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# PARADIGMS IN PHYSICS SYMMETRIES PH 320

OSU

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This 7 page Class Notes was uploaded by Mylene Russel on Monday October 19, 2015. The Class Notes belongs to PH 320 at Oregon State University taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/224533/ph-320-oregon-state-university in Physics 2 at Oregon State University.

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Date Created: 10/19/15

PH 320 Day 6 1 Quadrupoles Continue the visualization exercise from yesterday by considering the poten tial due to 4 charges arranged in a square7 either all positive7 or 2 dipoles placed together so that diagonally opposite charges are the same 2 Step functions This wasn t actually done until Day 7 Piecewise functions7 such as the potential from today7s air track derno7 are often written in the form km 7 cl2 0 lt lt cl fa 0 clltlt02 km 7 Cg2 Cg lt lt L An alternative is to use the Heaviside function 97 also called the step function7 Which is de ned to be 0 if lt 0 and 1 if gt 0 For instance7 the function above becomes for 0 lt lt L x km 7 cl2cl 7 km 7 022 7 02 Note the use of 9 to turn a function both on and oil7 since 9717 is a step down7 rather than up This expression also motivates the de nition although the value of 9 at 0 does not in fact rnatter PH 320 Day 12 1 A uniformly charged ring Yesterday we derived the formula 1 1 AW mm W for the potential due to a uniformly charged ring of radius R In cylindrical coordinates ldm Rda where R is the radius of the ring and we use primes to emphasize that we7re referring to W rather than F Thus 11 On the axis The above integral is quite dif cult in general so we consider special cases First of all let7s determine the potential along the axis In this case P and W form the legs of a right triangle with lengths m and Wl B respectively This implies that Wim xR222 which brings the integral to the form W 7quot 1 ARd 1 27MB 1 Q 2 0 47m MR2 22 47m MR2 22 47m MR 22 where Q 27TRA is the total charge on the ring It is instructive to consider the limiting case when 2 0 which yields 1 1 Q T 47TEO R which is what one expects since all of the charge is a distance R from the origin What about the limit as 2 a 0 lf 2 lt R we have V0 1 7 1 71 7122 324i xR2ZZTR 117131 2R2 8R4 and we have Q 1 122 324 V 7 7777 iii 2 4m R 2 R3 SR5 which of course agrees with our previous calculation for z 0 Consider the rst two terms of this expansion for small 2 There is no linear term which means that the potential has a max or a min at z 0 this is an equilibrium point Including the quadratic term results in a parabola 7 which opens up or down depending on the sign of Q Choosing Q gt 0 cor responds to a situation with like charges and like charges repel a positively charged particle placed along the z axis near the origin will be repelled away from the plane of the ring The origin is therefore an unstable equilibrium point the parabola opens downwards Conversely Q lt 0 corresponds to op posite charges which attract such a particle would be attracted towards the origin and would oscillate back and forth through the origin In this case the origin is a stable equilibrium point the parabola opens upwards Another interesting limit is z a 00 what happens then lf 2 gt R then 1 i 1 1 i 1 1 1R2 3R4 Wiz z 222 82 so that Q 1 1 R2 3 R4 V2 7 777 iii 47reo z 223825 The leading term is now 1 Vz 9 47TEO 2 which is just the potential due to the total charge to this order the ring structure is not apparent Furthermore the next order term is zero there is no dipole term which makes sense since the ring only has positive charge 2 A uniform disk What is the potential due to a uniformly charged disk First of all we need to generalize our previous expressions to surface densities Chop up the region7 determine the potential on each one7 and add them up7 obtaining 1 UdA 47TEO l 7 M W is But we have already seen that the potential on the axis due to a ring of radius r is VA 7 1 Qring F1119 47760 T2 22 and the Charge on such a ring of width dr is Qring 027TT dr Adding up all the rings which make up a disk of radius R7 we obtain V i R 039 27Tr dr Z 7 0 47m T222 R 7U 27nr 2 22 47TEO 0 L lt27nR2 22 7 2W2 47TEO This expression can be analyzed as before in terms of power series when 2 lt R or 2 gt R PH 320 Day 5 1 Dipoles Today was devoted to understanding the physics of the mathematical com putations done yesterday Do the terms have the correct dimensions Do the leading terms give you the behavior you expect Where is each series valid 2 Visualization of potentials The dipole examples were also used to motivate trying to visualize the equipo tential surfaces These examples are inherently 3 dimensional plotting func tions of 3 variables is di icult Think about the shape of the equipotential surfaces are for various potentials PH 320 Day 1 1 Introduction ldealizations are concepts like point charges in nitely thin wires massless rods in nitely sharp barriers etc 7 physics is full of them They are enormously useful but the real world is messier and we need to understand how to deal with this We engage in approximations all the time and we need to know when and how to approximate and when and how to idealize We7ll use electrostatics as an example of how to use approximations and well learn how to describe space with vectors Symmetry is an enormously powerful concept in physics the simple ob servation that a small stone dropped in a pond produces circular ripples leads to the powerful conclusion that the water is isotopic responds the same in all directions Nother7s theorems about translational and rotational invariance leading to conservation of energy and angular momentum is a fundamental statement about our universe group theory places limitations on the nature of atomic orbitals or the properties of fundamental particles based entirely on symmetry arguments We7ll use the concept of symmetry to calculate electric elds of charges and magnetic elds of currents 2 Electric and Gravitational Potentials Recall that the electric potential due to a point charge at the origin is given by 1q 747139607 Similarly7 the gravitational potential of a point mass is given by m I 7G 7quot These quantities have the following units Nrnz G 667 10 X kgz 12 02 60 885gtlt10 Nmz Nrn V N c I N Nm kg and dimensions G MTL 2L2 7 L3 M2 MT2 Q2 7 QZTZ ED MTL 2L2 ML3 V MTL 2L 7 ML2 Q 7 QT I MTL 2L 7 L2 M T2

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