Physics 1C at UCLA 0 Formula Sheet (1 of 2) Mechanics, Slides of Calculus

Download Physics 1C at UCLA 0 Formula Sheet (1 of 2) Mechanics and more Slides Calculus in PDF only on Docsity! Physics 1C at UCLA ♦ Formula Sheet (1 of 2) Mechanics Forces change the momentum: ~F = d~p dt = m~a = m d~x dt Forces can come from potentials: ~F = −~∇U Which is equivalent to the work: U = − ∫ ` d~̀ · ~F Momentum and energy are conserved if they don’t enter or leave the system. Maxwell’s Equations In differential form: ~∇ · ~E = ρ ε0 ~∇× ~E = −∂ ~B∂t ~∇ · ~B = 0 ~∇× ~B = µ0 ~J + 1 c2 ∂ ~E ∂t In integral form:∫ A d ~A · ~E = ∫ V ρ ε0 dV∮ ` d~̀ · ~E = − ∂ ∂t ∫ A d ~A · ~B = −∂ΦB ∂t∫ A d ~A · ~B = 0∮ ` d~̀ · ~B = ∫ A d ~A · ( µ0 ~J + 1 c2 ∂ ~E ∂t ) = µ0Iencl + 1 c2 ∂ΦE ∂t Vector Algebra The dot product, or overlap, is defined: ~u · ~v = uxvx + uyvy + uzvz = ||~u|| ||~v|| cos(θ) where θ is the angle between ~u and ~v. The cross product, or normal volume: ~u× ~v = ∣∣∣∣∣∣ x̂ ŷ ẑ ux uy uz vx vy vz ∣∣∣∣∣∣ where ∣∣... ... ... ∣∣ indicates the determinant. Vector Calculus The gradient of a scalar function f : ~∇f(x, y, z) = ∂f ∂x x̂+ ∂f ∂y ŷ + ∂f ∂z ẑ The divergence of a vector function ~f : ~∇· ~f(x, y, z) = ∂ ~f · x̂ ∂x + ∂ ~f · ŷ ∂y + ∂ ~f · ẑ ∂z The curl of a vector function ~f : ~∇× ~f(x, y, z) = ∣∣∣∣∣∣ x̂ ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z ~f · ~x ~f · ~y ~f · ~z ∣∣∣∣∣∣ Lorentz Force Law For a point charge: ~F = q ~E + q(~v × ~B) For part of a neutrally charged wire: d~F = I d~̀× ~B Note that magnetic fields do no work! Cyclotron Motion If ~B ⊥ ~v then a point charge will gyrate with period T = 2πR/||~v|| and radius: R = m ||~v|| |q| || ~B|| If ~B · ~v = || ~B|| ||~v|| cos θ, then the radius is R 7→ R/ sin(θ) and motion is helical. Electromagnetic Flux The electric flux through an area A: ΦE = ∫ A d ~A · ~E The magnetic flux through an area A: ΦB = ∫ A d ~A · ~B Biot-Savart Law The ~B-field of a moving charge is: ~B(~r) = µ0 4π q ~v × ~r ||~r||3 Or for a differential current element: d ~B(~r) = µ0 4π I d~̀× ~r ||~r||3 Ampere’s Loop Law Ampere’s Law is useful for constant currents with circular symmetry:∮ ` d~̀ · ~B = µ0Iencl Some results found via Ampere’s Law: Current Geometry Magnetic Field Long straight wire ~B = µ0I 2πr θ̂ Above circular loop ~B = µ0Ia 2 ẑ 2(z2+a2)3/2 Inside conductor ~B = µ0I 2π r R2 θ̂ Center of solenoid ~B = µ0NI ẑ Faraday Law & Inductance There is a voltage (emf) associated with changing magnetic fluxes: E = −dΦB dt Let the self-inductance of a current be: L = ΦB/I Energycanbe stored inmagneticfields: UL = 1 2LI 2 Circuit Elements The voltages of circuit components are: Ebattery = E0 Eresistor = IR Ecapacitor = Q/C Einductor = −LdI/dt if a capacitor is in a wire, I = dQ/dt. Kirchhoff’s Laws Voltage is single-valued, so the sum of voltages around any loop is zero:∑ E = 0 Charge is conserved (no accumulation), so at any junction currents sum to zero:∑ ⊥ I = 0 Kirchoff’s Laws form a system of linear ODEs/equations which can be solved. Characteristic Equations A powerful method to solve linear homogeneous differential equations is the method of characteristic equations. Consider the ODE: a d2q(t) dt2 + b dq(t) dt + c q(t) = 0 The general solution to this is: q(t) = q+e ω+t + q−e ω−t where: ω± = −b± √ b2 − 4ac 2a and q± are fixed by boundary values. Alternating Current AC circuits are periodically driven: Esource = E0 cos(ωt) Isource = I0 cos(ωt) Ohm’s Law with complex impedances: E = IZ Where the impedances are given by: Zresistor = R Zcapacitor = 1/iωC Zinductor = iωL Series and parallel impedances add as: Zseries = Z1 + Z2 + · · ·+ Zn Zparallel = 1 1 Z1 + 1 Z2 + · · ·+ 1 Zn If we only care about the amplitudes then we can consider the magnitude: |Z| = √ Re(Z) + Im(Z) Spenser Talkington C spenser.science B Winter 2021