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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 class="toc-currentpage">

  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>

  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>Example calculations for the potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_fp.html">Field in terms of the potential&emsp;<small>[ems.es.ep.fp]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep_PL.html">Poisson's and Laplace's Equations&emsp;<small>[ems.es.ep.PL]</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_ex" class="outline-5">

  1611. <h5 id="ems_es_ep_ex">Example calculations for the potential<a class="headline-permalink" href="./ems_es_ep_ex.html#ems_es_ep_ex"><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.ex</span></h5>

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

  1616. <div class="example div" id="orga8633d3">

  1617. <p>

  1618. <b>Spherical shell: via \({\bf E}\)</b>

  1619. </p>

  1620. 

  1621. <p>

  1622. Consider a spherical shell of radius \(R\) with uniform surface charge density

  1623. and total charge \(q\) (so \(\sigma = q/4\pi R^2\)). The shell is centered at the origin.

  1624. </p>

  1625. 

  1626. <p>

  1627. We want to know the potential throughout space. Starting from the field

  1628. as obtained from applying Gauss's law <a href="./ems_es_ef_Gl.html#Gl_i">Gl_i</a>,

  1629. </p>

  1630. 

  1631. <p>

  1632. \[

  1633. {\bf E} = \frac{1}{4\pi \varepsilon_0} q \frac{\hat{\bf r}}{r^2}, \hspace{5mm} r &gt; R, \hspace{2cm}

  1634. {\bf E} = 0, \hspace{5mm} r &lt; R.

  1635. \]

  1636. </p>

  1637. 

  1638. <p>

  1639. Adopting our usual convention of putting the reference point at infinity,

  1640. and using spherical symmetry, we get:

  1641. </p>

  1642. 

  1643. <ul class="org-ul">

  1644. <li>for \(r &gt; R\):</li>

  1645. </ul>

  1646. <p>

  1647. \[

  1648. \phi(r &gt; R) = -\int_{{\cal O}: \infty}^{\bf r} {\bf E} \cdot d{\bf l} = -\frac{1}{4\pi \varepsilon_0} \int_{\infty}^r q \frac{dr'}{r'^2}

  1649. = \frac{1}{4\pi \varepsilon_0} \frac{q}{r},

  1650. \]

  1651. </p>

  1652. 

  1653. <ul class="org-ul">

  1654. <li>for \(r &lt; R\): the integrand vanishes for the part \(r &lt; R\), and</li>

  1655. </ul>

  1656. <p>

  1657. \[

  1658. \phi(r &lt; R) = \phi(R) = \frac{1}{4\pi \varepsilon_0} \frac{q}{R}.

  1659. \]

  1660. </p>

  1661. 

  1662. </div>

  1663. 

  1664. <div class="example div" id="org9e191c6">

  1665. <p>

  1666. <b>Spherical shell: direct calculation</b>

  1667. </p>

  1668. 

  1669. <p>

  1670. We can also compute the potential for the uniformly-charged shell using

  1671. a direct integral, starting from <a href="./ems_es_ep_d.html#p_scd">p_scd</a>

  1672. </p>

  1673. 

  1674. <p>

  1675. \[

  1676. \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int da' \frac{\sigma}{|{\bf r} - {\bf r}'|}.

  1677. \]

  1678. </p>

  1679. 

  1680. <p>

  1681. Exploiting spherical symmetry, let us orient \({\bf r}\) along the \(\hat{\bf z}\) axis

  1682. so \({\bf r} = z \hat{\bf z}\). Our integration point \({\bf r}'\) is at position

  1683. \(R \sin \theta' \cos \varphi' \hat{{\bf x}} + R \sin \theta' \sin \varphi' \hat{\bf y} + R\cos \theta' \hat{\bf z}\) so

  1684. </p>

  1685. 

  1686. \begin{equation*}

  1687. |{\bf r} - {\bf r}'|^2 = R^2 \sin^2 \theta' + (R \cos \theta' - z)^2

  1688. = R^2 + z^2 - 2Rz \cos \theta'

  1689. \end{equation*}

  1690. 

  1691. <p>

  1692. Using the element of surface area (in spherical coordinates)

  1693. \(R^2 \sin \theta' d\theta' d\varphi'\), so

  1694. </p>

  1695. 

  1696. \begin{align}

  1697. 4\pi \varepsilon_0 \phi(z) &amp;= \sigma \int_0^{\pi} d\theta' \int_0^{2\pi} d\varphi'  \frac{R^2 \sin \theta'}{\sqrt{R^2 + z^2 - 2Rz \cos \theta'}} \nonumber \\

  1698. &amp;= 2\pi R^2 \sigma \int_0^{\pi} d\theta' \frac{\sin \theta'}{\sqrt{R^2 + z^2 - 2Rz \cos \theta'}} \nonumber \\

  1699. &amp;= 2\pi R^2 \sigma \left. \left( \frac{\sqrt{R^2 + z^2 - 2Rz \cos \theta'}}{Rz} \right)\right|_0^{\pi} \nonumber \\

  1700. &amp;= \frac{2\pi R\sigma}{z} \left(\sqrt{R^2 + z^2 + 2Rz} - \sqrt{R^2 + z^2 - 2Rz} \right) \nonumber \\

  1701. &amp;= \frac{2\pi R \sigma}{z} \left( \sqrt{(R + z)^2} - \sqrt{(R - z)^2} \right).

  1702. \end{align}

  1703. 

  1704. <p>

  1705. We must obviously choose the positive root \(\sqrt{(R + z)^2} = R+z\) and thus

  1706. </p>

  1707. \begin{align}

  1708. \phi(z) = \frac{R\sigma}{2\varepsilon_0 z} ((R+z) - (z-R)) = \frac{R^2 \sigma}{\varepsilon_0 z}, \hspace{1cm} \mbox{outside}, \nonumber \\

  1709. \phi(z) = \frac{R\sigma}{2\varepsilon_0 z} ((R+z) - (R-z)) = \frac{R \sigma}{\varepsilon_0}, \hspace{1cm} \mbox{inside}.

  1710. \end{align}

  1711. <p>

  1712. In terms of total charge on the shell, \(q = 4\pi R^2 \sigma\),

  1713. we thus obtain

  1714. \(\phi(z) = \frac{1}{4\pi \varepsilon_0} \frac{q}{z}\) outside,

  1715. and \(\phi(z) = \frac{1}{4\pi \varepsilon_0} \frac{q}{R}\) inside which indeed

  1716. correspond to our previous solution via \({\bf E}\).

  1717. </p>

  1718. 

  1719. </div>

  1720. </div>

  1721. </div>

  1722. 

  1723. 

  1724. 

  1725. 

  1726. <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_fp.html">Field in terms of the potential&emsp;<small>[ems.es.ep.fp]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep_PL.html">Poisson's and Laplace's Equations&emsp;<small>[ems.es.ep.PL]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul>

  1727. <br>

  1728. <hr>

  1729. <div class="license">

  1730. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  1731. target="_blank" class="m-2">

  1732. <img alt="Creative Commons License" style="border-width:0"

  1733. src="https://licensebuttons.net/l/by/4.0/80x15.png"/>

  1734. </a>

  1735. Except where otherwise noted, all content is licensed under a

  1736. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  1737. target="_blank">Creative Commons Attribution 4.0 International License</a>.

  1738. </div>

  1739. <div id="postamble" class="status">

  1740. <p class="author">Author: Jean-Sébastien Caux</p>

  1741. <p class="date">Created: 2022-03-24 Thu 08:42</p>

  1742. <p class="validation"></p>

  1743. </div>

  1744. 

  1745. 

  1746. </div>

  1747. </html>