Equipotential surfaces

Transcription

1 Equiotential surfaces dv E ds E d E y dy E z dz E ˆ V yˆ V zˆ V E y z E V V V V, E y, Ez or y z Equiotential surfaces are E V (r) const uniform field From the differential Ohm s law, V Jϱ, it follows that metal surfaces are equiotential in the absence of electric currents, and also that E Jϱ J σe 1

2 Dielectric media E 0 0 E olarized dioles At a boundary between two dielectrics: E 0 + χ e E E E 0 + E int D εe + P ε + 0 E + ε 0 χ e E + Polarized medium ε ε 0 (1+χ e ) ε 1ε 0 for silicon D 1 must be continuous ε 1 D ε The electric flu density across the interface D D 1 D E 1y D 1 E y E 1 E The surface charge contributes only to the normal comonent of the electric field E 1y E y

3 aacitors d +Q V A V ε -Q A roortionality coefficient Q V is the caacitance of the caacitor For a arallel late caacitor, Q V Φ V DA V εea Ed εa d Energy of a charged caacitor: W aacitors in arallel and in series: Q Vdq 0 Q 0 q dq Q W V

4 agnetic fields ( 16) Permanent magnets and magnetic field agnetic fields of currents: Amère s law and Biot-avart law agnetic flu and magnetic flu density Gauss law for magnetism Forces between magnets and currents agnetic materials Electric motors Force on a moving charge in a magnetic field 4

5 Permanent magnets and magnetic field Bar magnet magnetic diole agnetic oles are similar to ositive and negative charges creating a magnetic field unlike oles attract like oles reel N agnetic field force er unit magnetic charge F F 5

6 agnetic fields of currents: Amère s law F The mechanical work uon moving a ole around the wire W F dl πr and the magnetic field is πr (W qv for q) (intuitive) Amere s law: For any current and any loo around it, we always obtain F dl dl n terms of the electric current density J, J dl J da 6

7 7 Amère s law: An eamle of alication d d J l agnetic field inside a long solenoid: l N n l N l y z N turns l z J + + l y d d d d z J y ˆ 0 0 0

8 y P ϕ R r θ l 0 The Biot-avart law The element l should at P roduce a z-directional magnetic field with a strength l sinθ rojection of l onto For a straight long wire along l we must obtain r 1 similarly to E q q/(4πεr ) π / 1 sin θ R a a sinθ d a sinθ θ π d a R r R sin θ R 0 1 4π Thus we obtain the Biot-avart law: n a general form, we have d dl sinθ zˆ 4πr π dl rˆ 4 circuit r dl rˆ a b c c absinθ c 8

9 agnetic flu and magnetic flu density Total flu of the magnetic field is Φ. n, the roortionality coefficient is 1 and we have Φ [Wb] agnetic flu density B (the flu er unit surface area) must be roortional to and be a vector: dφ B bˆ µ da, where A is erendicular to B, and µ is the magnetic ermeability µ 0 4π 10-7 NA -1 Total flu through any surface area A is [T] B Φ B da A A dabcosα 9

10 Gauss law for magnetism agnetic field of a single magnetic ole: B, A 4 π Φ r B B 4πr 4πr 4πµ r indeendent of µ deends on µ Gauss law for magnetism: Φ B da i i inside total owever, indeendent magnetic oles (monooles) do not eist, and the flu inside the magnet makes the total flu around the ole equal to zero B da B da 0 always!

11 Between oles of bar magnets: agnetic forces F 1 1 F 4πµ r 4πµ r attractive, if 1 < 0 reulsive if, 1 > 0 Between a current element and a ole of a magnet: df d dl (imosed on ) z y r dl d dl dl ( rˆ) 4πr df dl rˆ 4πr rˆ B rˆ df dl B 4πµ r 4πr (on dl) 11

12 Torque on a bar magnet and a current loo Bar magnet in a magnetic field T r F l m B m l / µ is the diole moment urrent loo in a magnetic field -F - B l T F y. z F 1 l 1 B B A l F B onsider the grey stri of length and width y F F ˆ l1b sin 1 1 θ ˆ lbsin θ ˆ yb The torque on the stri is T 0.5 ( F1 F ˆ yb ) yˆ yb yˆ AB The total torque on the loo is therefore T yˆ BA f the loo surface forms an angle α with B, we have T yˆ BA sinα T A B T m B m A is the diole moment of the loo 1

13 agnetic materials 0, m e 0 0 ferromagnetic 0, m e 0 0, 0 aramagnetic 0, m e 0 0, m e 0 diamagnetic 13

14 Boundaries between magnetic media At a boundary between two magnetic media: + + B µ + + µ 0 + µ 0 χ m magnetized medium χ m, µ µ 0 (1+χ m ) B 1 must be continuous µ 1 B µ The magnetic flu density across the interface B magnetic dioles B 1 B 1y B 1 y 1 The magnetization affects only the normal comonent of the magnetic field 1y y 14

15 Electric motor Princile: F lb T A B A motor 15

16 Force on a moving charge in a magnetic field F l F l B ρva l B Vρv B Qv B ρva A l V Vρ Q Q A l F qv B f both magnetic and electric fields are resent, the force on a moving charge is F qe + qv B (Lorentz force) f ρe + J B The force density (local, er unit volume) is then. The motion trajectory: v F qvb m R R mv qb mass sectrometry article accelerators 16