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q1q2 F12 ke 2 rˆ r q E ke 2 rˆ r Fe qE E E.dA surface Attractive/repulsive force between charges Every charge generates an electric field Electric Flux qin E E.dA 0 Gauss’ Law Only charges within the Gaussian surface contribute to the electric flux through the surface U q0 E.d s B Electric Potential Energy A U V E.d s q0 A B Potential Difference In a Uniform Field For a Point Charge For a Charge Distribution V Ed q V ke r V Ex x V Ey y dq V ke r V Ez z There is no electric field inside a conductor. The field on the surface is perpendicular to the surface and proportional to the surface charge density. All charges reside on the surface. Electric potential is constant throughout the conductor. Q C V Parallel Plate Capacitor For series capacitors For parallel capacitors Q 0 A C V d n 1 1 Ceq j 1 C j n Ceq C j j 1 (C/V=F) Stored Energy in a Capacitor Q2 1 1 U QV C (V ) 2 2C 2 2 Capacitor with a dielectric C C0 dQ I (t ) dt J E l R A V R I 2 V P IV I 2 R R Kirchoff’s Rules V 0 I closed loop in junction I out junction 1 FB FB qv B B q +q v FB IL B For a closed loop of current For a straight wire in a uniform field τ IA B FB 0 μ IA τ μB U μ.B v2 FB qvB m r mv r qB v qB r m T 2r 2 2m v qB Cyclotron frequency 0 I d s rˆ B 4 r 2 B.ds 0 I For r >= R 0 I B 2a For r < R I B 0 02 r 2R B 0 nI Magnetic Flux B B.dA B.dA 0 Faraday’s Law d B E N dt d B E.ds N dt The polarity of the induced emf is such that it tends to produce a current that creates a magnetic flux to oppose the change in magnetic flux through the area enclosed by the current loop. Q E.dA 0 Gauss’ Law B.dA 0 Gauss’ Law for Magnetism – no magnetic monopoles d B E.ds dt d E B.ds 0 I 0 0 dt F qE qv B Faraday’s Law Ampère-Maxwell Law Lorentz Force Law Displacement Current c 1 0 0 2f E Emax coskx t B Bmax cos kx t E Emax c B Bmax k k c f 2 Speed of Light Angular Frequency Wavelength Wavenumber 1 S EB 0 I Sav cuav 2 Bmax 1 2 2 uav 0 E av 0 Emax 2 20 Complete Absorption Complete Reflection Momentum Pressure U p c F 1 dp S P A A dt c p 2U c P 2S c q1 q1 c n v n1 sin q1 n2 sin q 2 n2 sin q c n1 pq M 1 1 1 1 p q f R f 2 h q M h p n1 n2 n2 n1 p q R 1 1 1 p q f 1 1 1 n 1 f R1 R2 M h q h p The intensity observed at any point is proportional to the time average of the sum of the fields incident on that point. If the fields are coherent and monochromatic they can interfere with each other, creating bright and dark areas (fringes). Destructive or constructive interference depends on the relative phase of the waves with each other. 2 Path length difference d sin q Double Slit 2nt Thin Film Reflection n1 > n2 : no phase n1 < n2 : phase n1 n2 Relative phase change on reflections 0 m Const. Dest. (m+½) Dest. Const. Rays going through obstacles or openings can diffract around the edges. The amount of diffraction depends on the ratio of the wavelength of the wave to the size of the opening/obstacle. The diffracted beam interferes with itself to create a pattern of bright and dark fringes on a far away screen. sin q ym m a m 1,2,3,... L m a q min For circular apertures a q min 1.22 D A device for spectrally analyzing light sources d sin q m Polarization of a wave is the direction of its electric field vector. Ordinary light is unpolarized. A polarizer passes light only with polarization along its transmission axis. If light is incident on a surface at Brewster’s angle, the reflection is also polarized. q p q2 90 n tan q p I0 I0 / 2 (I0 / 2)cos2q