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  239. 

  240. <div id="content">

  241. <header>

  242. <h1 class="title">

  243. <a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>

  244. </h1>

  245. </header>

  246. <nav id="collapsed-table-of-contents">

  247. <details>

  248. <summary>

  249. Table of contents

  250. </summary>

  251. <ul>

  252. <li>

  253. 

  254. <details>

  255. <summary>

  256. <a href="./in.html#in">Introduction</a><span class="headline-id">in</span>

  257. 

  258. 

  259. </summary>

  260. <ul>

  261. <li>

  262. <a href="./in_p.html#in_p">Preface</a><span class="headline-id">in.p</span>

  263. 

  264. </li>

  265. <li>

  266. 

  267. <details>

  268. <summary>

  269. <a href="./in_t.html#in_t">Tips for the reader</a><span class="headline-id">in.t</span>

  270. 

  271. 

  272. </summary>

  273. <ul>

  274. <li>

  275. <a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>

  276. 

  277. </li>

  278. <li>

  279. <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>

  280. 

  281. </li>

  282. 

  283. </ul>

  284. </details>

  285. </li>

  286. 

  287. </ul>

  288. </details>

  289. </li>

  290. <li>

  291. 

  292. <details open="">

  293. <summary class="toc-open">

  294. <a href="./ems.html#ems">Electromagnetostatics</a><span class="headline-id">ems</span>

  295. 

  296. 

  297. </summary>

  298. <ul>

  299. <li>

  300. 

  301. <details open="">

  302. <summary class="toc-open">

  303. <a href="./ems_es.html#ems_es">Electrostatics</a><span class="headline-id">ems.es</span>

  304. 

  305. 

  306. </summary>

  307. <ul>

  308. <li>

  309. 

  310. <details>

  311. <summary>

  312. <a href="./ems_es_ec.html#ems_es_ec">Electric Charge</a><span class="headline-id">ems.es.ec</span>

  313. 

  314. 

  315. </summary>

  316. <ul>

  317. <li>

  318. <a href="./ems_es_ec_b.html#ems_es_ec_b">Basics</a><span class="headline-id">ems.es.ec.b</span>

  319. 

  320. </li>

  321. <li>

  322. <a href="./ems_es_ec_c.html#ems_es_ec_c">Conservation</a><span class="headline-id">ems.es.ec.c</span>

  323. 

  324. </li>

  325. <li>

  326. <a href="./ems_es_ec_q.html#ems_es_ec_q">Quantization</a><span class="headline-id">ems.es.ec.q</span>

  327. 

  328. </li>

  329. <li>

  330. <a href="./ems_es_ec_s.html#ems_es_ec_s">Structure</a><span class="headline-id">ems.es.ec.s</span>

  331. 

  332. </li>

  333. 

  334. </ul>

  335. </details>

  336. </li>

  337. <li>

  338. 

  339. <details>

  340. <summary>

  341. <a href="./ems_es_efo.html#ems_es_efo">Electric Force and Energy</a><span class="headline-id">ems.es.efo</span>

  342. 

  343. 

  344. </summary>

  345. <ul>

  346. <li>

  347. <a href="./ems_es_efo_cl.html#ems_es_efo_cl">Coulomb's Law</a><span class="headline-id">ems.es.efo.cl</span>

  348. 

  349. </li>

  350. <li>

  351. <a href="./ems_es_efo_ps.html#ems_es_efo_ps">Principle of Superposition</a><span class="headline-id">ems.es.efo.ps</span>

  352. 

  353. </li>

  354. <li>

  355. <a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>

  356. 

  357. </li>

  358. <li>

  359. <a href="./ems_es_efo_e.html#ems_es_efo_e">Energy in Systems of Point Charges</a><span class="headline-id">ems.es.efo.e</span>

  360. 

  361. </li>

  362. 

  363. </ul>

  364. </details>

  365. </li>

  366. <li>

  367. 

  368. <details>

  369. <summary>

  370. <a href="./ems_es_ef.html#ems_es_ef">Electrostatic Fields</a><span class="headline-id">ems.es.ef</span>

  371. 

  372. 

  373. </summary>

  374. <ul>

  375. <li>

  376. <a href="./ems_es_ef_pc.html#ems_es_ef_pc">Electrostatic Field of Point Charges</a><span class="headline-id">ems.es.ef.pc</span>

  377. 

  378. </li>

  379. <li>

  380. <a href="./ems_es_ef_ccd.html#ems_es_ef_ccd">Electrostatic Field of Continuous Charge Distributions</a><span class="headline-id">ems.es.ef.ccd</span>

  381. 

  382. </li>

  383. <li>

  384. <a href="./ems_es_ef_cE.html#ems_es_ef_cE">The Curl of \({\bf E}\)</a><span class="headline-id">ems.es.ef.cE</span>

  385. 

  386. </li>

  387. <li>

  388. <a href="./ems_es_ef_Gl.html#ems_es_ef_Gl">Gauss's Law: the divergence of \({\bf E}\)</a><span class="headline-id">ems.es.ef.Gl</span>

  389. 

  390. </li>

  391. 

  392. </ul>

  393. </details>

  394. </li>

  395. <li>

  396. 

  397. <details open="">

  398. <summary class="toc-open">

  399. <a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>

  400. 

  401. 

  402. </summary>

  403. <ul>

  404. <li>

  405. <a href="./ems_es_ep_d.html#ems_es_ep_d">Definition</a><span class="headline-id">ems.es.ep.d</span>

  406. 

  407. </li>

  408. <li>

  409. <a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>

  410. 

  411. </li>

  412. <li>

  413. <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>

  414. 

  415. </li>

  416. <li>

  417. <a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>

  418. 

  419. </li>

  420. <li class="toc-currentpage">

  421. <a href="./ems_es_ep_bc.html#ems_es_ep_bc">Electrostatic Boundary Conditions</a><span class="headline-id">ems.es.ep.bc</span>

  422. 

  423. </li>

  424. 

  425. </ul>

  426. </details>

  427. </li>

  428. <li>

  429. <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>

  430. 

  431. </li>

  432. <li>

  433. 

  434. <details>

  435. <summary>

  436. <a href="./ems_es_c.html#ems_es_c">Conductors</a><span class="headline-id">ems.es.c</span>

  437. 

  438. 

  439. </summary>

  440. <ul>

  441. <li>

  442. <a href="./ems_es_c_p.html#ems_es_c_p">Properties</a><span class="headline-id">ems.es.c.p</span>

  443. 

  444. </li>

  445. <li>

  446. <a href="./ems_es_c_ic.html#ems_es_c_ic">Induced Charges</a><span class="headline-id">ems.es.c.ic</span>

  447. 

  448. </li>

  449. <li>

  450. <a href="./ems_es_c_sc.html#ems_es_c_sc">Surface Charge and the Force on a Conductor</a><span class="headline-id">ems.es.c.sc</span>

  451. 

  452. </li>

  453. <li>

  454. <a href="./ems_es_c_cap.html#ems_es_c_cap">Capacitors</a><span class="headline-id">ems.es.c.cap</span>

  455. 

  456. </li>

  457. 

  458. </ul>

  459. </details>

  460. </li>

  461. 

  462. </ul>

  463. </details>

  464. </li>

  465. <li>

  466. 

  467. <details>

  468. <summary>

  469. <a href="./ems_ca.html#ems_ca">Calculating or Approximating the Electrostatic Potential</a><span class="headline-id">ems.ca</span>

  470. 

  471. 

  472. </summary>

  473. <ul>

  474. <li>

  475. 

  476. <details>

  477. <summary>

  478. <a href="./ems_ca_fe.html#ems_ca_fe">Fundamental Equations for the Electrostatic Potential</a><span class="headline-id">ems.ca.fe</span>

  479. 

  480. 

  481. </summary>

  482. <ul>

  483. <li>

  484. <a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>

  485. 

  486. </li>

  487. <li>

  488. <a href="./ems_ca_fe_g.html#ems_ca_fe_g">Green's Identities</a><span class="headline-id">ems.ca.fe.g</span>

  489. 

  490. </li>

  491. <li>

  492. <a href="./ems_ca_fe_uP.html#ems_ca_fe_uP">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span>

  493. 

  494. </li>

  495. 

  496. </ul>

  497. </details>

  498. </li>

  499. <li>

  500. 

  501. <details>

  502. <summary>

  503. <a href="./ems_ca_mi.html#ems_ca_mi">The Method of Images</a><span class="headline-id">ems.ca.mi</span>

  504. 

  505. 

  506. </summary>

  507. <ul>

  508. <li>

  509. <a href="./ems_ca_mi_isc.html#ems_ca_mi_isc">Induced Surface Charges</a><span class="headline-id">ems.ca.mi.isc</span>

  510. 

  511. </li>

  512. <li>

  513. <a href="./ems_ca_mi_fe.html#ems_ca_mi_fe">Force and Energy</a><span class="headline-id">ems.ca.mi.fe</span>

  514. 

  515. </li>

  516. <li>

  517. <a href="./ems_ca_mi_o.html#ems_ca_mi_o">Other Image Problems</a><span class="headline-id">ems.ca.mi.o</span>

  518. 

  519. </li>

  520. 

  521. </ul>

  522. </details>

  523. </li>

  524. <li>

  525. 

  526. <details>

  527. <summary>

  528. <a href="./ems_ca_sv.html#ems_ca_sv">Separation of Variables</a><span class="headline-id">ems.ca.sv</span>

  529. 

  530. 

  531. </summary>

  532. <ul>

  533. <li>

  534. <a href="./ems_ca_sv_car.html#ems_ca_sv_car">Cartesian Coordinates</a><span class="headline-id">ems.ca.sv.car</span>

  535. 

  536. </li>

  537. <li>

  538. <a href="./ems_ca_sv_cyl.html#ems_ca_sv_cyl">Cylindrical Coordinates</a><span class="headline-id">ems.ca.sv.cyl</span>

  539. 

  540. </li>

  541. <li>

  542. <a href="./ems_ca_sv_sph.html#ems_ca_sv_sph">Spherical Coordinates</a><span class="headline-id">ems.ca.sv.sph</span>

  543. 

  544. </li>

  545. 

  546. </ul>

  547. </details>

  548. </li>

  549. <li>

  550. 

  551. <details>

  552. <summary>

  553. <a href="./ems_ca_me.html#ems_ca_me">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span>

  554. 

  555. 

  556. </summary>

  557. <ul>

  558. <li>

  559. <a href="./ems_ca_me_a.html#ems_ca_me_a">Approximate Potential at Large Distance</a><span class="headline-id">ems.ca.me.a</span>

  560. 

  561. </li>

  562. <li>

  563. <a href="./ems_ca_me_md.html#ems_ca_me_md">Monopole and Dipole Terms</a><span class="headline-id">ems.ca.me.md</span>

  564. 

  565. </li>

  566. <li>

  567. <a href="./ems_ca_me_h.html#ems_ca_me_h">Higher Moments</a><span class="headline-id">ems.ca.me.h</span>

  568. 

  569. </li>

  570. <li>

  571. <a href="./ems_ca_me_Ed.html#ems_ca_me_Ed">The Electric Field of a Dipole</a><span class="headline-id">ems.ca.me.Ed</span>

  572. 

  573. </li>

  574. <li>

  575. <a href="./ems_ca_me_Eq.html#ems_ca_me_Eq">The Electric Field of a Quadrupole</a><span class="headline-id">ems.ca.me.Eq</span>

  576. 

  577. </li>

  578. 

  579. </ul>

  580. </details>

  581. </li>

  582. 

  583. </ul>

  584. </details>

  585. </li>

  586. <li>

  587. 

  588. <details>

  589. <summary>

  590. <a href="./ems_ms.html#ems_ms">Magnetostatics</a><span class="headline-id">ems.ms</span>

  591. 

  592. 

  593. </summary>

  594. <ul>

  595. <li>

  596. 

  597. <details>

  598. <summary>

  599. <a href="./ems_ms_lf.html#ems_ms_lf">Charges in Motion: the Lorentz Force Law</a><span class="headline-id">ems.ms.lf</span>

  600. 

  601. 

  602. </summary>

  603. <ul>

  604. <li>

  605. <a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>

  606. 

  607. </li>

  608. <li>

  609. <a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>

  610. 

  611. </li>

  612. 

  613. </ul>

  614. </details>

  615. </li>

  616. <li>

  617. <a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>

  618. 

  619. </li>

  620. <li>

  621. <a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>

  622. 

  623. </li>

  624. <li>

  625. 

  626. <details>

  627. <summary>

  628. <a href="./ems_ms_dcB.html#ems_ms_dcB">Divergence and Curl of \({\bf B}\)</a><span class="headline-id">ems.ms.dcB</span>

  629. 

  630. 

  631. </summary>

  632. <ul>

  633. <li>

  634. <a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>

  635. 

  636. </li>

  637. <li>

  638. <a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>

  639. 

  640. </li>

  641. <li>

  642. <a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>

  643. 

  644. </li>

  645. 

  646. </ul>

  647. </details>

  648. </li>

  649. <li>

  650. 

  651. <details>

  652. <summary>

  653. <a href="./ems_ms_vp.html#ems_ms_vp">The Vector Potential</a><span class="headline-id">ems.ms.vp</span>

  654. 

  655. 

  656. </summary>

  657. <ul>

  658. <li>

  659. <a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>

  660. 

  661. </li>

  662. <li>

  663. <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>

  664. 

  665. </li>

  666. <li>

  667. <a href="./ems_ms_vp_me.html#ems_ms_vp_me">Multipole Expansion of the Vector Potential</a><span class="headline-id">ems.ms.vp.me</span>

  668. 

  669. </li>

  670. <li>

  671. <a href="./ems_ms_vp_comp.html#ems_ms_vp_comp">Comparison of Electrostatics and Magnetostatics</a><span class="headline-id">ems.ms.vp.comp</span>

  672. 

  673. </li>

  674. <li>

  675. <a href="./ems_ms_vp_LC.html#ems_ms_vp_LC">The Levi-Civita Symbol</a><span class="headline-id">ems.ms.vp.LC</span>

  676. 

  677. </li>

  678. 

  679. </ul>

  680. </details>

  681. </li>

  682. 

  683. </ul>

  684. </details>

  685. </li>

  686. 

  687. </ul>

  688. </details>

  689. </li>

  690. <li>

  691. 

  692. <details>

  693. <summary>

  694. <a href="./emsm.html#emsm">Electromagnetostatics in matter</a><span class="headline-id">emsm</span>

  695. 

  696. 

  697. </summary>

  698. <ul>

  699. <li>

  700. 

  701. <details>

  702. <summary>

  703. <a href="./emsm_esm.html#emsm_esm">Electrostatics in matter</a><span class="headline-id">emsm.esm</span>

  704. 

  705. 

  706. </summary>

  707. <ul>

  708. <li>

  709. 

  710. <details>

  711. <summary>

  712. <a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>

  713. 

  714. 

  715. </summary>

  716. <ul>

  717. <li>

  718. <a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>

  719. 

  720. </li>

  721. <li>

  722. <a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</span>

  723. 

  724. </li>

  725. 

  726. </ul>

  727. </details>

  728. </li>

  729. <li>

  730. 

  731. <details>

  732. <summary>

  733. <a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>

  734. 

  735. 

  736. </summary>

  737. <ul>

  738. <li>

  739. <a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>

  740. 

  741. </li>

  742. <li>

  743. <a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>

  744. 

  745. </li>

  746. 

  747. </ul>

  748. </details>

  749. </li>

  750. <li>

  751. 

  752. <details>

  753. <summary>

  754. <a href="./emsm_esm_D.html#emsm_esm_D">The Electric Displacement</a><span class="headline-id">emsm.esm.D</span>

  755. 

  756. 

  757. </summary>

  758. <ul>

  759. <li>

  760. <a href="./emsm_esm_D_bc.html#emsm_esm_D_bc">Boundary Conditions</a><span class="headline-id">emsm.esm.D.bc</span>

  761. 

  762. </li>

  763. 

  764. </ul>

  765. </details>

  766. </li>

  767. <li>

  768. <a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>

  769. 

  770. </li>

  771. <li>

  772. 

  773. <details>

  774. <summary>

  775. <a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>

  776. 

  777. 

  778. </summary>

  779. <ul>

  780. <li>

  781. <a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>

  782. 

  783. </li>

  784. <li>

  785. <a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>

  786. 

  787. </li>

  788. <li>

  789. <a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>

  790. 

  791. </li>

  792. <li>

  793. <a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>

  794. 

  795. </li>

  796. 

  797. </ul>

  798. </details>

  799. </li>

  800. 

  801. </ul>

  802. </details>

  803. </li>

  804. <li>

  805. 

  806. <details>

  807. <summary>

  808. <a href="./emsm_msm.html#emsm_msm">Magnetostatics in matter</a><span class="headline-id">emsm.msm</span>

  809. 

  810. 

  811. </summary>

  812. <ul>

  813. <li>

  814. 

  815. <details>

  816. <summary>

  817. <a href="./emsm_msm_m.html#emsm_msm_m">Magnetization</a><span class="headline-id">emsm.msm.m</span>

  818. 

  819. 

  820. </summary>

  821. <ul>

  822. <li>

  823. <a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>

  824. 

  825. </li>

  826. <li>

  827. <a href="./emsm_msm_m_fdi.html#emsm_msm_m_fdi">Torques and Forces on Magnetic Dipoles</a><span class="headline-id">emsm.msm.m.fdi</span>

  828. 

  829. </li>

  830. <li>

  831. <a href="./emsm_msm_a.html#emsm_msm_a">Effect of Magnetic Field on Atomic Orbits</a><span class="headline-id">emsm.msm.a</span>

  832. 

  833. </li>

  834. 

  835. </ul>

  836. </details>

  837. </li>

  838. <li>

  839. 

  840. <details>

  841. <summary>

  842. <a href="./emsm_msm_fmo.html#emsm_msm_fmo">The Field of a Magnetized Object</a><span class="headline-id">emsm.msm.fmo</span>

  843. 

  844. 

  845. </summary>

  846. <ul>

  847. <li>

  848. <a href="./emsm_msm_fmo_bc.html#emsm_msm_fmo_bc">Bound Currents</a><span class="headline-id">emsm.msm.fmo.bc</span>

  849. 

  850. </li>

  851. <li>

  852. <a href="./emsm_msm_fmo_pibc.html#emsm_msm_fmo_pibc">Physical Interpretation of Bound Currents</a><span class="headline-id">emsm.msm.fmo.pibc</span>

  853. 

  854. </li>

  855. <li>

  856. <a href="./emsm_msm_fmo_fim.html#emsm_msm_fmo_fim">The Magnetic Field Inside Matter</a><span class="headline-id">emsm.msm.fmo.fim</span>

  857. 

  858. </li>

  859. 

  860. </ul>

  861. </details>

  862. </li>

  863. <li>

  864. 

  865. <details>

  866. <summary>

  867. <a href="./emsm_msm_H.html#emsm_msm_H">The H Field</a><span class="headline-id">emsm.msm.H</span>

  868. 

  869. 

  870. </summary>

  871. <ul>

  872. <li>

  873. <a href="./emsm_msm_H_A.html#emsm_msm_H_A">Ampère's Law in Magnetized Materials</a><span class="headline-id">emsm.msm.H.A</span>

  874. 

  875. </li>

  876. 

  877. </ul>

  878. </details>

  879. </li>

  880. <li>

  881. 

  882. <details>

  883. <summary>

  884. <a href="./emsm_msm_lnlm.html#emsm_msm_lnlm">Linear and Nonlinear Media</a><span class="headline-id">emsm.msm.lnlm</span>

  885. 

  886. 

  887. </summary>

  888. <ul>

  889. <li>

  890. <a href="./emsm_msm_lnlm_sp.html#emsm_msm_lnlm_sp">Magnetic Susceptibility and Permeability</a><span class="headline-id">emsm.msm.lnlm.sp</span>

  891. 

  892. </li>

  893. <li>

  894. <a href="./emsm_msm_lnlm_fm.html#emsm_msm_lnlm_fm">Ferromagnetism</a><span class="headline-id">emsm.msm.lnlm.fm</span>

  895. 

  896. </li>

  897. 

  898. </ul>

  899. </details>

  900. </li>

  901. 

  902. </ul>

  903. </details>

  904. </li>

  905. 

  906. </ul>

  907. </details>

  908. </li>

  909. <li>

  910. 

  911. <details>

  912. <summary>

  913. <a href="./emd.html#emd">Electromagnetodynamics</a><span class="headline-id">emd</span>

  914. 

  915. 

  916. </summary>

  917. <ul>

  918. <li>

  919. 

  920. <details>

  921. <summary>

  922. <a href="./emd_Fl.html#emd_Fl">Induction: Faraday's Law</a><span class="headline-id">emd.Fl</span>

  923. 

  924. 

  925. </summary>

  926. <ul>

  927. <li>

  928. <a href="./emd_Fl_Fl.html#emd_Fl_Fl">Faraday's Law</a><span class="headline-id">emd.Fl.Fl</span>

  929. 

  930. </li>

  931. <li>

  932. <a href="./emd_Fl_ief.html#emd_Fl_ief">The Induced Electric Field</a><span class="headline-id">emd.Fl.ief</span>

  933. 

  934. </li>

  935. <li>

  936. <a href="./emd_Fl_i.html#emd_Fl_i">Inductance</a><span class="headline-id">emd.Fl.i</span>

  937. 

  938. </li>

  939. <li>

  940. <a href="./emd_Fl_e.html#emd_Fl_e">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span>

  941. 

  942. </li>

  943. 

  944. </ul>

  945. </details>

  946. </li>

  947. <li>

  948. 

  949. <details>

  950. <summary>

  951. <a href="./emd_Me.html#emd_Me">Maxwell's Equations</a><span class="headline-id">emd.Me</span>

  952. 

  953. 

  954. </summary>

  955. <ul>

  956. <li>

  957. <a href="./emd_Me_ebM.html#emd_Me_ebM">Electrodynamics Before Maxwell</a><span class="headline-id">emd.Me.ebM</span>

  958. 

  959. </li>

  960. <li>

  961. <a href="./emd_Me_dc.html#emd_Me_dc">Maxwell's Correction to Ampère's Law; the Displacement Current</a><span class="headline-id">emd.Me.dc</span>

  962. 

  963. </li>

  964. <li>

  965. <a href="./emd_Me_Me.html#emd_Me_Me">Maxwell's Equations</a><span class="headline-id">emd.Me.Me</span>

  966. 

  967. </li>

  968. <li>

  969. <a href="./emd_Me_mc.html#emd_Me_mc">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span>

  970. 

  971. </li>

  972. 

  973. </ul>

  974. </details>

  975. </li>

  976. <li>

  977. 

  978. <details>

  979. <summary>

  980. <a href="./emd_ce.html#emd_ce">Charge and Energy Flows</a><span class="headline-id">emd.ce</span>

  981. 

  982. 

  983. </summary>

  984. <ul>

  985. <li>

  986. <a href="./emd_ce_ce.html#emd_ce_ce">The Continuity Equation</a><span class="headline-id">emd.ce.ce</span>

  987. 

  988. </li>

  989. <li>

  990. <a href="./emd_ce_poy.html#emd_ce_poy">Poynting's Theorem; the Poynting Vector</a><span class="headline-id">emd.ce.poy</span>

  991. 

  992. </li>

  993. <li>

  994. <a href="./emd_ce_mst.html#emd_ce_mst">Maxwell's Stress Tensor</a><span class="headline-id">emd.ce.mst</span>

  995. 

  996. </li>

  997. <li>

  998. <a href="./emd_ce_mom.html#emd_ce_mom">Momentum</a><span class="headline-id">emd.ce.mom</span>

  999. 

  1000. </li>

  1001. <li>

  1002. <a href="./emd_ce_amom.html#emd_ce_amom">Angular Momentum</a><span class="headline-id">emd.ce.amom</span>

  1003. 

  1004. </li>

  1005. 

  1006. </ul>

  1007. </details>

  1008. </li>

  1009. <li>

  1010. 

  1011. <details>

  1012. <summary>

  1013. <a href="./emd_emw.html#emd_emw">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span>

  1014. 

  1015. 

  1016. </summary>

  1017. <ul>

  1018. <li>

  1019. <a href="./emd_emw_we.html#emd_emw_we">The Wave Equation</a><span class="headline-id">emd.emw.we</span>

  1020. 

  1021. </li>

  1022. <li>

  1023. <a href="./emd_emw_mpw.html#emd_emw_mpw">Monochromatic Plane Waves</a><span class="headline-id">emd.emw.mpw</span>

  1024. 

  1025. </li>

  1026. <li>

  1027. <a href="./emd_emw_ep.html#emd_emw_ep">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span>

  1028. 

  1029. </li>

  1030. 

  1031. </ul>

  1032. </details>

  1033. </li>

  1034. 

  1035. </ul>

  1036. </details>

  1037. </li>

  1038. <li>

  1039. 

  1040. <details>

  1041. <summary>

  1042. <a href="./emdm.html#emdm">Electromagnetodynamics in Matter</a><span class="headline-id">emdm</span>

  1043. 

  1044. 

  1045. </summary>

  1046. <ul>

  1047. <li>

  1048. 

  1049. <details>

  1050. <summary>

  1051. <a href="./emdm_Me.html#emdm_Me">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me</span>

  1052. 

  1053. 

  1054. </summary>

  1055. <ul>

  1056. <li>

  1057. <a href="./emdm_Me_Mem.html#emdm_Me_Mem">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me.Mem</span>

  1058. 

  1059. </li>

  1060. <li>

  1061. <a href="./emdm_Me_bc.html#emdm_Me_bc">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span>

  1062. 

  1063. </li>

  1064. 

  1065. </ul>

  1066. </details>

  1067. </li>

  1068. <li>

  1069. 

  1070. <details>

  1071. <summary>

  1072. <a href="./emdm_emwm.html#emdm_emwm">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span>

  1073. 

  1074. 

  1075. </summary>

  1076. <ul>

  1077. <li>

  1078. <a href="./emdm_emwm_plm.html#emdm_emwm_plm">Propagation in Linear Media</a><span class="headline-id">emdm.emwm.plm</span>

  1079. 

  1080. </li>

  1081. <li>

  1082. <a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>

  1083. 

  1084. </li>

  1085. <li>

  1086. 

  1087. <details>

  1088. <summary>

  1089. <a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>

  1090. 

  1091. 

  1092. </summary>

  1093. <ul>

  1094. <li>

  1095. <a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>

  1096. 

  1097. </li>

  1098. <li>

  1099. <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>

  1100. 

  1101. </li>

  1102. 

  1103. </ul>

  1104. </details>

  1105. </li>

  1106. <li>

  1107. 

  1108. <details>

  1109. <summary>

  1110. <a href="./emdm_emwm_ad.html#emdm_emwm_ad">Absorption and Dispersion</a><span class="headline-id">emdm.emwm.ad</span>

  1111. 

  1112. 

  1113. </summary>

  1114. <ul>

  1115. <li>

  1116. <a href="./emdm_emwm_ad_c.html#emdm_emwm_ad_c">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span>

  1117. 

  1118. </li>

  1119. 

  1120. </ul>

  1121. </details>

  1122. </li>

  1123. <li>

  1124. 

  1125. <details>

  1126. <summary>

  1127. <a href="./emdm_emwm_wg.html#emdm_emwm_wg">Waveguides</a><span class="headline-id">emdm.emwm.wg</span>

  1128. 

  1129. 

  1130. </summary>

  1131. <ul>

  1132. <li>

  1133. <a href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw">Guided waves</a><span class="headline-id">emdm.emwm.wg.gw</span>

  1134. 

  1135. </li>

  1136. <li>

  1137. <a href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r">Rectangular Waveguides</a><span class="headline-id">emdm.emwm.wg.r</span>

  1138. 

  1139. </li>

  1140. <li>

  1141. <a href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span>

  1142. 

  1143. </li>

  1144. 

  1145. </ul>

  1146. </details>

  1147. </li>

  1148. 

  1149. </ul>

  1150. </details>

  1151. </li>

  1152. 

  1153. </ul>

  1154. </details>

  1155. </li>

  1156. <li>

  1157. 

  1158. <details>

  1159. <summary>

  1160. <a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>

  1161. 

  1162. 

  1163. </summary>

  1164. <ul>

  1165. <li>

  1166. <a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>

  1167. 

  1168. </li>

  1169. <li>

  1170. 

  1171. <details>

  1172. <summary>

  1173. <a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>

  1174. 

  1175. 

  1176. </summary>

  1177. <ul>

  1178. <li>

  1179. <a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>

  1180. 

  1181. </li>

  1182. <li>

  1183. <a href="./emf_g_Lg.html#emf_g_Lg">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span>

  1184. 

  1185. </li>

  1186. 

  1187. </ul>

  1188. </details>

  1189. </li>

  1190. 

  1191. </ul>

  1192. </details>

  1193. </li>

  1194. <li>

  1195. 

  1196. <details>

  1197. <summary>

  1198. <a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>

  1199. 

  1200. 

  1201. </summary>

  1202. <ul>

  1203. <li>

  1204. 

  1205. <details>

  1206. <summary>

  1207. <a href="./red_sr.html#red_sr">Special Relativity</a><span class="headline-id">red.sr</span>

  1208. 

  1209. 

  1210. </summary>

  1211. <ul>

  1212. <li>

  1213. <a href="./red_sr_p.html#red_sr_p">Postulates and their consequences</a><span class="headline-id">red.sr.p</span>

  1214. 

  1215. </li>

  1216. <li>

  1217. <a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>

  1218. 

  1219. </li>

  1220. <li>

  1221. <a href="./red_sr_4v.html#red_sr_4v">Covariant and Contravariant Four-Vectors</a><span class="headline-id">red.sr.4v</span>

  1222. 

  1223. </li>

  1224. 

  1225. </ul>

  1226. </details>

  1227. </li>

  1228. <li>

  1229. 

  1230. <details>

  1231. <summary>

  1232. <a href="./red_rm.html#red_rm">Relativistic Mechanics</a><span class="headline-id">red.rm</span>

  1233. 

  1234. 

  1235. </summary>

  1236. <ul>

  1237. <li>

  1238. <a href="./red_rm_pt.html#red_rm_pt">Proper Time and Proper Velocity</a><span class="headline-id">red.rm.pt</span>

  1239. 

  1240. </li>

  1241. <li>

  1242. <a href="./red_rm_rme.html#red_rm_rme">Relativistic Momentum and Energy</a><span class="headline-id">red.rm.rme</span>

  1243. 

  1244. </li>

  1245. <li>

  1246. <a href="./red_rm_Mf.html#red_rm_Mf">Relativistic version of Newton's Laws; the Minkowski Force</a><span class="headline-id">red.rm.Mf</span>

  1247. 

  1248. </li>

  1249. 

  1250. </ul>

  1251. </details>

  1252. </li>

  1253. <li>

  1254. 

  1255. <details>

  1256. <summary>

  1257. <a href="./red_rem.html#red_rem">Relativistic Electromagnetism</a><span class="headline-id">red.rem</span>

  1258. 

  1259. 

  1260. </summary>

  1261. <ul>

  1262. <li>

  1263. <a href="./red_rem_mre.html#red_rem_mre">Magnetism as a Relativistic Effect</a><span class="headline-id">red.rem.mre</span>

  1264. 

  1265. </li>

  1266. <li>

  1267. <a href="./red_rem_Ltf.html#red_rem_Ltf">Lorentz Transformation of Electromagnetic Fields</a><span class="headline-id">red.rem.Ltf</span>

  1268. 

  1269. </li>

  1270. <li>

  1271. <a href="./red_rem_Fmunu.html#red_rem_Fmunu">The Field Tensor</a><span class="headline-id">red.rem.Fmunu</span>

  1272. 

  1273. </li>

  1274. <li>

  1275. <a href="./red_rem_Me.html#red_rem_Me">Maxwell's Equations in Relativistic Notation</a><span class="headline-id">red.rem.Me</span>

  1276. 

  1277. </li>

  1278. 

  1279. </ul>

  1280. </details>

  1281. </li>

  1282. 

  1283. </ul>

  1284. </details>

  1285. </li>

  1286. <li>

  1287. 

  1288. <details>

  1289. <summary>

  1290. <a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>

  1291. 

  1292. 

  1293. </summary>

  1294. <ul>

  1295. <li>

  1296. <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>

  1297. 

  1298. </li>

  1299. 

  1300. </ul>

  1301. </details>

  1302. </li>

  1303. <li>

  1304. 

  1305. <details>

  1306. <summary>

  1307. <a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>

  1308. 

  1309. 

  1310. </summary>

  1311. <ul>

  1312. <li>

  1313. <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>

  1314. 

  1315. </li>

  1316. <li>

  1317. <a href="./d_ems_ca.html#d_ems_ca">Diagnostics: Calculating or Approximating the Electostatic Potential</a><span class="headline-id">d.ems.ca</span>

  1318. 

  1319. </li>

  1320. <li>

  1321. <a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>

  1322. 

  1323. </li>

  1324. <li>

  1325. <a href="./d_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</span>

  1326. 

  1327. </li>

  1328. <li>

  1329. <a href="./d_emsm_msm.html#d_emsm_msm">Diagnostics: Magnetostatics in Matter</a><span class="headline-id">d.emsm.msm</span>

  1330. 

  1331. </li>

  1332. <li>

  1333. <a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>

  1334. 

  1335. </li>

  1336. <li>

  1337. <a href="./d_emd_ce.html#d_emd_ce">Diagnostics: Conservation Laws</a><span class="headline-id">d.emd.ce</span>

  1338. 

  1339. </li>

  1340. <li>

  1341. <a href="./d_emd_emw.html#d_emd_emw">Diagnostics: Electromagnetic Waves</a><span class="headline-id">d.emd.emw</span>

  1342. 

  1343. </li>

  1344. <li>

  1345. <a href="./d_emf.html#d_emf">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span>

  1346. 

  1347. </li>

  1348. <li>

  1349. <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>

  1350. 

  1351. </li>

  1352. <li>

  1353. <a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>

  1354. 

  1355. </li>

  1356. 

  1357. </ul>

  1358. </details>

  1359. </li>

  1360. <li>

  1361. 

  1362. <details>

  1363. <summary>

  1364. <a href="./a.html#a">Appendices</a><span class="headline-id">a</span>

  1365. 

  1366. 

  1367. </summary>

  1368. <ul>

  1369. <li>

  1370. <a href="./a_l.html#a_l">Literature</a><span class="headline-id">a.l</span>

  1371. 

  1372. </li>

  1373. 

  1374. </ul>

  1375. </details>

  1376. </li>

  1377. <li>

  1378. 

  1379. <details>

  1380. <summary>

  1381. <a href="./c.html#c">Compendium</a><span class="headline-id">c</span>

  1382. 

  1383. 

  1384. </summary>

  1385. <ul>

  1386. <li>

  1387. 

  1388. <details>

  1389. <summary>

  1390. <a href="./c_m.html#c_m">Mathematics</a><span class="headline-id">c.m</span>

  1391. 

  1392. 

  1393. </summary>

  1394. <ul>

  1395. <li>

  1396. 

  1397. <details>

  1398. <summary>

  1399. <a href="./c_m_va.html#c_m_va">Vector Analysis</a><span class="headline-id">c.m.va</span>

  1400. 

  1401. 

  1402. </summary>

  1403. <ul>

  1404. <li>

  1405. <a href="./c_m_va_n.html#c_m_va_n">Notation and algebraic properties</a><span class="headline-id">c.m.va.n</span>

  1406. 

  1407. </li>

  1408. <li>

  1409. <a href="./c_m_va_sp.html#c_m_va_sp">Scalar product</a><span class="headline-id">c.m.va.sp</span>

  1410. 

  1411. </li>

  1412. <li>

  1413. <a href="./c_m_va_cp.html#c_m_va_cp">Cross product</a><span class="headline-id">c.m.va.cp</span>

  1414. 

  1415. </li>

  1416. <li>

  1417. <a href="./c_m_va_tp.html#c_m_va_tp">Triple Products</a><span class="headline-id">c.m.va.tp</span>

  1418. 

  1419. </li>

  1420. <li>

  1421. <a href="./c_m_va_pds.html#c_m_va_pds">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span>

  1422. 

  1423. </li>

  1424. 

  1425. </ul>

  1426. </details>

  1427. </li>

  1428. <li>

  1429. 

  1430. <details>

  1431. <summary>

  1432. <a href="./c_m_dc.html#c_m_dc">Differential Calculus</a><span class="headline-id">c.m.dc</span>

  1433. 

  1434. 

  1435. </summary>

  1436. <ul>

  1437. <li>

  1438. <a href="./c_m_dc_g.html#c_m_dc_g">Gradient</a><span class="headline-id">c.m.dc.g</span>

  1439. 

  1440. </li>

  1441. <li>

  1442. <a href="./c_m_dc_del.html#c_m_dc_del">The \({\boldsymbol \nabla}\) Operator</a><span class="headline-id">c.m.dc.del</span>

  1443. 

  1444. </li>

  1445. <li>

  1446. <a href="./c_m_dc_div.html#c_m_dc_div">The Divergence</a><span class="headline-id">c.m.dc.div</span>

  1447. 

  1448. </li>

  1449. <li>

  1450. <a href="./c_m_dc_curl.html#c_m_dc_curl">The Curl</a><span class="headline-id">c.m.dc.curl</span>

  1451. 

  1452. </li>

  1453. <li>

  1454. <a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>

  1455. 

  1456. </li>

  1457. <li>

  1458. <a href="./c_m_dc_d2.html#c_m_dc_d2">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span>

  1459. 

  1460. </li>

  1461. 

  1462. </ul>

  1463. </details>

  1464. </li>

  1465. <li>

  1466. 

  1467. <details>

  1468. <summary>

  1469. <a href="./c_m_ic.html#c_m_ic">Integral Calculus</a><span class="headline-id">c.m.ic</span>

  1470. 

  1471. 

  1472. </summary>

  1473. <ul>

  1474. <li>

  1475. <a href="./c_m_ic_lsv.html#c_m_ic_lsv">Line, Surface and Volume Integrals</a><span class="headline-id">c.m.ic.lsv</span>

  1476. 

  1477. </li>

  1478. <li>

  1479. <a href="./c_m_ic_ftc.html#c_m_ic_ftc">The Fundamental Theorem of Calculus</a><span class="headline-id">c.m.ic.ftc</span>

  1480. 

  1481. </li>

  1482. <li>

  1483. <a href="./c_m_ic_ftg.html#c_m_ic_ftg">The Fundamental Theorem for Gradients</a><span class="headline-id">c.m.ic.ftg</span>

  1484. 

  1485. </li>

  1486. <li>

  1487. <a href="./c_m_ic_gauss.html#c_m_ic_gauss">Gauss' Theorem</a><span class="headline-id">c.m.ic.gauss</span>

  1488. 

  1489. </li>

  1490. <li>

  1491. <a href="./c_m_ic_stokes.html#c_m_ic_stokes">Stokes' Theorem</a><span class="headline-id">c.m.ic.stokes</span>

  1492. 

  1493. </li>

  1494. <li>

  1495. <a href="./c_m_ic_ip.html#c_m_ic_ip">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span>

  1496. 

  1497. </li>

  1498. 

  1499. </ul>

  1500. </details>

  1501. </li>

  1502. <li>

  1503. 

  1504. <details>

  1505. <summary>

  1506. <a href="./c_m_cs.html#c_m_cs">Coordinate Systems</a><span class="headline-id">c.m.cs</span>

  1507. 

  1508. 

  1509. </summary>

  1510. <ul>

  1511. <li>

  1512. <a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>

  1513. 

  1514. </li>

  1515. <li>

  1516. <a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>

  1517. 

  1518. </li>

  1519. <li>

  1520. <a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>

  1521. 

  1522. </li>

  1523. 

  1524. </ul>

  1525. </details>

  1526. </li>

  1527. <li>

  1528. 

  1529. <details>

  1530. <summary>

  1531. <a href="./c_m_dd.html#c_m_dd">Dirac delta Distribution</a><span class="headline-id">c.m.dd</span>

  1532. 

  1533. 

  1534. </summary>

  1535. <ul>

  1536. <li>

  1537. <a href="./c_m_dd_div.html#c_m_dd_div">The Divergence of \(\hat{\bf r}/r^2\)</a><span class="headline-id">c.m.dd.div</span>

  1538. 

  1539. </li>

  1540. <li>

  1541. <a href="./c_m_dd_1d.html#c_m_dd_1d">The One-Dimensional Dirac Delta Function</a><span class="headline-id">c.m.dd.1d</span>

  1542. 

  1543. </li>

  1544. <li>

  1545. <a href="./c_m_dd_3d.html#c_m_dd_3d">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span>

  1546. 

  1547. </li>

  1548. 

  1549. </ul>

  1550. </details>

  1551. </li>

  1552. <li>

  1553. 

  1554. <details>

  1555. <summary>

  1556. <a href="./c_m_vf.html#c_m_vf">Vector Fields</a><span class="headline-id">c.m.vf</span>

  1557. 

  1558. 

  1559. </summary>

  1560. <ul>

  1561. <li>

  1562. <a href="./c_m_vf_helm.html#c_m_vf_helm">The Helmholtz Theorem</a><span class="headline-id">c.m.vf.helm</span>

  1563. 

  1564. </li>

  1565. <li>

  1566. <a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>

  1567. 

  1568. </li>

  1569. 

  1570. </ul>

  1571. </details>

  1572. </li>

  1573. <li>

  1574. 

  1575. <details>

  1576. <summary>

  1577. <a href="./c_m_uf.html#c_m_uf">Useful Formulas</a><span class="headline-id">c.m.uf</span>

  1578. 

  1579. 

  1580. </summary>

  1581. <ul>

  1582. <li>

  1583. <a href="./c_m_uf_cyl.html#c_m_uf_cyl">Cylindrical coordinates</a><span class="headline-id">c.m.uf.cyl</span>

  1584. 

  1585. </li>

  1586. <li>

  1587. <a href="./c_m_uf_sph.html#c_m_uf_sph">Spherical coordinates</a><span class="headline-id">c.m.uf.sph</span>

  1588. 

  1589. </li>

  1590. <li>

  1591. <a href="./c_m_uf_vi.html#c_m_uf_vi">Vector identities</a><span class="headline-id">c.m.uf.vi</span>

  1592. 

  1593. </li>

  1594. 

  1595. </ul>

  1596. </details>

  1597. </li>

  1598. 

  1599. </ul>

  1600. </details>

  1601. </li>

  1602. 

  1603. </ul>

  1604. </details>

  1605. </li>

  1606. 

  1607. </ul>

  1608. </details>

  1609. </nav>

  1610. <ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_es.html">Electrostatics</a></li><li><a class="breadcrumb-link"href="ems_es_ep.html">The Electrostatic Potential</a></li><li>Electrostatic Boundary Conditions</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_PL.html">Poisson's and Laplace's Equations&emsp;<small>[ems.es.ep.PL]</small></a></li><li>Next:&nbsp;<a href="ems_es_e.html">Electrostatic Energy from the Potential&emsp;<small>[ems.es.e]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul><div id="outline-container-ems_es_ep_bc" class="outline-5">

  1611. <h5 id="ems_es_ep_bc">Electrostatic Boundary Conditions<a class="headline-permalink" href="./ems_es_ep_bc.html#ems_es_ep_bc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">

  1612.   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>

  1613.   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>

  1614. </svg></a><span class="headline-id">ems.es.ep.bc</span></h5>

  1615. <div class="outline-text-5" id="text-ems_es_ep_bc">

  1616. <p>

  1617. For a surface, Gauss's law gives us that

  1618. \[

  1619. \oint_{\cal S} {\bf E} \cdot d{\bf a} = \frac{Q_{\mbox{enc}}}{\varepsilon_0} = \frac{1}{\varepsilon_0} \sigma A

  1620. \]

  1621. where \(A\) is the area of the Gaussian pillbox and \(\sigma\) the surface charge density,

  1622. which is here assumed to be constant.

  1623. The sides contribute nothing if the pillbox is made infinitesimally thin, in which case

  1624. we get

  1625. </p>

  1626. 

  1627. <p>

  1628. \[

  1629. {\bf E}^{\perp}_{\mbox{above}} - {\bf E}^{\perp}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0},

  1630. \label{Gr(2.31)}

  1631. \]

  1632. </p>

  1633. 

  1634. <p>

  1635. so the normal component of \({\bf E}\) is discontinuous at the boundary by an amount \(\sigma/\varepsilon_0\).

  1636. </p>

  1637. 

  1638. <p>

  1639. The tangential component is continuous:  from the curlless condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a>

  1640. applied to a small loop straddling the surface:

  1641. </p>

  1642. 

  1643. <p>

  1644. \[

  1645. {\bf E}^{\parallel}_{\mbox{above}} = {\bf E}^{\parallel}_{\mbox{below}}

  1646. \label{Gr(2.32)}

  1647. \]

  1648. </p>

  1649. 

  1650. <p>

  1651. We can unify both boundary conditions into a  single equation:

  1652. </p>

  1653. <div class="eqlabel" id="orgd6777ab">

  1654. <p>

  1655. <a id="Edisc"></a><a href="./ems_es_ep_bc.html#Edisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">

  1656.   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>

  1657.   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>

  1658. </svg></a>

  1659. </p>

  1660. <div class="alteqlabels" id="orgc09e9b8">

  1661. <ul class="org-ul">

  1662. <li>Gr (2.33)</li>

  1663. </ul>

  1664. 

  1665. </div>

  1666. 

  1667. </div>

  1668. 

  1669. <p>

  1670. \[

  1671. {\bf E}_{\mbox{above}} - {\bf E}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}

  1672. \tag{Edisc}\label{Edisc}

  1673. \]

  1674. with \(\hat{\bf n}\) a unit vector normal to the surface, pointing 'out'.

  1675. </p>

  1676. 

  1677. <p>

  1678. The potential is however continuous across any boundary:  since

  1679. \[

  1680. \phi_{\mbox{above}} - \phi_{\mbox{below}} = -\int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}

  1681. \]

  1682. where the path shrinks to zero, we conclude that

  1683. \[

  1684. \phi_{\mbox{above}} = \phi_{\mbox{below}}

  1685. \label{Gr(2.34)}

  1686. \]

  1687. The gradient however inherits the discontinuity of the electrostatic field, since

  1688. \({\bf E} = -{\boldsymbol \nabla} \phi\):

  1689. \[

  1690. {\boldsymbol \nabla} \phi_{\mbox{above}} - {\boldsymbol \nabla} \phi_{\mbox{below}} = -\frac{\sigma}{\varepsilon_0} \hat{\bf n}

  1691. \label{Gr(2.35)}

  1692. \]

  1693. or

  1694. </p>

  1695. <div class="eqlabel" id="orgb49ce52">

  1696. <p>

  1697. <a id="dpdisc"></a><a href="./ems_es_ep_bc.html#dpdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">

  1698.   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>

  1699.   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>

  1700. </svg></a>

  1701. </p>

  1702. <div class="alteqlabels" id="orge2712a7">

  1703. <ul class="org-ul">

  1704. <li>Gr (2.36)</li>

  1705. </ul>

  1706. 

  1707. </div>

  1708. 

  1709. </div>

  1710. <p>

  1711. \[

  1712. \frac{\partial \phi_{\mbox{above}}}{\partial n} - \frac{\partial \phi_{\mbox{below}}}{\partial n} = -\frac{\sigma}{\varepsilon_0}

  1713. \tag{dpdisc}\label{dpdisc}

  1714. \]

  1715. where

  1716. \[

  1717. \frac{\partial \phi}{\partial n} = {\boldsymbol \nabla} \phi \cdot \hat{\bf n}

  1718. \label{Gr(2.37)}

  1719. \]

  1720. is the <b>normal derivative</b> of the potential.

  1721. </p>

  1722. 

  1723. <p>

  1724. This is the kind of boundary condition that we need to fix a unique solution

  1725. to Poisson's equation <a href="./ems_es_ep_PL.html#Poi">🐟</a>:

  1726. our only problem is that <a href="./ems_es_ep_bc.html#dpdisc">dpdisc</a> gives the change of the normal derivative of \(V\), not

  1727. its value.  However, if we assume that there are no charges living outside

  1728. of our volume \({\cal V}\), we find that <a href="./ems_es_ep_bc.html#dpdisc">dpdisc</a> fully specifies the potential's normal derivative

  1729. if the surface charge is known.

  1730. </p>

  1731. </div>

  1732. </div>

  1733. 

  1734. 

  1735. <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_PL.html">Poisson's and Laplace's Equations&emsp;<small>[ems.es.ep.PL]</small></a></li><li>Next:&nbsp;<a href="ems_es_e.html">Electrostatic Energy from the Potential&emsp;<small>[ems.es.e]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul>

  1736. <br>

  1737. <hr>

  1738. <div class="license">

  1739. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  1740. target="_blank" class="m-2">

  1741. <img alt="Creative Commons License" style="border-width:0"

  1742. src="https://licensebuttons.net/l/by/4.0/80x15.png"/>

  1743. </a>

  1744. Except where otherwise noted, all content is licensed under a

  1745. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  1746. target="_blank">Creative Commons Attribution 4.0 International License</a>.

  1747. </div>

  1748. <div id="postamble" class="status">

  1749. <p class="author">Author: Jean-Sébastien Caux</p>

  1750. <p class="date">Created: 2022-03-24 Thu 08:42</p>

  1751. <p class="validation"></p>

  1752. </div>

  1753. 

  1754. 

  1755. </div>

  1756. </html>