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PHY49 Fall 15 – Acosta, Woodard Recalling that / () 01 q t CV t q qe==−t τ in which the dischargingconditions qCCV(0) (0)==E and q() 0∞= give q 0 =0 tand qCV 1 = (0), we have Vt V e( ) (0)= − /τThe third condition Vs V Ve(100 ) 15 (150 )==−(100 )/s τ determines τ to be 15 τ=100 /( ln ) ss− 150 =, which determines the voltage at all times, and so weSearch the world's information, including webpages, images, videos and more Google has many special features to help you find exactly what you're looking forShare your videos with friends, family, and the world



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5 Application of Ordinary Differential Equations Series RL Circuit RL circuit diagram The RL circuit shown above has a resistor and an inductor connected in series A constant voltage V is applied when the switch is closed The (variable) voltage across the resistor is given by V R = i R \displaystyle {V}_ { {R}}= {i} {R} V ROhm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship =, where I is the current through the conductor in units of amperes, V is the voltage measured across the conductor in2 0 9 0 0 N E 3 0 T H A V E , S T E 3 0 7 A V E N T U R A , F L 3 3 1 8 0 T C I I C A P I T A L C O M M A R C L E W I N 1 marc@tciicapitalcom O F F E R I N G S U M M A R Y Lease Rate Negotiable Number of Units 34 Available SF 9 2,113 SF Building Size 91,956 SF



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Have to solve this differential equation, to get position r and velocity (v= r˙) given E(r, t), B(r, t) Approach Start simple, gradually generalize 21 Uniform B field, E = 0 mv˙ = qv ∧ B (22) 211 Qualitatively in the plane perpendicular to B Accel is perp to v so particle moves in a circle whoseS a t o la h hyde pitt dare w ake duplin bladen pender bertie wilkes u nio carteret n ash robeson s ampson moore craven onslow h alif x beaufort columbus swain asheV R 1 R 2 C R 3 S i • Why?



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11 VECTOR SPACES 11 Examples Proofs will be given later let V = R3 = {(a,b,c) a,b,c ∈R} (11) U = {(a,b,0) a,b ∈R} Clearly U ⊂V and also U is a subspace of V let v1,v2 ∈R3 (12) W = {av1 bv2 a,b ∈R} W is a subspace of R3 In this case we say W is "spanned" by {v1,v2}In general, let S ⊂V,aSolution If the ammeter has zero resistance, the potential difference across it is zero, or nodes C and D are at equal potentials If I is the current through the battery, 1 2),if X has (join t) de nsit y f (x ) = 1 !



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