share/physics/dimension.tex

   1 % TeX code generated by batTeX. Don't edit this file; edit the
   2 % input file dimdem2.tex instead.
   3 \documentclass[12pt]{article}
   4 \usepackage{mathptm}
   5 %\usepackage{euler}
   6 \usepackage{battex}
   7 \usepackage{color}
   8 %\usepackage{fleqn}
   9
  10 \title{A new Maxima package for dimensional analysis}
  11  \author{Barton Willis \\
  12   University of Nebraska at Kearney \\
  13   Kearney Nebraska}
  14
  15 \begin{document}
  16
  17 \maketitle
  18
  19
  20 \subsubsection*{Introduction}
  21
  22 \noindent This document demonstrates some of the abilities
  23 of a new Maxima package for dimensional analysis.  Maxima
  24 comes with an older package dimensional  analysis that is
  25 similar to the one that was in the commercial Macsyma system.
  26 The software described in this document differs greatly from
  27 the older one.
  28
  29 The new dimensional analysis package was written by Barton Willis of
  30 the University of Nebraska at Kearney. It is released under the terms of
  31 the General Public License GPL. You may contact the author at
  32 \begin{verb} willisb@unk.edu \end{verb}.
  33
  34
  35 \subsubsection*{Installation}
  36
  37 To use this package, you must first download the file
  38 {\tt dimension.mac}; it may be found at
  39 \begin{verb} www.unk.edu/acad/math/people/willisb \end{verb}.
  40 After downloading, copy it into a directory that Maxima
  41 can find.
  42
  43
  44 \subsubsection*{Usage}
  45
  46 To use the package, you must first load it. From a Maxima prompt, this
  47 is done using the command
  48
  49 \vspace{0.1in}
  50
  51 \begin{mcline}{c1}
  52 load("dimension.mac");
  53 \end{mcline}
  54
  55 \begin{mdline}{d1}
  56 dimen1.mac
  57 \end{mdline}
  58
  59 \vspace{0.1in}
  60
  61 \noindent To begin, we need to assign  dimensions to the
  62 variables  we want to use. Use the {\tt qput} function to do this;
  63 for example,  to declare $x$ a length, $c$ a
  64 speed, and $t$ a time, use the commands
  65
  66 \vspace{0.1in}
  67
  68 \begin{mcline}{c2}
  69 qput(x, "length", dimension)$
  70 \end{mcline}
  71
  72 \begin{mcline}{c3}
  73 qput(c, "length" / "time", dimension)$
  74 \end{mcline}
  75
  76 \begin{mcline}{c4}
  77 qput(t, "time", dimension)$
  78 \end{mcline}
  79
  80
  81 \vspace{0.1in}
  82
  83 \noindent We've defined the dimensions length and time to be
  84 strings; doing so reduces the chance that they will conflict
  85 with other user variables. To declare a dimensionless  variable
  86 $\sigma$, use $1$ for the dimension. Thus
  87
  88 \vspace{0.1in}
  89
  90 \begin{mcline}{c5}
  91 qput(sigma,1,dimension)$
  92 \end{mcline}
  93
  94 \vspace{0.1in}
  95
  96 \noindent To find the dimension of an expression, use the
  97 {\tt dimension} function. For example
  98
  99 \vspace{0.1in}
 100
 101 \begin{mcline}{c6}
 102 dimension(4 * sqrt(3) /t);
 103 \end{mcline}
 104
 105 \begin{mdline}{d6}
 106 {{1}\over{time}}
 107 \end{mdline}
 108
 109 \begin{mcline}{c7}
 110 dimension(x + c * t);
 111 \end{mcline}
 112
 113 \begin{mdline}{d7}
 114 length
 115 \end{mdline}
 116
 117 \begin{mcline}{c8}
 118 dimension(sin(c * t / x));
 119 \end{mcline}
 120
 121 \begin{mdline}{d8}
 122 1
 123 \end{mdline}
 124
 125 \begin{mcline}{c9}
 126 dimension(abs(x - c * t));
 127 \end{mcline}
 128
 129 \begin{mdline}{d9}
 130 length
 131 \end{mdline}
 132
 133 \begin{mcline}{c10}
 134 dimension(sigma * x / c);
 135 \end{mcline}
 136
 137 \begin{mdline}{d10}
 138 time
 139 \end{mdline}
 140
 141 \begin{mcline}{c11}
 142 dimension(x * sqrt(1 - c * t / x));
 143 \end{mcline}
 144
 145 \begin{mdline}{d11}
 146 length
 147 \end{mdline}
 148
 149 \vspace{0.1in}
 150
 151 \noindent {\tt dimension} applies {\tt logcontract} to its
 152 argument; thus expressions involving a difference of logarithms
 153 with dimensionally equal arguments are dimensionless; thus
 154
 155 \vspace{0.1in}
 156
 157 \begin{mcline}{c12}
 158 dimension(log(x) - log(c*t));
 159 \end{mcline}
 160
 161
 162
 163 \begin{mdline}{d12}
 164 1
 165 \end{mdline}
 166
 167
 168 \vspace{0.1in}
 169
 170 \noindent {\tt dimension} is automatically maps over lists. Thus
 171
 172 \vspace{0.1in}
 173
 174
 175
 176 \begin{mcline}{c13}
 177 dimension([42, min(x,c*t), max(x,c*t), x^^4, x . c]);
 178 \end{mcline}
 179
 180
 181
 182 \begin{mdline}{d13}
 183 \left[ 1,length,length,length^4,{{length^2}\over{time}} \right]
 184 \end{mdline}
 185
 186
 187
 188 \noindent When an expression is dimensionally inconsistent,
 189 {\tt dimension} should signal an error
 190
 191
 192
 193 \begin{mcline}{c14}
 194 dimension(x + c);
 195 \end{mcline}
 196
 197 {\em Expression is dimensionally inconsistent.}
 198
 199
 200 \begin{mcline}{c15}
 201 dimension(sin(x));
 202 \end{mcline}
 203
 204 {\em Expression is dimensionally inconsistent.}
 205
 206 \vspace{0.1in}
 207
 208 \noindent An {\em equation\/} is dimensionally correct
 209 when either the dimensions of both sides match or
 210 when one side of the equation vanishes.  For example
 211
 212 \begin{mcline}{c16}
 213 dimension(x = c * t);
 214 \end{mcline}
 215
 216 \begin{mdline}{d16}
 217 length
 218 \end{mdline}
 219
 220 \begin{mcline}{c17}
 221 dimension(x * t = 0);
 222 \end{mcline}
 223
 224 \begin{mdline}{d17}
 225 length\,time
 226 \end{mdline}
 227
 228
 229 \vspace{0.1in}
 230
 231 \noindent When the two sides of an equation have
 232 different dimensions and neither side vanishes,
 233 {\tt dimension} signals an error
 234
 235 \vspace{0.1in}
 236
 237
 238
 239 \begin{mcline}{c18}
 240 dimension(x = c);
 241 \end{mcline}
 242
 243 {\em Expression is dimensionally inconsistent.}
 244
 245 \vspace{0.1in}
 246
 247 \noindent The function {\tt dimension} works with derivatives and
 248 integrals
 249
 250 \vspace{0.1in}
 251
 252 \begin{mcline}{c19}
 253 dimension('diff(x,t));
 254 \end{mcline}
 255
 256
 257
 258 \begin{mdline}{d19}
 259 {{length}\over{time}}
 260 \end{mdline}
 261
 262 \begin{mcline}{c20}
 263 dimension('diff(x,t,2));
 264 \end{mcline}
 265
 266
 267
 268 \begin{mdline}{d20}
 269 {{length}\over{time^2}}
 270 \end{mdline}
 271
 272 \begin{mcline}{c21}
 273 dimension('diff(x,c,2,t,1));
 274 \end{mcline}
 275
 276
 277
 278 \begin{mdline}{d21}
 279 {{time}\over{length}}
 280 \end{mdline}
 281
 282 \begin{mcline}{c22}
 283 dimension('integrate (x,t));
 284 \end{mcline}
 285
 286 \begin{mdline}{d22}
 287 length\,time
 288 \end{mdline}
 289
 290 \vspace{0.1in}
 291
 292 Thus far, any string may be used as a dimension; the other
 293 three functions in this package,
 294 \begin{verb} dimension_as_list  \end{verb},
 295 \begin{verb} dimensionless \end{verb}, and
 296 \begin{verb} natural_unit \end{verb} all require that each
 297 dimension is a member of  the list
 298 \begin{verb} fundamental_dimensions \end{verb}. The default value  is of
 299 this list is
 300
 301 \vspace{0.1in}
 302
 303 \begin{mcline}{c23}
 304 fundamental_dimensions;
 305 \end{mcline}
 306
 307
 308
 309 \begin{mdline}{d23}
 310 \left[ mass,length,time \right]
 311 \end{mdline}
 312
 313
 314 \noindent A user may insert or delete elements from this list.
 315 The function \begin{verb} dimension_as_list \end{verb} returns the dimension
 316 of an expression as a list of the exponents of the
 317 fundamental dimensions. Thus
 318
 319 \vspace{0.1in}
 320
 321
 322
 323 \begin{mcline}{c24}
 324 dimension_as_list(x);
 325 \end{mcline}
 326
 327
 328
 329 \begin{mdline}{d24}
 330 \left[ 0,1,0 \right]
 331 \end{mdline}
 332
 333 \begin{mcline}{c25}
 334 dimension_as_list(t);
 335 \end{mcline}
 336
 337
 338
 339 \begin{mdline}{d25}
 340 \left[ 0,0,1 \right]
 341 \end{mdline}
 342
 343 \begin{mcline}{c26}
 344 dimension_as_list(c);
 345 \end{mcline}
 346
 347
 348
 349 \begin{mdline}{d26}
 350 \left[ 0,1,-1 \right]
 351 \end{mdline}
 352
 353 \begin{mcline}{c27}
 354 dimension_as_list(x/t);
 355 \end{mcline}
 356
 357
 358
 359 \begin{mdline}{d27}
 360 \left[ 0,1,-1 \right]
 361 \end{mdline}
 362
 363 \begin{mcline}{c28}
 364 dimension_as_list("temp");
 365 \end{mcline}
 366
 367
 368
 369 \begin{mdline}{d28}
 370  \left[ 0,0,0 \right]
 371 \end{mdline}
 372
 373
 374 \vspace{0.1in}
 375
 376 \noindent In the last example, "temp" isn't an element of
 377 \begin{verb} fundamental_dimensions \end{verb}; thus,
 378 \begin{verb} dimension_as_list \end{verb}
 379 reports that "temp" is dimensionless. To correct this, append "temp" to the list
 380 \begin{verb} fundamental_dimensions \end{verb}
 381
 382 \vspace{0.1in}
 383
 384
 385
 386 \begin{mcline}{c29}
 387 fundamental_dimensions : endcons("temp", fundamental_dimensions);
 388 \end{mcline}
 389
 390
 391
 392 \begin{mdline}{d29}
 393 \left[ mass,length,time,temp \right]
 394 \end{mdline}
 395
 396
 397 \vspace{0.1in}
 398
 399 \noindent Now we have
 400
 401 \vspace{0.1in}
 402
 403
 404
 405 \begin{mcline}{c30}
 406 dimension_as_list(x);
 407 \end{mcline}
 408
 409
 410
 411 \begin{mdline}{d30}
 412 \left[ 0,1,0,0 \right]
 413 \end{mdline}
 414
 415 \begin{mcline}{c31}
 416 dimension_as_list(t);
 417 \end{mcline}
 418
 419
 420
 421 \begin{mdline}{d31}
 422 \left[ 0,0,1,0 \right]
 423 \end{mdline}
 424
 425 \begin{mcline}{c32}
 426 dimension_as_list(c);
 427 \end{mcline}
 428
 429
 430
 431 \begin{mdline}{d32}
 432 \left[ 0,1,-1,0 \right]
 433 \end{mdline}
 434
 435 \begin{mcline}{c33}
 436 dimension_as_list(x/t);
 437 \end{mcline}
 438
 439
 440
 441 \begin{mdline}{d33}
 442 \left[ 0,1,-1,0 \right]
 443 \end{mdline}
 444
 445 \begin{mcline}{c34}
 446 dimension_as_list("temp");
 447 \end{mcline}
 448
 449
 450
 451 \begin{mdline}{d34}
 452 \left[ 0,0,0,1 \right]
 453 \end{mdline}
 454
 455
 456 \vspace{0.1in}
 457
 458 \noindent To remove "temp" from
 459 \begin{verb} fundamental_dimensions \end{verb}, use the {\tt delete} command
 460
 461 \vspace{0.1in}
 462
 463
 464
 465 \begin{mcline}{c35}
 466 fundamental_dimensions : delete("temp", fundamental_dimensions)$
 467 \end{mcline}
 468
 469
 470 \vspace{0.1in}
 471
 472
 473 The function {\tt dimensionless} finds a {\em basis\/} for the
 474 dimensionless quantities that can be formed from a list of
 475 dimensioned  quantities.  For example
 476
 477 \vspace{0.1in}
 478
 479
 480
 481
 482 \begin{mcline}{c36}
 483 dimensionless([c,x,t]);
 484 \end{mcline}
 485
 486
 487 Dependent equations eliminated:  (1)
 488
 489
 490 \begin{mdline}{d36}
 491 \left[ {{c\,t}\over{x}},1 \right]
 492 \end{mdline}
 493
 494 \begin{mcline}{c37}
 495 dimensionless([x,t]);
 496 \end{mcline}
 497
 498
 499 Dependent equations eliminated:  (1)
 500
 501
 502 \begin{mdline}{d37}
 503 \left[ 1 \right]
 504 \end{mdline}
 505
 506
 507 \vspace{0.1in}
 508
 509 \noindent In the first example, every dimensionless quantity
 510 that can be formed as a product of powers of $c,x$, and $t$ is
 511 a power of $c t/x$; in the second example, the only
 512 dimensionless quantity that can be formed from
 513 $x$ and $t$ are the constants.
 514
 515 The function \begin{verb} natural_unit(e, [v1,v2,...,vn]) \end{verb}
 516  finds powers $p_1,p_2, \dots p_n$ such that
 517 \[
 518   \mbox{dimension}(e) = \mbox{dimension} (v_1^{p_1} v_2^{p_2} \dots v_n^{p_n}).
 519 \]
 520 Simple examples are
 521
 522 \vspace{0.1in}
 523
 524
 525
 526 \begin{mcline}{c38}
 527 natural_unit(x,[c,t]);
 528 \end{mcline}
 529
 530
 531 Dependent equations eliminated:  (1)
 532
 533
 534 \begin{mdline}{d38}
 535    \left[ c\,t \right]
 536 \end{mdline}
 537
 538 \begin{mcline}{c39}
 539    natural_unit(x,[x,c,t]);
 540 \end{mcline}
 541
 542
 543 Dependent equations eliminated:  (1)
 544
 545
 546 \begin{mdline}{d39}
 547    \left[ x \right]
 548 \end{mdline}
 549
 550
 551 \vspace{0.1in}
 552
 553 Here is a more complex example; we'll study the Bohr model of
 554 the hydrogen atom using dimensional analysis.  To make things
 555 more interesting, we'll include the magnetic moments of the
 556 proton and electron as well as the universal gravitational
 557 constant in with our list of physical quantities.
 558 Let  $\hbar$ be Planck's constant, $e$ the electron charge, $\mu_e$ the
 559 magnetic moment of the electron, $\mu_p$ the magnetic
 560 moment of the proton, $m_e$ the mass of the electron, $m_p$
 561 the mass of the proton, $G$ the universal gravitational constant,
 562  and $c$ the speed of light in a vacuum.  For this problem, we might
 563 like to display the square root as an exponent instead  of as a radical;
 564 to do this, set {\tt sqrtdispflag} to false
 565
 566 \vspace{0.1in}
 567
 568
 569
 570 \begin{mcline}{c40}
 571      SQRTDISPFLAG : false$
 572 \end{mcline}
 573
 574
 575 \vspace{0.1in}
 576
 577 \noindent  Assuming a system of units where Coulomb's law is
 578 \[
 579   \mbox{force} = \frac{\mbox{product of charges}}{\mbox{distance}^2},
 580 \]
 581 we have
 582
 583 \vspace{0.1in}
 584
 585
 586
 587 \begin{mcline}{c41}
 588 qput(%hbar, "mass" * "length"^2 / "time",dimension)$
 589 \end{mcline}
 590
 591 \begin{mcline}{c42}
 592 qput(%%e, "mass"^(1/2) * "length"^(3/2) / "time",dimension)$
 593 \end{mcline}
 594
 595 \begin{mcline}{c43}
 596 qput(%mue, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$
 597 \end{mcline}
 598
 599 \begin{mcline}{c44}
 600 qput(%mup, "mass"^(1/2) * "length"^(5/2) / "time",dimension)$
 601 \end{mcline}
 602
 603 \begin{mcline}{c45}
 604 qput(%me, "mass",dimension)$
 605 \end{mcline}
 606
 607 \begin{mcline}{c46}
 608 qput(%mp, "mass",dimension)$
 609 \end{mcline}
 610
 611 \begin{mcline}{c47}
 612 qput(%g, "length"^3 / ("time"^2 * "mass"), dimension)$
 613 \end{mcline}
 614
 615 \begin{mcline}{c48}
 616 qput(%c, "length" / "time", dimension)$
 617 \end{mcline}
 618
 619 \vspace{0.1in}
 620
 621 \noindent  The numerical values of these quantities may
 622 defined using {\tt numerval}.  We have
 623
 624 \vspace{0.1in}
 625
 626 \begin{mcline}{c49}
 627 numerval(%%e, 1.5189073558044265d-14*sqrt(kg)*meter^(3/2)/sec)$
 628 \end{mcline}
 629
 630 \begin{mcline}{c50}
 631 numerval(%hbar, 1.0545726691251061d-34*kg*meter^2/sec)$
 632 \end{mcline}
 633
 634 \begin{mcline}{c51}
 635 numerval(%c, 2.99792458d8*meter/sec)$
 636 \end{mcline}
 637
 638 \begin{mcline}{c52}
 639 numerval(%me, 9.1093897d-31*kg)$
 640 \end{mcline}
 641
 642 \begin{mcline}{c53}
 643 numerval(%mp, 1.6726231d-27*kg)$
 644 \end{mcline}
 645
 646
 647 \vspace{0.1in}
 648
 649 \noindent To begin, let's use only the variables $e, c, \hbar, m_e$, and
 650 $m_p$ to find the dimensionless quantities.  We have
 651
 652 \begin{mcline}{c54}
 653 dimensionless([%hbar, %me, %mp, %%e, %c]);
 654 \end{mcline}
 655
 656 \begin{mdline}{d54}
 657 \left[ {{m_e}\over{m_p}},{{c\,\hbar}\over{e^2}},1 \right]
 658 \end{mdline}
 659
 660
 661 \vspace{0.1in}
 662
 663 \noindent The second element of this list is the reciprocal of the fine
 664 structure constant. To find numerical values, use {\tt float}
 665
 666 \vspace{0.1in}
 667
 668 \begin{mcline}{c55}
 669 float(%);
 670 \end{mcline}
 671
 672 \begin{mdline}{d55}
 673 \left[ 5.4461699709874866 \times 10^{-4},137.035990744505,1.0
 674   \right]
 675 \end{mdline}
 676
 677
 678 \vspace{0.1in}
 679
 680 The natural units of energy are given by
 681
 682 \begin{mcline}{c56}
 683       natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %me, %mp, %%e, %c]);
 684 \end{mcline}
 685
 686
 687
 688 \begin{mdline}{d56}
 689    \left[ c^2\,m_e,{{c^3\,\hbar\,m_p}\over{e^2}} \right]
 690 \end{mdline}
 691
 692
 693 \noindent Let's see what happens when we include
 694 will include $\mu_e, \mu_p$, and  $G$.  We have
 695
 696 \vspace{0.1in}
 697
 698
 699
 700 \begin{mcline}{c57}
 701      dimensionless([%hbar, %%e, %mue, %mup, %me, %mp, %g, %c]);
 702 \end{mcline}
 703
 704
 705
 706 \begin{mdline}{d57}
 707    \left[ {{\mu_p}\over{\mu_e}},{{c^2\,m_e\,\mu_e}\over{e^3}},{{c^2
 708  \,m_p\,\mu_e}\over{e^3}},{{e^4\,G}\over{c^4\,\mu_e^2}},{{c\,
 709  \hbar}\over{e^2}},1 \right]
 710 \end{mdline}
 711
 712
 713 \vspace{0.1in}
 714
 715
 716 To find the natural units of mass, length, time,
 717 speed, force, and energy, use  the commands
 718
 719 \vspace{0.1in}
 720
 721
 722
 723 \begin{mcline}{c58}
 724     natural_unit("mass", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]);
 725 \end{mcline}
 726
 727
 728
 729 \begin{mdline}{d58}
 730    \left[ m_p,{{c^2\,m_e^2\,\mu_e}\over{e^3}},{{c^2\,m_e^2\,\mu_p
 731  }\over{e^3}},{{G\,m_e^3}\over{e^2}},{{c\,\hbar\,m_e}\over{e^
 732  2}} \right]
 733 \end{mdline}
 734
 735 \begin{mcline}{c59}
 736     natural_unit("length", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]);
 737 \end{mcline}
 738
 739
 740
 741 \begin{mdline}{d59}
 742    \left[ {{e^2\,m_p}\over{c^2\,m_e^2}},{{\mu_e}\over{e}},{{\mu_p
 743  }\over{e}},{{G\,m_e}\over{c^2}},{{\hbar}\over{c\,m_e}} \right]
 744 \end{mdline}
 745
 746 \begin{mcline}{c60}
 747    natural_unit("time", [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]);
 748 \end{mcline}
 749
 750
 751
 752 \begin{mdline}{d60}
 753    \left[ {{e^2\,m_p}\over{c^3\,m_e^2}},{{\mu_e}\over{e\,c}},{{
 754  \mu_p}\over{e\,c}},{{G\,m_e}\over{c^3}},{{\hbar}\over{c^2\,m_e
 755  }} \right]
 756 \end{mdline}
 757
 758 \begin{mcline}{c61}
 759     natural_unit("mass" * "length" / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]);
 760 \end{mcline}
 761
 762
 763
 764 \begin{mdline}{d61}
 765    \left[ {{c^4\,m_e\,m_p}\over{e^2}},{{c^6\,m_e^3\,\mu_e}\over{
 766  e^5}},{{c^6\,m_e^3\,\mu_p}\over{e^5}},{{c^4\,G\,m_e^4}\over{
 767  e^4}},{{c^5\,\hbar\,m_e^2}\over{e^4}} \right]
 768 \end{mdline}
 769
 770 \begin{mcline}{c62}
 771     natural_unit("mass" * "length"^2 / "time"^2, [%hbar, %%e, %me, %mp, %mue, %mup, %g, %c]);
 772 \end{mcline}
 773
 774
 775
 776 \begin{mdline}{d62}
 777    \left[ c^2\,m_p,{{c^4\,m_e^2\,\mu_e}\over{e^3}},{{c^4\,m_e^2\,
 778  \mu_p}\over{e^3}},{{c^2\,G\,m_e^3}\over{e^2}},{{c^3\,\hbar\,
 779  m_e}\over{e^2}} \right]
 780 \end{mdline}
 781
 782 \vspace{0.1in}
 783
 784 \noindent The first element of this list is the rest mass energy of the
 785 proton.
 786
 787
 788 The dimension package can handle vector operators such as
 789 dot and cross products, and the vector operators div, grad, and curl.
 790 To use the vector operators, we'll first declare them
 791
 792
 793 \begin{mcline}{c63}
 794 prefix(div)$
 795 \end{mcline}
 796
 797 \begin{mcline}{c64}
 798 prefix(curl)$
 799 \end{mcline}
 800
 801 \begin{mcline}{c65}
 802 infix("~")$
 803 \end{mcline}
 804
 805
 806 \noindent Let's work with the electric and magnetic fields;
 807 again assuming a system of units where Coulomb's law is
 808 \[
 809   \mbox{force} = \frac{\mbox{product of charges}}{\mbox{distance}^2}
 810 \]
 811 the dimensions of the electric and magnetic field are
 812
 813
 814 \begin{mcline}{c66}
 815 qput(e, sqrt("mass") / (sqrt("length") * "time"), dimension)$
 816 \end{mcline}
 817
 818 \begin{mcline}{c67}
 819 qput(b, sqrt("mass") / (sqrt("length") * "time"),dimension)$
 820 \end{mcline}
 821
 822 and the units of charge density $\rho$ and current density $j$ are
 823
 824
 825 \begin{mcline}{c68}
 826 qput(rho, sqrt("mass")/("time" * "length"^(3/2)), dimension)$
 827 \end{mcline}
 828
 829 \begin{mcline}{c69}
 830 qput(j, sqrt("mass") / ("time"^2 * sqrt("length")), dimension)$
 831 \end{mcline}
 832
 833 Finally, declare the speed of light $c$ as
 834
 835
 836 \begin{mcline}{c70}
 837 qput(c, "length" / "time", dimension);
 838 \end{mcline}
 839
 840
 841
 842 \begin{mdline}{d70}
 843 {{length}\over{time}}
 844 \end{mdline}
 845
 846 \noindent Let's find the dimensions of
 847 $\| \mathbf{E} \|^2, \mathbf{E} \cdot \mathbf{B},
 848 \| \mathbf{B} \|^2$, and $\mathbf{E} \times \mathbf{B} / c$.  We have
 849
 850 \begin{mcline}{c71}
 851 dimension(e.e);
 852 \end{mcline}
 853
 854
 855
 856 \begin{mdline}{d71}
 857 {{mass}\over{length\,time^2}}
 858 \end{mdline}
 859
 860 \begin{mcline}{c72}
 861 dimension(e.b);
 862 \end{mcline}
 863
 864
 865
 866 \begin{mdline}{d72}
 867 {{mass}\over{length\,time^2}}
 868 \end{mdline}
 869
 870 \begin{mcline}{c73}
 871 dimension(b.b);
 872 \end{mcline}
 873
 874
 875
 876 \begin{mdline}{d73}
 877 {{mass}\over{length\,time^2}}
 878 \end{mdline}
 879
 880 \begin{mcline}{c74}
 881 dimension((e ~ b) / c);
 882 \end{mcline}
 883
 884
 885
 886 \begin{mdline}{d74}
 887 {{mass}\over{length^2\,time}}
 888 \end{mdline}
 889
 890
 891 \vspace{0.1in}
 892
 893 \noindent The physical significance of these quantities becomes more apparent
 894 if they are integrated over $\mathbf{R^3}$.  Defining
 895
 896 \vspace{0.1in}
 897
 898 \begin{mcline}{c75}
 899 qput(v, "length"^3, dimension);
 900 \end{mcline}
 901
 902
 903
 904 \begin{mdline}{d75}
 905 length^3
 906 \end{mdline}
 907
 908 \vspace{0.1in}
 909
 910 \noindent We now have
 911
 912
 913 \begin{mcline}{c76}
 914 dimension('integrate(e.e, v));
 915 \end{mcline}
 916
 917
 918
 919 \begin{mdline}{d76}
 920 {{length^2\,mass}\over{time^2}}
 921 \end{mdline}
 922
 923 \begin{mcline}{c77}
 924 dimension('integrate(e.b, v));
 925 \end{mcline}
 926
 927 \begin{mdline}{d77}
 928 {{length^2\,mass}\over{time^2}}
 929 \end{mdline}
 930
 931 \begin{mcline}{c78}
 932 dimension('integrate(b.b, v));
 933 \end{mcline}
 934
 935 \begin{mdline}{d78}
 936 {{length^2\,mass}\over{time^2}}
 937 \end{mdline}
 938
 939 \begin{mcline}{c79}
 940  dimension('integrate((e ~ b) / c,v));
 941 \end{mcline}
 942
 943
 944 \begin{mdline}{d79}
 945 {{length\,mass}\over{time}}
 946 \end{mdline}
 947
 948 \noindent It's clear that $\| \mathbf{E} \|^2, \mathbf{E} \cdot \mathbf{B}$
 949 and $\| \mathbf{B} \|^2$ are energy densities while
 950 $\mathbf{E} \times \mathbf{B} / c$ is a momentum density.
 951
 952 Let's also check that the Maxwell equations are
 953 dimensionally consistent.
 954
 955 \begin{mcline}{c80}
 956 dimension(DIV(e)= 4*%pi*rho);
 957 \end{mcline}
 958
 959 \begin{mdline}{d80}
 960 {{mass^{{{1}\over{2}}}}\over{length^{{{3}\over{2}}}\,time}}
 961 \end{mdline}
 962
 963 \begin{mcline}{c81}
 964 dimension(CURL(b) - 'diff(e,t) / c = 4 * %pi * j / c);
 965 \end{mcline}
 966
 967 \begin{mdline}{d81}
 968 {{mass^{{{1}\over{2}}}}\over{length^{{{3}\over{2}}}\,time}}
 969 \end{mdline}
 970
 971 \begin{mcline}{c82}
 972 dimension(CURL(e) + 'diff(b,t) / c = 0);
 973 \end{mcline}
 974
 975 \begin{mdline}{d82}
 976 {{mass^{{{1}\over{2}}}}\over{length^{{{3}\over{2}}}\,time}}
 977 \end{mdline}
 978
 979 \begin{mcline}{c83}
 980 dimension(DIV(b) = 0);
 981 \end{mcline}
 982
 983 \begin{mdline}{d83}
 984 {{mass^{{{1}\over{2}}}}\over{length^{{{3}\over{2}}}\,time}}
 985 \end{mdline}
 986
 987
 988 \subsubsection*{Conclusion and Future directions}
 989
 990 Algorithmically, the dimensional analysis package is straightforward;
 991 nevertheless, there are many details, such as correctly setting
 992 option variables for linsolve, that need to be tended to.  Let me know
 993 when you find a bug; I'll try to fix it.  There may be some operators that
 994 aren't handled; again, let me know what is missing and I'll try
 995 to fix it.
 996
 997 Eventually, I hope that this package will work smoothly with the new physical
 998 constants package.
 999
1000 I could add predefined dimensions for derived units such as momentum, charge,
1001 density, etc.; however, given the plethora of schemes for
1002 electromagnetic units, I'm hesitant to do this.
1003
1004 This documentation was processed by bat\TeX, a Maxima preprocessor
1005 for \TeX. The bat\TeX software is also available from the author's
1006 web page.
1007
1008 \end{document}
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1026 (D84)                                DONE
1027 (C84)