 Uqi KEF UqF • Sketch • How fast is the cart moving when very far (infinity) from the fixed charge? • How fast is the cart moving when 2.0 m from the fixed charge? 𝑚𝑣 2 2 𝑣= −2 ∙ ∆𝑈𝑞 𝑚 28.46 𝑚 𝑠 ∆𝑈𝑞 = −∆𝐾𝐸 𝑘 ∙ 𝑞1 ∙ 𝑞2 300 216 𝑣= Think: Conservation of energy • A 0.4 kg cart with charge +4.0x10-5 C starts at rest on a horizontal frictionless surface 0.50 m from a fixed object with charge +2.0x10-4 C. When the cart is released, it moves away from the fixed object. 400 1 1 − 10 4 ∆𝑈𝑞 = −324 𝐽 WHITEBOARD 5/6 (PROBLEM 46) 500 1 1 − 𝑟𝐹 𝑟𝑖 𝑣= 𝑟𝐹 = ∞ 1 1 𝑚𝑣 2 − =− 𝑟𝐹 𝑟𝑖 2 −2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2 1 1 − 𝑟𝐹 𝑟𝑖 𝑚 1 =0 𝑟𝐹 𝑣= 26.83 𝑚 𝑠 20 12/1/2016 ∆𝑈𝑞 = −∆𝐾𝐸 𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑣= 𝑟𝐹 = 2 𝑚 Van de Graaff generator 1 1 𝑚𝑣 2 − =− 𝑟𝐹 𝑟𝑖 2 −2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2 1 1 − 𝑟𝐹 𝑟𝑖 𝑚 𝑣= Wimshurst machine • The Wimshurst, invented in the 1880s, consists of two plastic disks that rotate in opposite directions. It can produce large charge separations. 23.24 𝑚 𝑠 © 2014 Pearson Education, Inc. A photocopier • In a copy machine, a drum gets electrically charged with an image of the page being copied. Dark toner particles stick to the places where the drum is charged. 21