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These are your formula sheets DO NOT TURN IT IN Derivatives Integrals d axn an xn 1 dx d sin ax a cos ax dx d cos ax a sin ax dx Z a xn dx a Z Z Z d ax e aeax dx dx ln x x 1 sin ax dx cos ax a cos ax dx Z d 1 ln ax dx x Z xn 1 n 1 1 sin ax a 1 eax dx eax a dx x arcsin a a 2 x2 Z dx 2 2 ln x a x x2 a 2 Z dx 1 x arctan 2 2 x a a a Z 1 dx x 2 3 2 2 a x a2 x2 a2 Z 1 x dx 3 2 2 x a2 x2 a2 Constants 0 8 8542 10 12 C2 N m2 0 4 10 7 Wb A m c 2 9979 108 m s DO NOT TURN THESE SHEETS IN Physics 208 Formula Sheet for Final Exam Do NOT turn in these formula sheets Electromagnetic waves Maxwell s equations predict the existence of electromagnetic waves that propagate in vacuum with the electric and magnetic fields perpendicular and with ratio E cB The waves travel with velocity c where 1 c 0 0 Energy in Electromagnetic waves The energy flow rate power per unit area of an elec tromagnetic wave is given by the Poynting vector S 1E B S 0 is called The magnitude of the time averaged value of S the intensity of the wave I 1 Emax Bmax E2 1 2 max 0 cEmax 2 0 2 0 c 2 Speed of light in materials When light propagates through a material its speed is lower than the speed in free space space by a factor called the index of refraction c v n Reflection and refraction At a smooth interface the incident reflected and refracted rays and the normal to the interface all lie in a single plane The angle of incidence and angle of reflection measured from the normal are equal r a and the angle of refraction is given by Snell s law na sin a nb sin b Polarization A polarizing filter passes waves that are linearly polarized along its polarizing axis When polarized light of intensity Imax is incident on a polarizing filter used as an analyzer the intensity I of the light transmitted depends on the angle between the polarization direction of the incident light and the polarizing axis of the analyzer I Imax cos2 Spherical Mirrors Object and image distances 1 1 1 s s0 f where f R 2 Thin Lenses Object and