Physics Formulas Educational Materials

! Energy U # Physics 2326 - Formula List ! Integrals un$1 n $C ! u "u # n$1 "u %n & '1( a2 'u2 "u u # arcsin   $ C a # ln u $ u2 $a2 1 au "u 1 u au ! 1 "u # 1 $ C  2 2 # arctan   $ C a a a u $a 1 "u 1 u $C ! sin %a u( " u # ' cos %a u( $ C  2 2 342 # 2 a a u $a  u2 $ a2 1 u "u 1 $C ! cos %a u( " u # sin %a u( $ C  2 2 342 # ' a u $a  u 2 $ a2  u # ln -u. $ C 2 "u   u2 $ a2  $ C ! Formulas from Geometry Sphere: Volume # 4 Π r3 , Area # 4 Π r2 3 8 Q1 7 Q2 r F 21 # ke Q1 Q2 122 where 8  Point Charge: E # ke Q 2 # ke Q 3 , E r r r r 8  r r Discrete: E # ke ; i Qi i2 # ke ; i Qi i3 ri 8  E # ke Q z0 8 z R2 $z20 342 r -Q. # ke r2 ?0 Qenclosed ! Potential and Potential Energy V # U 4 q0 , @ U # Q @ V For a charge distribution: U # ke A iB j Qi Q j ri j Conservation of Energy: Ki $ Ui # K f $ U f , where K # ! Potential due to Charges Point Charge: V # ke Discrete: V # ke ; i Qi ri , Continuous: V # ke  1 m v2 2 Q CV Cx and for y and z. also Er # ' r CV Cr ! Conductors in Electrostatics At surface: E D surface and E # Σ 4 ?0 Inside: E # 0 , voltage is const., no excess charge. ! Capacitance Q # C V, C is the capacitance. C # Κ C0 , where C0 # empty cap. and dielectric const. # Κ G 1 ? A C0 # 0 !H plate", C0 # d 1 ke  '  1 1 a b (sph.), I # n -q. vd A , J # n q vd ! Drift Velocity n# L of charge carriers vol. Σ # conductivity, I # J A (for a wire) , q # charge of charge carriers, vd # drift vel. ΡL ! Ohm's Law V # I R, R # O # V I, Ρ# 1 Σ A , J #ΣE # resistivity For a resistor: O # V I # I 2 R # Node Reduction: RPi j # Ri R j RH , where RH # ! Combinations of Capacitors Series: Ceq # V2 R '1 1 1 $ $ ... R1 R2 1 1 1 '1 $ $ R1 R2 R3 '1 1 1 $ $ ... , Parallel: Ceq # C1 $C2 $ ... C1 C2 C0 # 2 Π ?0 I ln%b4a( 8 8 8 x y z 8 A y Az 8 A x Az 8 A x A y A % B # A x A y Az # x 'y $z B y Bz B x Bz Bx By B x B y Bz 8 ! Magnetic Force on Particle F # Q v K B v D to uniform B R circle with r # ! Magnetic Force on Wire  (cyl.) mv Q B F # I ! " r K B F # I I K B (straight segment, uniform field) VHall # vd B L %Hall Voltage( ! Magnetic Dipoles Dipole Moment: Μ # N I A (N is # of turns) Torque: Τ # Μ K B , Τ # Μ B sin Θ, Pot. Energy: U # 'Μ # B ! Biot-Savart Law r "q ! Potential and Electric Field @V # '! E # " r , @V # 'E # @ r (uniform E) Ex # ' (current through surface) J is the current density. I # !surface J # " A, 8 Dot Product: A # B # A B cos Θ # Ax Bx $Ay By $Az Bz > E#$ A # "t ! Cross or Vector Product A % B # u A B sin Θ , right hand rule Q u ! Electric Flux < # ! E # " A , uniform E and flat surface: < # E # A ! Gauss's Law "Q ! Current and Current Density I # ! Kirchhoff's Rules Junctions: ;Iin # ;Iout , Loops: 0 # ;@ V ri (z0 from center of uniform ring) 1 Pot. Energy: U # ' p # E Series: Req # R1 $ R2 $ ... , Parallel: Req # Continuous: E # ke  2 " q # ke  3 " q r r r Torque: Τ # p K E, Τ # p E sin Θ, ! Combinations of Resistors ! Electric Field E # F 4 q0 or E # Lim F 4 q0 , F # Q E (force on Q) q0 :0 ! Electric Dipoles Dipole Moment: p # Q d, d from -Q to Q ! Power (magnitude of force) r2 8  r12 r # 12 2 r12 r12 3 and where r12 is the vector from Q1 to Q2 . Charge quantization: Q # n e, n is an integer. r12 (energy density in a field) @ R # Α R0 @ T, R # R0 %1 $ Α@ T ( , @Ρ # Α Ρ0 @ T, Ρ # Ρ0 %1 $ Α@ T ( Right Cylinder: Volume # Area 6 height F " ke Volume ! Temperature Dependence @ T # T 'T0 Circle: Area # Π r2 , Circumference # 2 Π r ! Coulomb's Law ? u # 0 E2 # Q2 1 1 C V2 # # Q V (energy in a cap.) 2 2 2C Energy 8 Μ I r B # 0  " s K 2 4Π r 8 Μ0 I Θ (at center of arc in xy-plane) 4ΠR R2 8 Μ I (z from center of circle) B#z 0 2 2 R $z2 342 B#z Μ I Μ I B # 0 %sin Φ2 ' sin Φ1 ( # 0 4Πa 4Πa x2 x22 $a2 Μ I B # 0 (distance r from long wire) 2Πr '  > B # " r # Μ0 Ienclosed L of turns B # Μ0 n I, n # (inside long solenoid) length ! Ampere's Law x1 x12 $a2 (segment) E#" A # 1 Qenc , ?0  > B # " r # Μ0 Ienc $ Μ0 ?0 " B#" A # 0  > E#"r # ' " I1 Long Solenoid: L # Μ0 n2 A I # Μ0 Energy in Inductor: U # ! Energy Density u # u # ue $ um # ?0 2 E2 $ 1 L I2 2 "t I "