SUMMARY Phys 2523 (University Physics II) Compiled by Prof. Erickson. F e (r )=q E(r ) dq r 2 ˆr = k e E = V. V (r )=k e r = k q i. r i r.

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1 SUMMARY Phys 53 (University Physics II) Compiled by Prof. Erickson q 1 q Coulomb s Law: F 1 = k e r ˆr where k e = 1 4π = N m /C, and = C /(N m )isthepermittivity of free space. Generally, F e (r )=q E(r ) Electric Field (N/C=V/m): q i E(r )=k e r ˆr = k r i r e qi r i r 3 (distribution of point charges) i i dq E(r )=k e r ˆr = k e q i r r dq r r 3 (continuous distribution) where dq = ρd 3 r, and ρ is the charge density. In steady-state, the electrical force is conservative, and the electric field can be derived from a potential V: E = V where is the gradient operator, = ˆx d dx + ŷ d dy + ẑ d dz in rectangular coordinates. Potential Energy (J): B U = q E ds A Electric Potential (V=J/C): V = U/q ; V = U/q q i V (r )=k e r = k q i e r i r i i dq V (r )=k e r = k e dq r r (distribution of point charges) (continuous distribution) Electric charge is quantized in units of e = C. An electron volt, ev = J. Electric Flux (N m /C=V m): Φ E = E da surface

2 Gauss s Law: Φ E = E da = Q inside surface (integral form) E = ρ (differential form) A conductor in electrostatic equilibrium has the following properties: 1. The electric field is zero everywhere inside the conductor.. Any net charge on the conductor resides entirely on its surface. 3. The electric field just outside the conductor is perpendicular to its surface and has a magnitude σ/,whereσ is the surface charge density at that point. 4. On an irregularly shaped conductor, the surface charge density is greatest where the radius of curvature of the surface is the smallest. 5. The potential is constant everywhere inside a conductor and equal to its value at the surface. Capacitance (F): C Q V and the energy stored in a capacitor is U = 1 C( V ). A dielectric material is composed of electric dipoles of magnitude p = aq, wherea is the dipole separation. A background electric field exerts a torque τ = p E on the dipoles. The potential energy of the system of an electric dipole in an electric field is U = p E. The potential energy of the dipoles reduces the potential difference in a capacitor, V = V κ, and C = κc where κ is the dielectric constant. The alignment of its dipoles results in an induced electric field (E ind ) that reduces the background electric field (E ) inside the dielectric to a value E = E E ind. A charge density σ ind = κ 1 κ σ is induced on its surface. Its dielectric strength is the maximum electric field that can be applied to the dielectric before its insulating properties break down and it begins to conduct. The capacitance of a parallel-plate capacitor is C = κ A d where A is the area of each plate, and d is the plate separation. Energy Density (J/m 3 ): u = E is the energy density contained in an electric field, where = κ.

3 Electrical Current (A=C/s): I = dq dt Ohm s Law: J = σe where J = nqv d is the current density, and σ is the conductivity. The resistivity ρ =1/σ = m e /(nq τ) in units of Ω m, where τ is the mean time between collisions. Over a limited temperature range, the resistivity of a conductor varies approximately linearly with temperature, ρ = ρ [1 + α(t T )], where α = 1 ρ ρ T is the temperature coefficient of resistivity. Resistance, R can be expressed similarly. Resistance (Ω=V/A): R V I Note that this equation, V = IR, is also referred to as Ohm s law. In terms of the resistivity, R = ρl/a, wherel is the length and A is the cross-sectional area of the material. Electrical Power (W): P = I V = I R = ( V ) R The emf of a battery is the maximum possible voltage that the battery can provide between its terminals. The terminal voltage will be V = Ir,wherer is the internal resistance of the battery. Combinations of Circuit Elements: 1 = 1 C eq i C i R eq = R i i (series combination) (series combination) C eq = C i i 1 = 1 R eq i R i (parallel combination) (parallel combination) Kirchhoff s Rules: Iin = I out (junction rule) V =0 closed loop (loop rule) 3

4 RC Circuits: Charging: q = q 1 e t/τ ; I = I e t/τ where q is the charge on the capacitor at time t, q = C( V ) is the equilibrium (fully charged) value of the charge, I = V/R is the current at t = 0, and τ = RC is the time constant of the dc circuit. Discharging: q = q e t/τ ; I = q RC e t/τ where q is the charge at t = 0. Magnetic Force on Particle: F B = qv B; F B = vb sin θ Gyroradius: r g = mv q B where the sense of rotation for a negatively charged particle follows the right-hand rule. When placed in a force field, F, in addition to a magnetic field, B, a charged particle s gyrocenter will drift with a velocity: v drift = F B qb ; v drift = F q B sin θ in order that the net force on the particle be zero. For example, when the additional force is the electrical force, a charged particle will drift such that F net = q(e + v B) = 0. Solving for the velocity of its gyrocenter: v drift = E B B ; v drift = E B sin θ Force on a Current Segment: F = IL B; F = ILB sin θ Torque on a Current Loop: where the magnetic moment is µ = NIA. τ = NIA B = µ B; τ = NIAB sin θ 4

5 Magnetic Field Due to Currents (T): Long, straight wire: Center of a current loop: Center of a partial current loop: Interior of a solenoid: B = µ I πr B = Nµ I R B = Nµ Iθ 4πR B = µ ni = µ NI where µ =4π 10 7 T m/a, N is the number of turns, and the direction of B is found using the right-hand rule. Ampere s Law: B d = µ I enclosed Biot-SavartLaw: db = µ I 4π d ˆr r Energy Density (J/m ): u = B µ Magnetic Flux (Wb=T m ): Φ B = B A = BAcos θ Faraday s Law (V): Motional Emf: = N dφ B dt = vbl Electrical generator: = NBAω sin ωt = max sin ωt where ω =πf is the angular frequency, and max = NBAω Lenz s law is stated as follows: The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change. This statement is a consequence of the law of conservation of energy. 5

6 Inductance (H): Self Inductance: L = NΦ B I = L I t Energy stored in inductor: U = 1 LI Mutual Inductance: M = N SΦ S I P = M I P t Transformers: V S V P = N S N P I S I P = N P N S where the subscript P refers to the primary coil and subscript S refers to the secondary coil. AC Circuits: The output of an AC generator is sinusoidal, the voltage varies as V = V max sin ωt where ω = πf is the angular frequency, and f is the linear frequency. voltage are The rms current and I rms = I max ; V rms = V max The voltage and current across a resistor are in phase V R,rms = I rms R The voltage across a capacitor lags the current by 90, and where the X C = 1 ωc V C,rms = I rms X C is the capacitive reactance. The voltage across an inductor leads the current by 90, and V L,rms = I rms X L where the X C = ωl is the inductive reactance. 6

7 RLC Series Circuit: Ohm s law for the LRC series circuit reads V max = I max Z where Z = R +(X L X C ) is the impedence of the circuit. The phase angle φ between the current and the voltage obeys tan φ = X L X C R The power delivered by the generator is dissipated in the resistor; there is no power loss in the ideal capacitor or inductor. P ave = I rmsr = I rms V rms cos φ where the voltage across the resistor is V R = V rms cos φ. Resonance in a Series RLC Circuit: According to Ohm s law, the rms current is maximum when the impedence is a minimum. The minimum value of the impedence is R and occurs when X L = X C. This occurs at the resonant frequency: f = 1 π LC Maxwell s Equations: E = ρ B =0 E = db dt B = µ j + µ de dt Electromagnetic Waves/Light: c = 1 µ = m/s v = fλ = 1 µ 7

8 Average energy density: ū = 1 E max + 1 µ B max = E rms = 1 µ B rms where E = vb, E rms = E max / and B rms = B max / Intensity (Poynting vector) (W/m ): S = 1 µ E B; S = vu, S = vū Doppler effect: f obs = f source 1 ± v rel c Reflection: θ i = θ r Index of refraction: n = c v 1; λ n = λ n Snell s Law of Refraction: n 1 sin θ 1 = n sin θ Total Internal Reflection: sin θ c = n n 1, where (n 1 >n ) Single-slit diffraction: sin θ dark = mλ,m= ±1, ±,... W Double-slit diffraction: sin θ bright = mλ,m=0, ±1, ±,... d sin θ dark = m + 1 λ,m=0, ±1, ±,... d Diffraction grating: sin θ bright = mλ,m=0, ±1, ±,... d Additional properties: Interference, Fermat s Principle, Dispersion, Huygens Principle 8

9 Quantum Effects: (h = Planck s constant = J s) Photon: E = hf, p = h λ = hf c Photoelectric effect: hf = KE max + W, where W = work function debroglie wavelength: λ = h p Heisenberg Uncertainty Principle: p x x h 4π, E t h 4π Bohr model (1 valence electron): r n = r 1n Z eff, where r 1 = m = Bohr radius E n = (13.6 ev) Z eff n Quantum numbers: Principal Quantum Number: n = 1,, 3,... (E n = (13.6 ev) Z eff n ) Orbital Quantum Number: l = 0, 1,,..., (n 1) (L = l(l + 1) h π ) Magnetic Quantum Number: m l = l,..., 1, 0, 1,,..., l (L z = m l h π ) Spin Quantum Number: m s = ± 1 The Pauli Exclusion Principle states that no two electrons in an atom may have the same set of quantum numbers. 9