These are your formula sheetsDO NOT TURN IT IN!Derivatives:ddxaxn= an xn−1ddxsin ax = a cos axddxcos ax = −a sin axddxeax= aeaxddxln ax =1xIntegrals:Za xndx = axn+1n + 1Zdxx= ln xZsin ax dx = −1acos axZcos ax dx =1asin axZeaxdx =1aeaxZdx√a2− x2= arcsinxaZdx√x2+ a2= ln³√x2+ a2+ x´Zdxx2+ a2=1aarctanxaZdx(x2+ a2)3/2=1a2x√x2+ a2Zx dx(x2+ a2)3/2= −1√x2+ a2DO NOT TURN THESE SHEETS IN!Physics 208 — Formula Sheet for Exam 3Do NOT turn in these formula sheets!Forces:The force on a charge q moving with velocity ~v in amagnetic field~B is~F = q~v ×~Band the force on a differential segment d~l carrying currentI isd~F = Id~l ×~BMagnetic Flux:Magnetic flux is defined analogously to electric flux(see formula sheet 1)ΦB=Z~B · d~AThe magnetic flux through a closed surface seems to bezeroI~B · d~A = 0Magnetic dipoles:A current loop creates a magnetic dipole ~µ = I~A whereI is the current in the loop and~A is a vector normal tothe plane of the loop and equal to the area of the loop.The torque on a magnetic dipole in a magnetic field is~τ = ~µ ×~BBiot-Savart Law:The magnetic field d~B produced at point P by a dif-ferential segment d~l carrying current I isd~B =µ04πI d~l × ˆrr2where ˆr points from the segment d~l to the point P .Magnetic field produced by a moving charge:Similarly, the magnetic field produced at a point P bya moving charge is~B =µ04πq ~v × ˆrr2Ampere’s Law: (without displacement current)I~B · d~l = µ0IenclFaraday’s Law:The EMF produced in a closed loop dep ends on thechange of the magnetic flux through the loopE = −dΦBdtWhen an EMF is produced by a changing magnetic fluxthere is an induced, nonconservative, electric field~E suchthatI~E · d~l = −ddtZA~B · d~AMutual Inductance:When a changing current i1in circuit 1 causes a chang-ing magnetic flux in circuit 2, and vice-versa, the inducedEMF in the circuits isE2= −Mdi1dtand E1= −Mdi2dtwhere M is the mutual inductance of the two loopsM =N2ΦB2i1=N1ΦB1i1where Niis the number of loops in circuit i.Self Inductance:A changing current i in any circuit generates a chang-ing magnetic field that induces an EMF in the circuit:E = −Ldidtwhere L is the self inductance of the circuitL = NΦBiFor example, for a solenoid of N turns, length l, area A,Ampere’s law gives B = µ0(N /l)i, so the flux is ΦB=µ0(N/l)iA, and soL = µ0N2lALR Circuits:When an inductor L and a resistance R app ear in asimple circuit, exponential energizing and de-energizingtime dependences are found that are analogous to thosefound for RC-circuits. The time constant τ for energizingan LR circuit isτ =LRLC Circuits:When an inductor L and a capacitor C appear in asimple circuit, sinusoidal current oscillation is found withfrequency f such that2πf =1√LCPhysics 208 — Formula Sheet for Exam 2Do NOT turn in these formula sheets!Capacitance:A capacitor is any pair of conductors separated by aninsulating material. When the conductors have equal andopposite charges Q and the potential difference betweenthe two conductors is Vab, then the definition of the ca-pacitance of the two conductors isC =QVabThe energy stored in the electric field isU =12CV2If the capacitor is made from parallel plates of area Aseparated by a distance d, where the size of the plates ismuch greater than d, then the capacitance is given byC = ²0A/dCapacitors in series:1Ceq=1C1+1C2+ ...Capacitors in parallel:Ceq= C1+ C2+ ...If a dielectric material is inserted, then the capacitanceincreases by a factor of K where K is the dielectric con-stant of the materialC = KC0Current:When current flows in a conductor, we define the cur-rent as the rate at which charge passes:I =dQdtWe define the current density as the current per unit area,and can relate it to the drift velocity of charge carriersby~J = nq~vdwhere n is the number density of charges and q is thecharge of one charge carrier.Ohm’s Law and Resistance:Ohm’s Law states that a current density J in a materialis proportional to the electric field E. The ratio ρ = E/Jis called the resistivity of the material. For a conductorwith cylindrical cross section, with area A and length L,the resistance R of the conductor isR =ρLAA current I flowing through the resistor R produces apotential difference V given byV = IRResistors in series:Req= R1+ R2+ ...Resistors in parallel:1Req=1R1+1R2+ ...Power:The power transferred to a component in a circuit bya current I isP = V Iwhere V is the potential difference across the component.Kirchhoff ’s rules:The algebraic sum of the currents into any junctionmust be zero:XI = 0The algebraic sum of the potential differences aroundany loop must be zero.XV = 0RC Circuits:When a capacitor C is charged by a battery with EMFgiven by E in series with a resistor R, the charge on thecapacitor isq(t) = CE³1 − e−t/RC´where t = 0 is when the the charging starts.When a capacitor C that is initially charged withcharge Q0discharges through a resistor R, the chargeon the capacitor isq(t) = Q0e−t/RCwhere t = 0 is when the the discharging starts.Physics 208 — Formula Sheet for Exam 1Do NOT turn in these formula sheets!Force on a charge:An electric field~E exerts a force~F on a charge q givenby:~F = q~ECoulomb’s law:A point charge q located at the coordinate origin givesrise to an electric field~E given by~E =q4π²0r2ˆrwhere r is the distance from the origin (spherical coor-dinate), ˆr is the spherical unit vector, and ²0is the per-mittivity of free space:²0= 8.8542 × 10−12C2/(N · m2)Superposition:The principle of superposition of electric fields statesthat the electric field~E of any combination of chargesis the vector sum of the fields caused by the individualcharges~E =Xi~EiTo calculate the electric field caused by a continuous dis-tribution of charge, divide the distribution into small el-ements and integrate all these elements:~E =Zd~E =Zqdq4π²0r2ˆrElectric flux:Electric flux is a measure of the “flow” of electric fieldthrough a surface. It is equal to the product of thearea element and the perpendicular component of~E in-tegrated over a surface:ΦE=ZE cos φ dA =Z~E · ˆn dA =Z~E · d~Awhere φ is the angle from the electric field~E to the sur-face normal ˆn.Gauss’ Law:Gauss’ law states that the total electric flux throughany closed surface is determined by the charge enclosedby that surface:ΦE=I~E · d~A =Qencl²0Electric conductors:The electric field inside a conductor is zero. All ex-cess charge on a conductor resides on the surface of