Magnetostatics. Magnetic Forces. = qu. Biot-Savart Law H = Gauss s Law for Magnetism. Ampere s Law. Magnetic Properties of Materials. Inductance M.

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Franklin Copeland
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1 Magnetic Foces Biot-Savat Law Gauss s Law fo Magnetism Ampee s Law Magnetic Popeties of Mateials nductance F m qu d B d R 4 R B B µ 0 J Magnetostatics M.

The magnetic flux density B at a point in space is defined in tems of the magnetic

2 Magnetic Foces The electic field E at a point in space has been defined as the electic foce F e pe unit chage acting on a test chage when place at that point. The magnetic flux density B at a point in space is defined in tems of the magnetic foce F m that would be exeted on a chaged paticle moving with a velocity u. F m qu B M.

3 Magnetic Foces on Cuent-Caying Conducto A cuent flowing though a conducting wie consists of chaged paticles difting though the mateial of the wie. Theefoe, when a cuent-caying wie is place in a magnetic field, it will expeience a foce. df m F m d l B l c d B Demonstation: M5. Q-Q3, M5. M.3

4 Biot-Savat Law Used fo calculating the magnetic field intensity d due to a cuent flowing in an element dl. d R d A / 4 R m The esultant magnetic field intensity is found by integating d fo all cuent elements 4 l d R R M.4

5 Gauss s Law fo Magnetism Fo electic fields, the net outwad flux of the electic flux density D though a closed suface enclosing a net chage Q is equal to Q. The magnetic analogue to a point chage is a magnetic pole, but magnetic poles do not exist in isolation. --- they occu in pais. B B ds 0 s 0 may be called law of nonexistence of isolated monoploes o the law of consevation of magnetic flux and Gauss s law fo magnetism M.5

6 Ampee s Law The line integal of aound any closed path C is equal to the cuent enclosed by the path (in a ight-handed sense). c J dl C dl Not enclosed M.6

7 M.7 Example: Magnetic Field of a Long Wie (a) a Choose the Ampeian contou C to be a cicula path of adius C dl ( ) 0 d d C l LS: The cuent flowing though the aea enclosed by C is equal to a a

8 M.8 Example: Magnetic Field of a Long Wie (b) a a > Choose the Ampeian contou C which encloses all the cuent ( ) d d C 0 l

9 Example: Magnetic Field of a Long Wie a M.9

10 Example: Magnetic Field inside a Tooidal Coil A tooidal coil (also called tous o tooid) is a doughnut-shaped stuctue with closely spaced tuns of wie wapped aound it. M.0

11 M. Example: Magnetic Field inside a Tooidal Coil Applying the Ampee s law along the contou C: ( ) ( ) b a N N d d C < < 0 l

12 Magnetic Popeties of Mateials Magnetization in a mateial is associated with atomic cuent loops geneated by Obital motions of the electons aound the nucleus Electon spin The magnetization vecto M of a mateial is defined as the vecto sum of the magnetic dipole moments of the atoms contained in a unit volume of the mateial B µ + µ µ µ o o o o ( + M ) ( + χ ) µ µ o m M Magnetic susceptibility Q M χ m Relative pemeability M.

13 Feomagnetic Mateials Feomagnetic mateials, which include ion, nickel, and cobalt, exhibit stong magnetic popeties due to the fact that thei magnetic moments tend to align eadily along the diection of the extenal magnetic field. such mateial emain patially magnetized even afte the emoval of the extenal field. M.3

14 Feomagnetic Mateials Unmagnetized domains Magnetized domains M.4

15 Feomagnetic Mateials Residual flux density Satuation Typical hysteesis cuve fo a feomagnetic mateial M.5

16 Summay Static Electic field Static Magnetic field Souce Chages Cuent Divegence equation (Gauss s law) D ρ v B 0 Dds Q B ds 0 s s Cul equation E 0 J c Edl 0 c dl M.6

17 Φ ij nductance L ij flux linking due to cuent in S i cuent in Si L is called the self-inductance; L is called the mutualinductance Φ B ds s L Φ M.7

18 M.8 Example: nductance of a Coaxial Tansmission Line Due to the cuent in the inne conducto, the magnetic field geneated inside the coaxial tansmission line is given by d d c µ 0 B l C

19 Example: nductance of a Coaxial Tansmission Line Ove the plana suface S, B is pependicula to the suface. ence, the flux though S is Φ l b a Bd b µ l d a µ l ln( b / a) The inductance pe unit length is L Φ / l µ ln( b / a) M.9

Unit 7: Sources of magnetic field

Unit 7: Sources of magnetic field Unit 7: Souces of magnetic field Oested s expeiment. iot and Savat s law. Magnetic field ceated by a cicula loop Ampèe s law (A.L.). Applications of A.L. Magnetic field ceated by a: Staight cuent-caying