INTRO ELECTRICITY & MAGNETISM
INTRO ELECTRICITY & MAGNETISM PHYS 260
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This 10 page Class Notes was uploaded by Eleazar Batz on Wednesday September 30, 2015. The Class Notes belongs to PHYS 260 at Western Kentucky University taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/216684/phys-260-western-kentucky-university in Physics 2 at Western Kentucky University.
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Date Created: 09/30/15
Dr Barzilov s MWF Section Physics A Math Review Math Tools Q What do you get if you cross an apple with a coconut A Apple coconut sine theta Resultant Vector I Problem You wish to find the vector sum of vectors A and E Pictorially this is shown in the figure on the rig Mathematically you want to break the vector into x y and maybe 2 components and find the resultant vector Let A 62 7 9 And if 122 69 HAEHJlt612V 7 62 y 7 6 tan 9 7 612 System of Eguations I Let6x7y 15 I And 4X 3y 9 I Now find X and y which satisfy these equations Use method of minorsquot o M 9 39 Wantincn El System of Equations cont d The solution for x is found by creating a minor wherein the constants in the equation are substituted in place of the x value and the value of the minor is found It is then divided by the value of the determinant 15 7 9 3 Z 153 79 det 46 Then back substitute the value of x into my initial equation and solve for y x 2347 6 2347 7y 15 and solve for y y 1304 OR use the minors again 6 15 4 9 269 415 det 46 01304 Larger Systems Larger systems are broken down into their resultant minors Forexampe 6X7y102 12 9X15y2260 5x12y10215 6 7 10 15 2 9 2 9 15 det 9 15 2 6 7 10 2 3434 12 10 5 10 5 12 5 12 1 12 7 10 6015 2 15 12 10 X det Spherical Coordinates I Cartesian coordinates X y z Spherical coordinates r 6 Cylindrical Coordinates Eylindric 3 Coordinates Z Cartesian coordinates X y z Spherical coordinates r g z xrcos yrsin zz 1 rx2 yzE lx2 y2 tan 1l X Partial Derivatives I So what is the difference between quotd andB the variable X d like ddX means the function only contains When the function contains not just X but may be y and 2 we use the partial differential a For example fxyzxyz 31i Bx Bx xyz W Note that the variables y and z are held constant when the differential operator acts on the function What is the solution to 3m 7 a i 2 3y SK 0 Introduction to Del You can now make a special differential operator called del Del is defined as lt1 ll Treat del as a vector and thus you can apply the dot and cross products to them But first let s recall the dot and cross product The dot or Scalar Product The scalar product is A 1135 ay az2 defined as the multiplication of two and vectors in such a way l b 21 Ab 2 that result is a vector 1 yy z Then 3 E 1le ayby 1le I 0is the angle between A and B or 31 cos 6 Cross or Vector Product The vector product is the multiplication of two vectors such that the result is a vector and furthermore the resulting vector is perpendicular to the either of the two original vectors The best way to find a vector product is to set it up as a determinant as shown on the right 22 9 2 AX aX ay az bx by bz gtl x E aybz azby y c axbz lt1sz 9 axby aybx Xx 9 0 His the angle between A and B First application of del Gradient The gradient is defined as the shortest or steepest path up a mountain or down into a valley Let s go back to fgtltyz then Vf VW yzf xz xy2 You see that grad 1 makes a vector which points in a particular direction Also note that grad 1 takes a scalar function and makes a vector of it A particle which travels through a region of space wherein the potential energy Ugtltyz varies as a function of space has a force exerted on it equivalent to a 8U 8U 8U F VU 7 7 7 l 623 ayjyl azjz The Scalar Product and V I We can apply V to the scalar product ie o VA where A is some vector I VA is called the divergence of A or divAquot I Geometrically we are discussing if A is diverging from some central point A is diverging is m from a central dlvergmg from a central point potnt so so DivA is equal to DivA is equal some value to zero The Vector Product and V I We can apply V to the vector product ie o VxA where A is some vector I VXA is called the curl of A or curlA I Geometrically we are discussing if A is curling around some central 1 I A is curlin A Is not curling around a 9 around a central central point potnt so so curlA is curlA IS equal equal to some to zero value What about A V and A x V I These two products do not describe the geometrical properties I A V is not equal to V A due to the nature of the differential operator I A VU would be equivalent to A F where F is a force described by VU I Likewise for A x VU Two Special Integrals I Integrating over a closed loop d I Integrating over a closed surface B dEi Integrating over a closed loop The loop can be circular or rectangular O d rd6 lSEd Blt2zrrgt From 0 to 27 Looping from Point A to Point D B using straight line segments C d dsAfrdsB idscfrdsD i Closed Surface Integral The vector nhat is normal to the surface This means that da must consist of the differential distance in the phidirection multiplied by the differential distance in Theta is integrated from 0to 7 the theta direction so and phi is integrated from 0 to 27 do Rsin 6d Rd Th f if Ed d I PI 39 ere ore 9P9 5 on y on R2 sm 6 d6 W5 the E d5 E47z1 2
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