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# General Physics II PHY 114

WFU

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This 14 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 114 at Wake Forest University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/230739/phy-114-wake-forest-university in Physics 2 at Wake Forest University.

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Date Created: 10/28/15

Test Information for Test 3 Formulas to memorize Impedance Freguency amp 73 IAV wavelength Inductors 73 2 a Z f 8 L lt gtImsR 1T d 1mX AVmaXZ f kl 27239 Speed of Light Transformers wk fdb V 5quot AVi AV2 Re ection amp Refraction c300x10 ms N1 N2 t2 6 IIAV1 IZAV2 n sint9 r17 sin 9 Power amp Pressure Absorption 73 Sgt039 F P0 Re ection F 2P0 Formulas to know how to use but you need not memorize Constant LC Circuits RL Circuits 0 4nx10 7 TmA a1LC rLR Q Q0 cos wt 1 087 EM waves 1 1 eel139 E0 630 Impedance R 1 X lCa X La S E X B is 2 L 2 RLC Circuits 0 Z R XL Xc SgtcBO22u0 PSC tan 1XL XcR mo lJL C AVAV max AC Circuits sin wt I 1mX sinat gt l R2 a LC 4L2 Q Qoe39RtZL cos wt quotquot5 AV ms RMS values max 45 Other things to know re ectingtransparentabsorbing The order of the seven types of electromagnetic radiation 0 The order of at least six visible colors ROYGBV Inductors oppose changes in current capacitors in voltage What type of current gets through a capacitor Through an inductor How are E B and the direction of a wave related How does the power absorbed and pressure change if the target is partly That when you are in vacuum or air 71 1 for other materials 71 gt 1 Material for test 2 Chapter 32 Inductance Chapter 33 AC Circuits Chapter 34 Electromagnetic Waves Chapter 35 Re ection and Refraction Organization of the Test Part 1 Multiple Choice 20 points For each question choose the best answer 2 points each questions 110 Part 11 Short answer 20 points Choose two of the following questions and give a short answer 13 sentences 10 points each questions 1113 Part III Calculation 60 points Choose three of the following four questions and perform the indicated calculations 20 points each questions 1417 Test Information for Test 2 Formulas to memorize Magnetlc Force K h if L Ampere s Law Resistors FqEVXB M quE 1 1 B39dS 0inzuogo AV IR leLxB Z Z on Cl d 73 IAV 0 2 AV series RC Circuits 1 p Gauss s Law Faraday s R R1 R2 1 2 RC W1th Capac1t0rs Magnetism Law Parallel Idedt CJSB dA0 gdqs i i i Units AV QC s dt R R1 R2 V Nn C A Cs T NNm Formulas to know how to use but you need not memorize M m Hall Effect Loops BiotSavart el602gtlt103919C nMotion IB 1IAgtltB yo dsxf 712 2 2 RB AV B J 80 8854gtlt10 C Nm quot1quot q H mq U IAB 4n 2 yo4nx104 TmA mz m Solen01d Field from W1re RC Circutitrs Bin IL ONIg B L0 cos 91 COS 92 Q Q09 47m 1 C8 l e39W B Q 27m General Comments on Working Out Problems The test isn t exactly like webassign and therefore the problems won t be quite the same First and this is important I do take credit off for not showing units Keeping track of units is an important part of physics and often helps us recognize and check our mistakes In Webassign keeping calculations to two or three digits is always correct On tests you should generally follow the standard rules of signi cant digits However I don t mind if you keep an extra digit or even two beyond those given in the problem What I don t want to see is answers like E 36180359 Vm Rewrite answers like this in scienti c notation 362 X 107 Vm or better still 362 MVm Material for test 2 Chapter 28 DC Circuits Chapter 29 Magnetic Field and Forces Chapter 30 Sources of Magnetic Fields Chapter 31 Faraday s Law Chapter 32 Inductance Organization of the Test Part 1 Multiple Choice 20 points For each question choose the best answer 2 points each questions 110 Part 11 Short answer 20 points Choose two of the following questions and give a short answer 13 sentences 10 points each questions 1113 Part III Calculation 60 points Choose three of the following four questions and perform the indicated calculations 20 points each questions 1417 Math Review Geometrx Much of what you learned in high school geometry is either pretty intuitive or will not be needed for this class but many formulas you learned earlier for area or volume will come up a lot Three twodimensional shapes come up a lot rectangles triangles and circles For each of these shapes the perimeter P is the distance around it and the areaA is the total size of the content For polygons like the triangle and the rectangle the perimeter is just the sum of the sides while for the circle it is 27rtimes the radius We will also be working with threedimensional objects A lot of our shapes will be generalized cylinders which are formed by taking any twodimensional shape and stretching it vertically These objects have a volume that is equal to their base area times their height VAh and a lateral surface area Sm Ph39 however they also have additional surface areas at each end The cases we will most commonly encounter are rectangular prisms boxes and circular cylinders for which the formulas below apply The cylinder will have an additional surface area of 7W2 on each end and the surface area on any face of the box can be calculated from the rectangle area formula For a cube of size a the volume is V a3 and the total surface area is S a2 One of the most common shapes we will encounter is a sphere and for this you simply memorize the formula for the volume and surface area We may also occasionally encounter a cone for which the formula for the volume is simple but the lateral surface area is kind of complicated V 7rr2h r For the cone there is an additional surface area of 7W2 coming from the at surface Algebra Most of algebra is pretty straightforward One thing that comes up fairly often in physics is the quadratic formula that is nding the solutions of the equation ax2 bx c 0 This has solution 7 7b i b2 74616 2a One other common expression that comes up is W lxla not x Powers and exponents come up a lot Some trivial formulas that you should remember are l x l xlx x12 and xquot7 m Jewquot and 96 xm Very often especially when working with calculus we get expressions with e raised to various powers where e is the base of the natural logarithm The three rules above apply in particular to ex ex y and e y a One other useful fact to remember is that the inverse function of ex is the natural logarithm lnx so we have ex y ltgt x ln y Trigonomet In trigonometry we very often need to find the angle and hypotenuse in a right triangle from the legs or alternatively given the angle and hypotenuse we need to find the legs The following formulas c can help find these quickly czazb2 sin 2 b or 6 tan 0 a a a cos 0 c The three trigonometric relations are often memorized by the mnemonic SOHCAH TOA which means sine is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite over adjacent From these we can very easily prove the very useful identities sin2 0 cos2 0 1 and tan 0 sin cos 0 For these three basic trigonometric functions certain values come up often enough that it is helpful to know an 16 111 COS tan their values Rather than memorizing all of these it is 0 0 0 1 0 easier to memorize the pattern for sine xZZ for n 0 30 g f l 2 3 4 and then memorize that the cosine is the same 450 g g 1 thing backwards and tangent is the ratio In addition if D j i you remember that sine and tangent are odd functions 60 3 73 7 NE while cosine is even you can get the values for the 90 1 0 00 negatives of each of these sin7x isinx cos 7x cosx and tan7x itanx In addition to the three standard trigonometric functions above there are three others cos0 i l sec0 csc0 and cot i cos0 s1n0 s1n0 tan0 Some day you should memorize these as well but they don t come up as often We also often encounter the inverse functions typically written as sin 1 x and so on These can be evaluated with a calculator or for certain special values by using the table above for example sin 1 g In trigonometry there are two standard ways of measuring angles in degrees or radians There are 360 in a circle and 211 radians Degrees are most commonly used when you are talking about a physical angle Radians are always used when you are working with calculus Most calculators can be used to calculate using either of these units Make sure to check which one your calculator is set for before you begin a calculation The sum of angles formula tends to come up a lot in physics so let s lay it out for cosine and sine cos x i y cos xcos y I sin xsiny sin xi y sin xcos y i cos xsin y I don t expect you to memorize these but I will use them occasionally when I need them From these we can easily prove the double angle formulas as well cos2x cos2 xisin2 x 2cos2 x71172sin2 x sin 2x 2 sin xcos x Vectors Vectors come up so often in physics that most physicists learn them in physics classes not math A vector is a quantity that has both a magnitude and a direction like displacement velocity force or acceleration A quantity with a magnitude but no direction is called a scalar In this class I will denote vectors by putting them in bold face V though when I write it I normally draw some sort of arrow over it 7 Scalars will be generally denoted by math italic font 3 In three dimensions a vector has three components vxvyvz or VVivyjvzk where i j and 1 are unit vectors in the x y and z directions respectively The little roofs over the symbols are read hat and signify a vector of unit length The length of a vector V can be determined using the 3D equivalent of the Pythagorean theorem It is denoted by V or just plain v and can be computed using lvlzvz lv v vzz It is very common in two dimensions to discuss the length v and the angle 9 of a vector V The angle is normally measured counterclockwise starting from the xaXis With the help of the trigonometric formulas above it is not too difficult to convert from components to directions and vice versa For example suppose we were given the vector V 72i 7 3i and asked to compute the magnitude v and direction 9 of this vector The magnitude would be VWJ 3606 n 6 AI To find the direction make a sketch of the vector as shown at right The angle a at the origin can be seen to be a tan 1 32 5631quot The angle 9 however should start from the xaXis and is therefore 180 degrees more than this for a total of 23631quot Adding and subtracting vectors is pretty straightforward To add two vectors you simply make a little parallelogram out of them by copying each vector and placing its tail on the head of the other vector as sketched at right In components the vectors can be added component by component that is vw vwvwivy wyjvzwzk Subtracting vectors in component notation is similarly easy Multiplying with vectors is a bit more complicated First of all you can multiply or divide a vector V by a scalar s in a straightforward manner Geometrically SV points in the same direction as V but is 3 times longer In components SV xvi svyj svzk Such multiplication is commutative SV E VS associative rsv rs V and distributive rsV rVSV and SVw svsw Somewhat trickier is vector multiplication It turns out there are two ways to multiply two vectors called the dot and cross product and it is important to keep them straight The dot product is written Vw pronounced v dot w and produces a scalar quantity and is de ned by V V V w vw cos 0 where Bis the angle between the two vectors In components it is easy to calculate Vw vxw VyWy vzwz The other way to multiply two vectors is called a cross product written V gtltw pronounced v cross w and produces a vector quantity Like all vectors it will have a magnitude and a direction The magnitude is given by VgtltW vwsin The direction is de ned to be perpendicular to both V andw and chosen in accordance with the righthand rule The righthand rule works as follows Put your right hand out straight but with your thumb pointed out and make your ngers point in the direction of the vectorV Now twist your wrist so that when you start to curl your ngers your ngers will end up pointing in the directionw At this point your thumb is pointing in the direction of V gtltw The only ambiguity occurs when V and w point in the same directions parallel or exactly opposite directions antiparallel but in this case sin 0 0 and V gtltw 0 In the picture above the crossproduct points out ofthe paper In components the cross product can be computed using the determinant i j k Vgtltwdet vx vy VZ vywzivzwy1vzwxivxwz3vxwy ivywxk w wy wz It s messy but occasionally it is useful When you combine dot or cross products with scalar multiplication or vector addition it is easy to show that it is still associative and distributive so that we have SVwsVw VwxVXwx xVwxVxw svgtltwsVgtltw VwgtltxVgtltxwgtltx xgtltVwxgtltVxgtltw Also the dot product is commutative However the cross product is anticommutative VwwV but VXWWgtltV Triple vector products involving three vectors combined with dot and crossproducts are messier and we generally will not encounter them in this class Differentiation Most of physics is written in terms of differential equations and it is important to be able to take derivatives of even complicated functions quickly Fortunately it turns out that a few rules allow you to take the derivative of any function fx no matter how complicated The derivative of a function x is denoted either by d I x E x dx f lt f lt Higher derivatives can be denoted by similar notation d 2fltxgtfquotltxgt dxl xEf39quotx etc 613 abc3 It is important to know the derivatives of a few simple functions from which you can build up the derivatives of arbitrarily complicated functions The basic functions you need to know to get derivatives of more complicated ones are a and n are arbitrary real numbers d d d l a0 xquotnxquot 1 e e lnx dx dx dx dx x You should memorize all four of these In addition the following derivatives of trigonometric and inverse trigonometric functions come up a lot 61 d d 2 s1nxcosx cosxis1nx tanxsec x dx dx dx isin lx icos lxi d tan lxi 1 dx 17x2 dx hiya dx lx2 You should memorize at least the derivative of sine and cosine To take the derivative of more complicated functions you need rules for taking the derivative of sums differences products and quotients of functions as well as for functions of functions I I I I I 7 I fig f i g39 fg f g fg39 gg fg fg f gg Keep in mind that in mathematics only the variable will be denoted by a letter like x but in physics there will generally be a lot of constants denoted by letters as well As an illustration let s find the derivative i NeAsinkx2d7 Nielsinkxz NeAsinkx2 iASinkx2 dx dx dx iNAeASinWw cos 1ch l 72NAkxeAsmWw cos chZ 1 Integration There are in fact two different types of integration called de nite and inde nite integration If f x is an arbitrary function of x then the inde nite integral F x sometimes called an antiderivative is de ned to be that function F whose derivative is f that is to say F 39 f It is denoted by putting no limits on the integration symbol d F x x dx ltgt F x x H f W H m Because the derivative of a constant is zero the inde nite integral F is de ned only up to a constant and hence in proper formalism the answer to an inde nite integration should always look likeF x C where C is an unspeci ed constant of integration Often this constant can be ignored Make sure you keep the differential dx in your integration if you ever change variables this factor can be important A de nite integral has limits of integration x a and b and is de ned as the area under the curve fx starting from the point x a to the point x b The fundamental theorem of calculus relates the de nite integral to the antiderivative namely rmwmw Because the de nite integral involves the dz rence of F between the two endpoints the constant of integration C always cancels out and is therefore irrelevant in a de nite integral In contrast to differentiation there are no simple rules to perform integration Generally you do your best to manipulate your integration into a relatively simple form then you either immediately recognize the integral or you look it up in an integral table or better yet learn how to use Maple to do integration for you Many integrals cannot be written in simple closed form in which case modern computers can numerically calculate the result often to high accuracy for most realistic problems A few rules that allow you to nd indefinite integrals will help you If a is an arbitrary constant and f x and gx are functions whose integrals are F x and Gx respectively then it is not hard to show that fafltxgtdxaffxdxaFx ffltxagtdxFxa flfltxgtigltxgtldxffltxgtdxifgltxgtdxFxiGx Note also that it is commonly possible to do integration by substitution namely let x be 17 a EFb7Fa where F39x any function of a new variable y so x g y and then substitute this in However note that the differential dx transforms to dx d g y g39y dy so the new integral becomes ffxdxffgydgyffgyg39ydy Integral Tables The first four integrals on the left side and perhaps the integrals of cosine and sine should be memorized For more complicated integrals I recommend that you use Maple to assist you with the integral or you can look it up in the following table Note that all inde nite integrals have an implied C which can be ignored whenever you are performing a definite integral In all expressions below it is assumed that a b and n are real nonzero constants fdxx n1 fxquotdx x nil n1 flnxl femdx gear fxemdx 7i2ea a a cos ax dx lsinax a ftan ax dx illn lcos a fsin2 ax dx i 7 Sm zax 2 4a f cos2 ax dx i Sm zax 2 4a an ax tan2 ax dxt 7x f lt gt a ln melx2 61 f fi fitan li agt0 x2a J J dx i 1 x7 fxziaiz lnxjr agt0 fL elixir2 aVaixZ d f dez 7 aixz agt0 aix fxfixalnx2al f xdx l x2a32 x2a d l elixir2 aixz agt0 x bia xsin dx ficos ax izsin a a xcos dx isin ax Licos ax a a Partial Derivatives and Multiple Integrations In mathematics it is common to work with only one variable which we typically call x but in physics it is common to have at least three dimensions x y and z and sometimes four including time t Hence quantities are commonly functions of several variables at once we might write such a function as f x y z for example When differentiating it is then common that we need to take derivatives in more than one direction and in such cases we need a notation of partial derivatives A partial derivative is just like an ordinary derivative except we treat every variable except the one we are differentiating with respect to as a constant For example the partial derivative with respect to x is written as fxhyz7fxyz h Just as with ordinary derivatives we rarely use this definition instead just using our ordinary rules for differentiating For example suppose we had the function 8 Efxyz3 mg f x y z Alt2xy 7 z2 and we were asked to take various partial derivatives of it When taking the x derivative for example we would treat y and 2 as constants and hence the first term would yield a derivative of My while the second term would yield nothing since it is constant So the three partial derivatives of this would be BfBx2Ay BfBy2Ax and 8f8272Az It is also very common that in physics we need to perform a multipledimensional integration these will always be defmite integrals In such circumstances the integration should be worked from the inside out that is you first need to do the innermost integral then work your way out to the outermost integral One of the hardest parts of doing such an integration is setting it up in the first place since depending on the shape of the region you are integrating over it may be very difficult to figure out the limits of integration Often the limits of the inner integration will depend on the value of the variable in the outer integration When doing the innermost integrations all of the outer variables can be treated as constants Let s do an example to see how this works Suppose we are faced with a a 2 2 2 f0 dxfo dylta x y We start by performing the y integral since that is inside We treat a and x both as constants and hence we have Ladya2x2y2a2yxzy y3 gahraxz We then can easily finish the xintegration jjdxjjdyltaz 362 y2 foaabcga3 axz ga3x ax3a 1 0 3 614 ulp t a H I a Many multidimensional integrals can be simplified greatly when there are symmetries involved For example suppose you have to perform an integration over a disk of radius R but the integral is such that the integral depends only on the distance r from the center of the disk The integration can be performed by dividing the disk into thin annuli basically a slightly thickened circle of radius r and thickness dr The thin annulus can be thought of as a rectangle of length 2727 and width dr that has been bent into a circle and therefore has area 272mlquot Hence the area differential dA can be replaced by 272761quot For example to calculate the integral of 1 Ir2 a2 over a disk we would have R R fif 2w r2 612 27r R2 612 7a 0 V e V This method can be used to calculate volume integrals for cylinders cones and spheres as well However for spheres the most common situation is one where you must perform an integration over a spherical volume and the integral depends again only on the distance r from the center of the sphere In this case the most efficient way to do the computation is in terms of thin spherical shells having a radius of r and thickness dr The volume of this thin shell is the area of the shell 47272 times the thickness so we replace the volume element deith 47272dr For example if you are told that the total charge density of a sphere of radius R is given by pr Arz then the total charge of the sphere is R4WAR5 5 0 Q fpdV 2Ar24m2dr 4 A

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