Title PHY formulas.pdf ✅

PHYSICS FORMULAS 2426 Electron = -1.602 19 × 10-19 C = 9.11 × 10-31 kg Proton = 1.602 19 × 10-19 C = 1.67 × 10-27 kg Neutron = 0 C = 1.67 × 10-27 kg 6.022 × 1023 atoms in one atomic mass unit e is the elementary charge: 1.602 19 × 10-19 C Potential Energy, velocity of electron: PE = eV = 1⁄2mv 2 1V = 1J/C 1N/C = 1V/m 1J = 1 N·m = 1 C·V 1 amp = 6.21 × 1018 electrons/second = 1 Coulomb/second 1 hp = 0.756 kW 1 N = 1 T·A·m 1 Pa = 1 N/m2 Power = Joules/second = I 2R = IV [watts W] Quadratic Equation: x b b ac a = - ± - 2 4 2 Kinetic Energy [J] KE = mv 1 2 2 [Natural Log: when e b = x, ln x = b ] m: 10-3 m: 10-6 n: 10-9 p: 10-12 f: 10-15 a: 10-18 Addition of Multiple Vectors: r r r r R = A + B + C Resultant = Sum of the vectors r r r r Rx = Ax + Bx + Cx x-component A A x = cos q r r r r Ry = Ay + By + Cy y-component Ay = A sin q R = Rx + Ry 2 2 Magnitude (length) of R qR y x R R = - tan 1 or tanqR y x R R = Angle of the resultant Multiplication of Vectors: Cross Product or Vector Product: i ́ j = k j ́ i = -k i ́ i = 0 Positive direction: i j k Dot Product or Scalar Product: i × j = 0 i × i = 1 a × b = abcosq k i j Derivative of Vectors: Velocity is the derivative of position with respect to time: v = + + k = i + j + k d dt x y z dx dt dy dt dz dt ( i j ) Acceleration is the derivative of velocity with respect to time: a = + + k = i + j + k d dt v v v dv dt dv dt dv dt x y z x y z ( i j ) Rectangular Notation: Z = R ± jX where +j represents inductive reactance and -j represents capacitive reactance. For example, Z = 8 + j6W means that a resistor of 8W is in series with an inductive reactance of 6W. Polar Notation: Z = M Ðq, where M is the magnitude of the reactance and q is the direction with respect to the horizontal (pure resistance) axis. For example, a resistor of 4W in series with a capacitor with a reactance of 3W would be expressed as 5 Ð-36.9° W. In the descriptions above, impedance is used as an example. Rectangular and Polar Notation can also be used to express amperage, voltage, and power. To convert from rectangular to polar notation: Given: X - jY (careful with the sign before the ”j”) Magnitude: X Y M 2 2 + = Angle: tanq = - Y X (negative sign carried over from rectangular notation in this example) Note: Due to the way the calculator works, if X is negative, you must add 180° after taking the inverse tangent. If the result is greater than 180°, you may optionally subtract 360° to obtain the value closest to the reference angle. To convert from polar to rectangular (j) notation: Given: M Ðq X Value: M cosq Y (j) Value: M sinq In conversions, the j value will have the same sign as the q value for angles having a magnitude < 180°. Use rectangular notation when adding and subtracting. Use polar notation for multiplication and division. Multiply in polar notation by multiplying the magnitudes and adding the angles. Divide in polar notation by dividing the magnitudes and subtracting the denominator angle from the numerator angle. X M Magnitude q Y
ELECTRIC CHARGES AND FIELDS Coulomb's Law: [Newtons N] F k q q r = 1 2 2 where: F = force on one charge by the other[N] k = 8.99 × 109 [N·m2 /C2 ] q1 = charge [C] q2 = charge [C] r = distance [m] Electric Field: [Newtons/Coulomb or Volts/Meter] E k q r F q = 2 = where: E = electric field [N/C or V/m] k = 8.99 × 109 [N·m2 /C2 ] q = charge [C] r = distance [m] F = force Electric field lines radiate outward from positive charges. The electric field is zero inside a conductor. + - Relationship of k to Î0: k = Î 1 4p 0 where: k = 8.99 × 109 [N·m2 /C2 ] Î0 = permittivity of free space 8.85 × 10-12 [C2 /N·m2 ] Electric Field due to an Infinite Line of Charge: [N/C] E r k r = Î = l p l 2 2 0 E = electric field [N/C] l = charge per unit length [C/m} Î0 = permittivity of free space 8.85 × 10-12 [C2 /N·m2 ] r = distance [m] k = 8.99 × 109 [N·m2 /C2 ] Electric Field due to ring of Charge: [N/C] E kqz z R = ( + ) 2 2 3/2 or if z >> R, E kq z = 2 E = electric field [N/C] k = 8.99 × 109 [N·m2 /C2 ] q = charge [C] z = distance to the charge [m] R = radius of the ring [m] Electric Field due to a disk Charge: [N/C] E z z R = Î - + æ è ç ö ø ÷ s 2 1 0 2 2 E = electric field [N/C] s = charge per unit area [C/m2 } Î0 = 8.85 × 10-12 [C2 /N·m2 ] z = distance to charge [m] R = radius of the ring [m] Electric Field due to an infinite sheet: [N/C] E = Î s 2 0 E = electric field [N/C] s = charge per unit area [C/m2 } Î0 = 8.85 × 10-12 [C2 /N·m2 ] Electric Field inside a spherical shell: [N/C] E kqr R = 3 E = electric field [N/C] q = charge [C] r = distance from center of sphere to the charge [m] R = radius of the sphere [m] Electric Field outside a spherical shell: [N/C] E kq r = 2 E = electric field [N/C] q = charge [C] r = distance from center of sphere to the charge [m] Average Power per unit area of an electric or magnetic field: W m E c B c m m / 2 2 0 2 2 2 0 = = m m W = watts Em = max. electric field [N/C] m0 = 4p × 10-7 c = 2.99792 × 108 [m/s] Bm = max. magnetic field [T] A positive charge moving in the same direction as the electric field direction loses potential energy since the potential of the electric field diminishes in this direction. Equipotential lines cross EF lines at right angles. Electric Dipole: Two charges of equal magnitude and opposite polarity separated by a distance d. z -Q p d +Q E k z = 2 3 p E z = Î 1 2 0 3 p p when z » d E = electric field [N/C] k = 8.99 × 109 [N·m2 /C2 ] Î0 = permittivity of free space 8.85 × 10-12 C2 /N·m2 p = qd [C·m] "electric dipole moment" in the direction negative to positive z = distance [m] from the dipole center to the point along the dipole axis where the electric field is to be measured Deflection of a Particle in an Electric Field: 2 2 2 ymv = qEL y = deflection [m] m = mass of the particle [kg] d = plate separation [m] v = speed [m/s] q = charge [C] E = electric field [N/C or V/m L = length of plates [m]

Charge per unit Area: [C/m2 ] s = q A s = charge per unit area [C/m2 ] q = charge [C] A = area [m 2 ] Energy Density: (in a vacuum) [J/m3 ] u = Î E 1 2 0 2 u = energy per unit volume [J/m3 ] Î0 = permittivity of free space 8.85 × 10-12 C2 /N·m2 E = energy [J] Capacitors in Series: 1 1 1 Ceff C1 C2 = + ... Capacitors in Parallel: Ceff = C1 + C2 ... Capacitors connected in series all have the same charge q. For parallel capacitors the total q is equal to the sum of the charge on each capacitor. Time Constant: [seconds] t = RC t = time it takes the capacitor to reach 63.2% of its maximum charge [seconds] R = series resistance [ohms W] C = capacitance [farads F] Charge or Voltage after t Seconds: [coulombs C] charging: q Q( e ) t = - - 1 /t V V ( e ) S t = - - 1 /t discharging: q Qe t = - /t V V e S t = - /t q = charge after t seconds [coulombs C] Q = maximum charge [coulombs C] Q = CV e = natural log t = time [seconds] t = time constant RC [seconds] V = volts [V] VS = supply volts [V] [Natural Log: when e b = x, ln x = b ] Drift Speed: I ( ) Q t = = nqvd A D D DQ = # of carriers × charge/carrier Dt = time in seconds n = # of carriers q = charge on each carrier vd = drift speed in meters/second A = cross-sectional area in meters2 RESISTANCE Emf: A voltage source which can provide continuous current [volts] e = IR + Ir e = emf open-circuit voltage of the battery I = current [amps] R = load resistance [ohms] r = internal battery resistance [ohms] Resistivity: [Ohm Meters] r = E J r = RA L r = resistivity [W · m] E = electric field [N/C] J = current density [A/m2 ] R = resistance [W ohms] A = area [m 2 ] L = length of conductor [m] Variation of Resistance with Temperature: r - r0 = r0a - 0 (T T ) r = resistivity [W · m] r0 = reference resistivity [W · m] a = temperature coefficient of resistivity [K -1] T0 = reference temperature T - T0 = temperature difference [K or °C] CURRENT Current Density: [A/m2 ] i = × d ò J A if current is uniform and parallel to dA, then: i = JA J = ne Vd ( ) i = current [A] J = current density [A/m2 ] A = area [m 2 ] L = length of conductor [m] e = charge per carrier ne = carrier charge density [C/m3 ] Vd = drift speed [m/s] Rate of Change of Chemical Energy in a Battery: P = ie P = power [W] i = current [A] e = emf potential [V] Kirchhoff’s Rules 1. The sum of the currents entering a junctions is equal to the sum of the currents leaving the junction. 2. The sum of the potential differences across all the elements around a closed loop must be zero. Evaluating Circuits Using Kirchhoff’s Rules 1. Assign current variables and direction of flow to all branches of the circuit. If your choice of direction is incorrect, the result will be a negative number. Derive equation(s) for these currents based on the rule that currents entering a junction equal currents exiting the junction. 2. Apply Kirchhoff’s loop rule in creating equations for different current paths in the circuit. For a current path beginning and ending at the same point, the sum of voltage drops/gains is zero. When evaluating a loop in the direction of current flow, resistances will cause drops (negatives); voltage sources will cause rises (positives) provided they are crossed negative to positive—otherwise they will be drops as well. 3. The number of equations should equal the number of variables. Solve the equations simultaneously.