Physics 123Concepts- Summary

Last updated October 3, 2010, 1:35 am

Note: Although I have done my best to check for typos and list the formulas correctly, you should verify the formulas are correct before using them. Make sure you know what all the variables represent in any particular formula. Some letters are used in different formulas from different chapters and may represent different things. - Dr. Nazareth

Please excuse the mess while this document is being updated.

The concept summary notes were originally created following the section numbers, outline, and symbols of a different textbook than we are using this quarter (Fall 2010). The notes are good and can be utilized right away. Just be careful of the section numbers listed until they are updated. As the quarter progresses, I will be updating these notes to reflect the textbook we are using for this class: Physics (Volume 2), 4th edition, by James S. Walker (Addison-Wesley, San Francisco). I will also be adding notes for the chapters on modern physics.

Chapter 19 (Electric charges, forces and fields) – updated for current textbook – 9/22/10

Electric charge (19.1)

  • Intrinsic property of matter
  • Two types: positive and negative
  • Magnitude of charge on an electron or a proton = e
  • SI units = coulomb (C)
  • e = 1.60 x 10-19 C
  • electric charge is quantized – it can only be an integer multiple of e

Electric conductors and insulators (19.2)

  • insulator: material where charges are not free to move
  • conductor: materials that allow charges to move somewhat freely
  • semiconductor: material with properties in between conductors and insulators

Coulomb’s Law - Electrostatic Force (19.3)

  • Law of conservation of electric charge
  • “Like charges repel and unlike charges attract each other.”
  • Electrostatic: all charges are at rest
  • Coulomb’s Law
  • (magnitude only)
  • note absolute value of point charges used
  • r = distance between the point charges
  • SI units = Newton (N)
  • direction – acts along line between the two point charges and is attractive for oppositely charged point charges and repulsive for like charged point charges
  • k = 8.99 x 109 N·m2/C2
  • Similar in form to Newton’s law of gravitation, except, force may attract or repel, depending on signs of the charges
  • If more than two charges, than total force on a charge is the vector sum of the forces from each pair. Break problem into parts and calculate the net force (vector sum). See chapter 19, examples 19-2 and 19-3, pgs. 661-663
  • If total charge of Q is distributed over surface of a sphere, then treat sphere as a “point” located at the center of the sphere.
  • Q = σA(σ = surface charge density; A = surface area of sphere)
  • A = 4πR2(R = radius of sphere)
  • (q is located outside sphere at distance r from center)

Electric Field (19.4-19.5, some 19.6)

  • Definition
  • E = F/q0
  • SI units = Newton per coulomb (N/C)
  • Vector quantity – direction same as direction of electric force on positive test charge
  • Test charge, q0
  • Small enough not to disturb surrounding charges
  • Positive
  • “the electric field is the force per unit charge at a given location” pg 665
  • If you know E, then force felt by charge q is F = qE
  • Direction of force depends on sign of charge q
  • If q = + then F same direction as E
  • If q = - then F opposite direction as E
  • Point charge
  • (magnitude only - direction depends on whether q is + or -)
  • Points radial out for q = + and radial in for q = -
  • If have more than one point charge, then the electric field, E, is just the vector sum of the electric fields due to each charge separately, at that particular location
  • See example 19-5, pg 668-669
  • Electric field lines = lines of force
  • Point from + charges to – charges
  • Do not stop or start midspace
  • Start at positive charges or infinity
  • End at negative charges or infinity
  • Density is proportional to field strength (more lines per area when field is stronger)
  • Parallel plate capacitor
  • See figure 19.17
  • (magnitude only; only between plates away from edges)
  • σ = q/A = charge per unit area = charge density
  • E points from positively charged plate to negatively charged plate
  • Inside a conductor (19.6)
  • At equilibrium (electrostatic conditions)
  • any excess charge is on surface of conductor
  • E = 0 at any point insidethe material of the conductor (not a cavity within the conductor)
  • E just outside conductor is perpendicular to surface
  • Conductor shields inside from outside charges, but doesn’t shield outside world from charges enclosed within
  • Sharp point in a conductor – charges more densely packed here so the electric field is more dense outside sharp point

Charging by Induction (19.6)

  • Charge an object without making direct physical contact
  • How to:
  • connect object to ground using a grounding wire
  • bring charged rod nearby – the like charge is repelled away down the grounding wire (now object has net charge)
  • remove grounding wire while charged rod still in place
  • now remove rod and excess charge distributes itself about the object
  • NOTE: induced charge is opposite charge of that on charged rod (object charged by touch has the same charge as the charged rod)

Gauss’s Law (19.7)

  • Electric flux: Φ=EA cosθ
  • E = electric field magnitude
  • A = area of surface
  • θ= angle between direction of E and the perpendicular to the area A
  • think of electric field lines “flowing” through the surface of area A
  • SI unit: N m2/C
  • if surface A is closed (like a sphere - a rectangle would be open)
  • flux is positive if E field lines are leaving the enclosed surface
  • flux is negative if E field lines are entering the enclosed surface
  • Permittivity of free space, ε0 = 1/(4πk) = 8.85 x 10-12 C2/(N·m2)
  • Gauss’s law: if charge q is enclosed by any arbitrary surface, Φ = q/ε0
  • shape of surface doesn’t have to be a sphere!!!
  • Use Gauss’s law to find the electric field in highly symmetric situations

Chapter 20 (Electric Potential and Electric Potential Energy) – updated for current textbook – 10/3/10

  • NOTE: textbook uses U for electric potential energy and I use EPE

Electric Potential Energy and the Electric potential (20.1)

  • As charge moves from A to B, work WAB is done by electric force:
  • WAB = EPEA - EPEB
  • EPE = electric potential energy
  • SI units = joule (J) = N·m
  • For a positive test charge, q0, moving upward a distance, d, in a downward pointing uniform electric field
  • W = -q0Ed
  • Since ΔPE = -W, then ΔPE = q0Ed
  • Electric force does negative work to move positive charge upward so the change in potential energy is positive (it gets larger)
  • Compare this to lifting a ball upward in the gravitational field … the potential energy gets larger as you lift the ball higher
  • For a negative test charge, q0, moving upward a distance, d, in a downward pointing uniform electric field, the change in potential energy is negative (gets smaller) because the electric force does positive work to raise the negative charge upward
  • CHANGE IN ELECTRICAL POTENTIAL ENERGY DEPENDS ON SIGN OF CHARGE AND ITS MAGNITUDE
  • Electric potential, or simply, potential
  • SI units = volt (V) = joule/coulomb (J/C)
  • Not a vector quantity, but can be positive or negative
  • Electric potential and electric potential energy are NOT the same thing
  • Cannot determine V or EPE in the absolute sense because can only measure the differences, ΔV and ΔEPE, in terms of the work, WAB
  • just like the gravitational potential energy is always relative to a reference level (e.g., ground level = 0 gravitational potential energy)
  • Potential difference
  • “a positive charge accelerates from a region of higher electric potential toward a region of lower electric potential”
  • Electron volt: energy change an electron has when it moves through a potential difference of 1 V.
  • 1 eV = e(1V) = (1.60 x 10-19C) (1 V) = 1.60 x 10-19J
  • Connecting electric field and rate of change of electric potential difference
  • SI units = volts/meter = V/m
  • “the electric field depends on the rate of change of the electric potential with position.” Pg. 693, Physics, 4th ed., J.S. Walker, 2010.
  • Can think of V like height of a hill and E as the slope of that hill
  • Electric potential decreases as you move in same direction as the electric field
  • Can think like going downhill … potential decreases
  • In general, only gives component of E along displacement, Δs
  • ΔV = -ExΔx(displacement in x-direction)
  • ΔV = -EyΔy(displacement in y-direction)

Energy Conservation (20.2)

  • Total energy nowEtotal = KEtranslational + KErotational + PEgravitational + PEspring + PEelectrical
  • Etotal = ½ mv2 + ½ Iω2 + mgh + ½ kx2 + EPE
  • EPE = qV
  • If no work is done by non-conservative forces, then energy is conserved
  • Initial Energy = Final Energy= E0 = Ef
  • Electrostatic (electric) force is conservative
  • “Positive charges accelerate in the direction of decreasing electric potential.” pg 696, Walker
  • (can think: positive charges speed up rolling “downhill”)
  • “Negative charges accelerate in the direction of increasing electric potential.” pg 696, Walker
  • For both positive and negative charges, as they accelerate, they move to a region of lower electric potential energy

Electric Potential Difference from point charges (20.3)

  • SI units = volt, V
  • V above not absolute, but rather how potential differs at a distance, r, as compared to a distance of infinity from the point charge.
  • Assumes V = 0 at r = ∞
  • So a positive q, puts potential everywhere above the zero reference value.
  • So a negative q, puts potential everywhere below the zero reference value.
  • Can add the potential from multiple point charges at a location
  • Its an algebraic sum (meaning signs matter), NOT a vector sum
  • See chapter 20, examples 20-3 and 20-4
  • Electric potential energy for point charges q and q0 separated by distance, r
  • EPE = q0V = kq0q/rSI units = Joule, J
  • Note: r = distance NOT displacement so r is always positive

Equipotential surfaces(20.4)

  • Potential is same everywhere on an equipotential surface
  • “The net electric force does no work as a charge moves on an equipotential surface.”
  • Electric field is
  • always perpendicular to an equipotential surface
  • points in direction of decreasing potential
  • “Ideal conductors are equipotential surfaces; every point on or within such a conductor is at the same potential.” Pg. 703, Physics, 4th ed., J.S. Walker, 2010
  • electric field lines meet the conductors surface at right angles

Capacitors (20.5)

  • Stores electric charge, thus it stores energy
  • Capacitance
  • Q = CV
  • SI units of capacitance = farad (F) = coulomb/volt (C/V)
  • Depends on the geometry of the capacitor plate (or conductors) and the dielectric constant of material between the plates
  • Parallel plate capacitor without a dielectric,
  • A dielectric can be inserted between the plates of a capacitor to increase the capacitance
  • Reduces electric field between plates in the dielectric. This increases the amount of charge that can be stored for a given electric potential difference between the two capacitor plates.
  • Dielectric constant, κ = E0/E(unitless)
  • κ > 1
  • Parallel plate capacitor with a dielectric,
  • C = κC0(applies to any capacitor, not just parallel plate)
  • Dielectric breakdown: when the electric field applied is large enough to force the dielectric to conduct electricity
  • Dielectric strength: maximum e-field before breakdown
  • See table 20-2, pg 711, Physics, 4th, J.S. Walker, 2010

Electrical Energy Storage (20.6)

  • Energy stored – work done to charge up plates, increasing potential difference
  • This work “stored as electric potential energy in the capacitor”
  • Where is energy of a capacitor stored? In the electric field between the plates
  • Energy density = uE=
  • True for any electric field whether in capacitor or not
  • κ=1 if no dielectricuE = (½)ε0E2

The following has not yet been updated to reflect the current textbook (10/3/2010)

Electromotive “force” and current (20.1)

  • Electromotive “force”, emf = maximum potential difference between the terminals of a generator or battery in a circuit
  • Electric current,
  • SI units = ampere (A) = coulomb/second (C/s)
  • Direct current(dc) – charge moves around circuit in same direction all the time
  • Batteries produce dc current
  • Alternating current (ac) –charges move first one way then the other, then back, and so forth
  • Conventional current – hypothetical flow of positive charges in the circuit
  • Flows from positive terminal of battery through circuit to negative terminal
  • Flows from higher potential to lower potential (hence the positive → negative)
  • In reality, negatively charged electrons flow in the circuit and go the opposite direction of the conventional current

Ohms’s Law, resistance, and resistivity (20.2-20.3)

  • electrical resistance is voltage applied across a piece of material/current thru the material
  • SI units = ohm (Ω) = volts/ampere
  • If V/I is constant (the same) for all values of voltage and current (at a given temperature) then the material follows Ohm’s Law (not really a “law” … it’s an observed relationship)
  • Ohm’s Law: V/I = R = constant
  • Resistance of a material depends on geometry and resistivity (a material property)
  • Resistivity is an inherent property of the material and depends on the temperature
  • ρ = ρ0 [1 + α(T-T0)]
  • SI units = Ω·m = Ohm·meter
  • α = temperature coefficient of the resistivity (unit = 1/temperature)
  • If α > 0, then ρ increases with temperature (e.g., metals)
  • If α < 0, then ρ decreases with temperature (e.g., semiconductors)
  • R = R0[1 + α(T-T0)](R, R0 are resistances at temperatures T, T0, respectively)

Electric Power (20.4)

  • In a circuit with voltage, V, and current, I, electric power delivered to the circuit is
  • P = IV
  • SI units = Watt (W) = Joule/second (J/s)
  • For a resistor, the power dissipated in the resistance is

Alternating Current (20.5) – only considering circuits with resistors (more in chapter 23)

  • Current that changes both magnitude and direction as a function on time
  • usually has a sinusoidal shape
  • Caused by a voltage that is a sinusoidal function of time: V = V0sin2πft
  • V0 = peak value of the voltage
  • f = frequency at which voltage oscillates
  • In a circuit containing only resistance: I = I0sin2πft
  • I0 = peak value of the current
  • I0 =V0/R
  • Root mean square (a type of average)
  • Voltage:
  • Current:
  • Power in circuit oscillates with time because current and voltage oscillate with time
  • Average power in the circuit, Pave = IrmsVrms
  • Average power dissipated in a resistor

Series and Parallel Wiring (20.6-20.8, 20.12)

  • Series – devices connected one after the other so the same current passes through each
  • Equivalent resistance, Rs = R1 + R2 + R3 + ···
  • dissipates the same total power as the series combination
  • reciprocal of the equivalent capacitance,
  • Carries the same amount of charge as any one of the capacitors in a combination
  • Stores the same total energy as the series combination
  • Parallel – devices connected so that the same voltage is applied across each device
  • reciprocal of the equivalent resistance,
  • dissipates the same total power as the parallel combination
  • Equivalent capacitance, Cp = C1 + C2 + C3 + ···
  • Each individual capacitor carries different amount of charge
  • Equivalent capacitance carries same total charge as parallel combination
  • Equivalent capacitance stores same total energy as parallel combination
  • If circuit is wired partially in series and partially in parallel, often times the circuit can be analyzed part by part, each section following the rules of series or parallel wiring as applies.

Internal Resistance (20.9)

  • Although we often assume perfect conductors, things like batteries and generators have resistance in the materials that make them up. We call this internal resistance.
  • This internal resistance causes the voltage between the terminals of the battery (or generator) to be less than the emfwhen current is drawn from the battery.
  • Terminal voltage = emf – Ir(r = internal resistance)

Kirchhoff’s Rules (20.10)

  • Junction rule: sum of the current in = sum of the current out
  • adding magnitudes only
  • Loop rule: around any closed circuit loop, the sum of the potential drops = the sum of potential rises
  • Reasoning strategy
  • First, decide on the direction of the current on each segment of the circuit.
  • Second, mark resistors with + and – signs. (Current flows from + to -).
  • Third, choose a direction (clockwise or counterclockwise) and go around a complete loop, adding up the potential drops and potential rises.
  • V = IR across a resistor.
  • If you solve for the current and you get a negative current, then you have chosen the current direction incorrectly. The current flows the opposite direction on that segment than what you originally chose.

Measurement of Current and Voltage (20.11)

  • Ammeter – measures current
  • Must be inserted in the circuit (in series)
  • Analog version includes a galvanometer and a shunt resistor connected in parallel
  • Shunt resistor extends the range by providing a bypass for the current exceeding the galvanometer’s full scale limit
  • Voltmeter – measures voltage between two points in a circuit
  • Must be connected across the part of the circuit to be measured (connected in parallel)
  • Analog version includes a galvanometer and an external resistor connected in series (so it is in series with the resistance in the galvanometer coil)
  • External resistor extends the range of the galvanometer by splitting the voltage between the coil resistance and the external resistance

RC Circuits (20.13)

  • Circuit with resistors and capacitors
  • Once circuit complete (time, t = 0), charge starts flowing and capacitor charges up until the charge on the plates reaches its equilibrium value, q0 = CV0
  • (V0 = voltage from battery)
  • q = q0[1 – e-t/(RC)]capacitor charging
  • e = the “e” from the natural logarithm (i.e., e = 2.718…)
  • τ = RC = time constant of the circuit (in seconds)
  • τ = time for capacitor to gain ~63.2% of its total charge
  • q = q0[e-t/(RC)]capacitor discharging (no battery in the circuit)
  • τ = time for capacitor to lose ~63.2% of its total charge
  • Voltage across the capacitor at time t: V = q/C

Safety and the Physiological Effects of Current (20.14)

  • What causes electric shock to the body? … the current or the voltage?
  • The damage comes from the current passing through the body
  • But current flows through the body because there is a potential differencebetweentwo parts of the body
  • This is why there are so many warnings about high voltage.
  • Safety feature – electrical grounding
  • Third prong in a plug connected to a copper rod in the earth/ground
  • Copper has lower resistance than the body so the current “prefers” to flow through the copper rod into the ground and not your body → safe!

Magnetic Fields (21.1)

  • Magnetic force = force due to moving electrical charges
  • North and south magnetic poles
  • Like poles repel and unlike poles attract
  • Magnetic field surrounds the magnet
  • A vector field has both a magnitude and a direction at every point surrounding the magnet
  • Magnetic field lines (lines of force) point from north pole to south pole
  • Strength of field (magnitude) proportional to lines per unit area
  • Stronger where lines closer together
  • Weaker where lines farther apart
  • Strongest at the poles of the magnet

Magnetic force on a moving charge (21.2)

  • A charge in a magnetic field experiences a magnetic force if …
  • Charge is moving AND
  • Velocity of charge must have some component perpendicular to the direction of the magnetic field
  • Right Hand Rule Number 1(find direction of magnetic force)
  • Point fingers in direction of magnetic field
  • Point thumb in direction of the velocity of the charge
  • Palm faces direction of magnetic force on a positive particle
  • If the particle has a negative charge, the force points in the opposite direction.
  • Magnetic force (magnitude)
  • F = q0Bvsinθ
  • 0 ≤θ ≤ 180°
  • Magnetic field (a vector field)
  • (magnitude)
  • Direction: determine using a small compass needle
  • SI units: telsa (T) = N·s/(C·m) = N/(A·m)
  • Earth’s magnetic field near the earth’s surface ~10-4 T = ~1 gauss (not a SI unit)

Motion of charged particle in magnetic field and mass spectrometer (21.3-4)