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Transcript
What is the scope of the class? Physics 202 Helmut G. Katzgraber Scope: Electricity and magnetism Light and optical instruments Quantum effects due to light Particle–wave duality Atomic structures Radioactivity and nuclear physics. Why should anyone care? After all, it’s all boring physics! Understanding the underlying physical processes can help a lot. Knowing a bit about E&M and circuitry is useful for lab work. I guess I do not need to elaborate on the radioactive part... FS 10 Chapter 17: Electric charge & field What will we learn in this chapter? Interactions between electric charges Coulomb’s law Electric charges are quantized, i.e., the total charge is a multiple of e Electric charges obey a conservation law Notion of a “field” Have we felt charges before? Yes! Scuffing feet Exiting car Combing hair Wiping certain materials ... How many charges exist? Did we “create” charges? No. The experiment shows that charge was transferred from the fur/ silk onto/off the plastic/glass. We cannot create electric charge, but we can move negativelycharged electrons from one object to the other leaving positivelycharged “holes” behind. The total electric charge of both cloth and bar does not change. This is a simple example of a conservation law. fur – plastic silk – glass rubbed plastic – rubbed glass The simple experiment shows that there are only two types of charges: postivie (+) and negative (–). Same charges repel (+ + and – –), opposite attract (+ –). Physical basis of electric charge Neat experiment: Take a transparent balloon and add some tiny styrofoam pellets. Inflate and rub the balloon. The pellets will charge and form a mesh in space. Physical basis of electric charge contd. The interactions responsible for the structure and stability of atoms, molecules and matter are mainly electrical interactions between charged particles. nucleus 100000 times smaller (~2000 lighter than protons) Note: The charge of the proton and electron are equal. Net charge of a neutral atom is zero. Number of protons: Atomic number. The process of having nonzero charge is called ionization. Nuclei are not free to move in solids (charge due to electrons). Conductors vs insulators Conductors vs insulators contd. Conductors: Metals (outer electrons can move freely trough the lattice). Ionic liquids (e.g., NaCl+H20) and charged gases (charges can move freely). Note: Ionic solutions play a crucial role in biology (e.g., neurons). Insulators: Non-metals (electrons are bound to the nucleus). Conductors: permit charges to move. Insulators: do not transfer charges. Oddballs: Semiconductors (between conductors and insulators). Superconductors (perfect conductors – no resistivity). In general, good insulators can be “charged,” whereas conductors “spread” the charge. Induction Idea: charge an object without losing any charge. One can imagine “positive” charges (holes) moving. A electron deficiency corresponds to a positive charge. Quick review of some concepts…. e- buildup If you have a charged ball, where are the charges located? Evenly distributed? Center of the ball? Surface? Keep in mind that same charges repel each other! 3 identical balls are mounted on insulators. Ball A has a charge Q, balles B and C are uncharged. Ball A touches ball B, and then touches ball C. What is the charge on ball A? Q/2 Q/3 Q/4 ! Understanding air purifiers (not a scam) Charged objects can exert forces on neutral things: Balloons stick to walls Paper sticks to comb But first a comb: The charge on the comb attracts the electrons in the paper, inducing a net charge. Induced charge effect, the system polarizes. Air purifiers: Charged plates make dust and pollen stick. Conductors: electrons move around the system. Insulators: tiny shifts of electron clouds around atoms (adds up!). Conservation & quantization of charge In a closed system: The algebraic sum of all electric charges in any closed system is constant. Charge can be transferred from one object to another, and that is the only way that an object can acquire net charge. Notes: This is a universal conservation law. Even in colliders, where particles are created/destroyed, charge is conserved. LHC The magnitude of the electron/proton charge is a natural unit. Any amount of observable charge is a multiple of it (there are no half cents). Electric forces hold atoms and molecules together. Other applications: Car paint. Laser printers. ... Coulomb’s law Coulomb studied forces between charged objects with a torsion balance. Result: The force depends ! ! only on the distance ! ! and the amount of ! ! charge. C. A. de Coulomb (1736-1806) The magnitude F of the force between two point charges q1 and q2 is directly proportional to their product and inversely proportional to the square of their distance: 1 q1 q2 �0 = 8.854 × 10−12 C2 /(Nm2 ) F = 4π�0 r2 force measurement due to torsion of the wire Polarization Coulomb’s law contd. How many Newtons for one Coulomb? Notes: Coulomb forces obey Newton’s third law. Forces between 2 point charges act along a line and are equal in magnitude. Coulomb’s law is strictly valid for point charges in vacuum only. The principle of superposition for more than one charge holds: When adding several forces, you need to use vector sums. Unit of charge: Coulomb. The charge of an electron is e = 1.602 × 10−19 C Electric fields and forces Remember: 1 q1 q2 4π�0 r2 e = 1.602 × 10−19 C F = What does this all mean? Two balls with 1C charge at 1m distance repel with 9 109 N: 1 9 k= = 8.9875 109 Nm2 /C2 → F ≈ 8.98 10 N 4πε0 Typically we encounter values of 10-9 to 10-6C. One gram of material has approximately 1023 particles. This means we have potentially 104 – 105C. Electrical neutrality can only be disturbed with huge forces. Definition: electric field Remove charge q’ and replace by point P. We say that A causes an electric field E at point P. When a “test charge” is placed at P, it feels a force F’. Since P can move, the electric field E exists around A. When a charged particle with charge q’ at point P is acted upon by an � at that electric force F� , the electric field E � � =F point is defined as E q� If q’ is positive, the force and field point in the same direction; if negative in opposite directions.!! “Propensity to generate a force” q� > 0 To see if there is a field at point P, we place a test charge at P and measure the force. � Note: F� → E q� < 0 Units (SI): 1 N/C (Newton/Coulomb) Note: the force acting on q’ depends on its location in space. Thus the electric field is not a simple vector, but a vector field; i.e., a vector � associated with every point in space. Thus each component of E � depends on spatial coordinates, i.e., E = (Ex (�r), Ey (�r), Ez (�r)). Calculating electric fields Principle of superposition: The total electric field at any point due to two or more charges is the vector sum of the fields that would be produced at that point by the individual charges. �1 S Example: 3 charges at source points S. q1 �3 What is the field at field point P? S �2 S � q3 Compute individual fields Ei acting on q2 a test charge at P. P � =E �1 + E �2 + E �3 The total field at P is then: E � at point P due to a point The magnitude of the electric field E = |E| charge q at S at a distance r from P is 1 |q| E= 4π�0 r2 By definition, if q > 0 the field points away from the charge, if q < 0 points towards the point charge. Electric field lines In general, only animals can feel electric fields (e.g., birds). Visualizing electric fields: Electric field lines are imaginary lines for which the tangent at each point is the electric field. � at each point. Field lines show the direction of E The spacing between the lines gives an idea of the magnitude. The electric field is unique, hence only one field line passes through one point: lines never cross. Convention: field lines point away from positive charges. weak strong Special case: spherical point charges For spherically-symmetric charge distributions (e.g., an insulating sphere) the produced field outside the distribution is the same as if the charge were concentrated in P. To obtain the field outside a sphere, one can assume it collapsed to a point P with the total charge of the sphere Q. This is useful when computing fields between e.g., spheres and walls. P Examples of typical field patterns into negative charges positive point charge tangent to the field dipole close when strong field two positive point charges Note: These are cross-sections of the 3D patterns. Uniform field (needed in applications): parallel plate capacitor. Inside, the field is nearly uniform. Gauss’s law Historical interlude: Carl Friedrich Gauss Equivalent to Coulomb’s law. Provides an alternative approach to calculating electric fields. Allows us to find the field at point P caused by a single point charge q. How do we deal with extended charge distributions? We represent them as sums (integrals) over point charges (use the superposition principle here). What is the concept behind Gauss’s law? Surround a charge distribution with an imaginary closed surface. Study the electric field at various points on the surface. Gauss’s law relates the field on all points on the surface and the total charge enclosed. Electric flux C. F. Gauss (1777 - 1855) Some things you will encounter: Gauss distribution Gauss’s law Gauss’s law Think of liquid flowing trough a surface. Definition: �. Consider a small area A perpendicular to E The electric flux is then given by ΦE = EA . If the area is not perpendicular, we only consider the � , E⊥ = E cos(φ) : ΦE = EA cos(φ) perpendicular component of E ΦE = EA German mathematician. Possibly one of the most influential ever. Contributions in statistics, calculus, linear geometry, geodesy, electrostatics, astronomy, …. Did not like to publish his work. Had he published all his results at once, he would have advanced science back then by approx. 50 years! � 100 x Child prodigy: in primary school added in second. x=1 ΦE = EA cos(φ) ΦE = EA cos(90◦ ) = 0 The total electric flux ΦE coming out of a closed surface is proportional to the total electric charge Qencl inside the surface, according to the relation � E⊥ ∆A = 4πkQencl = Qencl /�0 The sum represents the operation of dividing a surface A into small elements ∆A and then computing the small flux contributions. k = 1/4π�0 Note: If Qencl = 0 the total flux must be zero.� � A � = Qencl /�0. Ed In the continuum limit, we would write S Gauss’s law is the “integral form” of one of Maxwell’s equations. These 4 equations can be used to derive all E&M equations you will learn in this course! Derivation of Gauss’s law for a point charge Assume a single positive point charge q. Place it at the center of an imaginary sphere of radius R. The magnitude of the electric field E at every point of the surface is given by (why?) q E=k 2 k = 1/4π�0 R � is always perpendicular to the surface with total On the sphere E 2 area A = 4πR . Thus we obtain for the flux q ΦE = EA = k 2 · 4πR2 = 4πkq (spherical surface) R The flux is independent of R, it only depends on the charge q! A generalization to arbitrary surfaces follows by dividing the surface into small elements ∆A. Faraday ice pail Experiment to prove with high precision Gauss’s / Coulomb’s laws. Place charged ball inside conducting bucket. Close the lid; charges are induced in the walls of the container. Let the ball touch the bucket. The ball becomes part of the cavity surface. If Gauss’s law is correct, the charge of the inner surface (bucket+ball) must be zero. The ball must lose all its charge. Pull out the ball and measure. The charge is zero. Charges on conductors So far we have learned: In electrostatic situations (no charge motion) the electric field at every point within a conductor is zero (otherwise the charges would move). In a solid conductor, charges distribute on the surface. solid conductor with charge q’ solid conductor with charge q’ with a cavity an isolated charge q is placed inside the cavity Gaussian surface charges are entirely on the surface � = 0 inside E because the field is zero within the conductor, the field on the Gaussian surface must be zero. there cannot be charge on the cavity surface for the field to be zero at all points on the Gaussian surface, the cavity surface must have a net charge –q. Application: Electromagnetic shielding Why: To protect sensitive instruments. To prevent students from using their cellphones. Protects people from lightning when inside cavities (planes, cars). (Faraday cage) How? Surround the object to be protected with conducting material. The field redistributes the charges in the conductor, causing another electric field. Hence the field inside the box must be zero. movie