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Lecture Notes for Chapters 7
- Reading Memo Insights: To Be Added
- Summary of Important Equations to understand for the HW:
- I = Q/t
- V = PE/Q
- Ohm's law: V = IR
- Power: P = VI = I2R
- Rserial = R1 + R2
- 1/Rparallel = 1/R1 + 1/R2
- Series: I → same; V → divides up.
- Parallel: I → divides up; V → same
- What is charge?
- In Mechanics, we examined the basic property of matter called mass
- In ElectroDynamics, the basic concept of matter we will examine is called charge
- In mechanics, we never really knew just what mass is, only how it behaves; in the same way, classical E&M only tells us how charge behaves, but not what it really is.
- Charge is responsible for electrical & magnetic phenomena
- Represented by q or Q and unit of measure: Coulomb (C)
- Possessed by e-, p+, etc. (+ and -)
- Charge on each is e = 1.6 x 10-19 C
- Transparency 1: Atom becomes ionized when it loses
or gains e- (Fig 7.3, p. 249)
- E.g., when rubbed, friction transfers electrons
- Electric Force and Coulomb's Law:
- Electric force of p+ keeps e- in orbit (like gravitational)
- F = kQ1Q2/r2, where k = 9 x 109 N-m2/C2
- Force on Q1 is equal and opposite to force on Q2
- Like charges repel (Force is +), unlike attract (Force is -)
- Transparency 2: Just like gravity, distance halved, Force quadrupled (Fig 7.7, p. 250)
- Mass and charge are both fundamental properties of matter so not surprising that gravity and coulombic force have similar forms
- Electric force between e- and p+ is 1039 times as large as gravitational force between them
- Transparency 3: charged object can attract uncharged object (separation of charges) (Fig 7.8, p. 251)
- The Electric Field
- Just like gravity, electrostatic force is "action at a distance"
- Just like gravitational field, helpful to invoke idea of an Electric Field
- Field is the "agent" of the electrostatic force
(actually virtual photons)
- In mechanics, g determines gravitational field
- Transparency 4: Each line is a line of
force and indicates
direction and magnitude of force that the field would exert on
a positive charge (Fig 7.9, p. 251)
- Action at a distance is inherently unpalatable... how does it work? How is force actually transmitted across empty space and time? Is it instantaneous? That would allow transmission of information faster than speed of light!
- So instead, let's just imagine a field, an abstract, mathematical idea
- Field itself exerts the force
- Place test charge (or mass) at various points in space around object and map both the direction and magnitude of the Force... that's the field
- Turns out, this idea of field is actually real (e.g., gravitational force is actually a warping of space-time by mass); just like Copernicus initially proposed heliocentric universe as an abstract, mathematical simplification only ("We know Earth is the center, but if we just use this mathematical model with the Sun at the center, we can make better predictions" and, lo and behold, it really is a truer idea)
- Positive charges follow lines of force; Negative charges move against lines of force
- Field is the "agent" of the electrostatic force
(actually virtual photons)
- Field strength indicated by density of field lines
- E = F/Q -- Force Density (just like Pressure is also a Force Density!)
- The Electric Field itself is exactly what exerts the force on charged objects (can be produced without electric charges, a la magnetic fields)
- Electric Current & Resistance
- An electric current is simply a flow of charged particles
- I = Q/t → C/sec = Ampere (A)
- Either + or - charges can comprise a current (+ charge in one direction same as - charge in the opposite direction)
- Conventional current is flow of positive charges (actual current: electrons)
- Insulators have tightly bound e- (Electric Field not strong enough to rip them free)
- Conductors (like metals) have loosely bound
e- that move from atom to atom when electric field
is present
- Good conductors of electricity also good conductors of heat
- Solid insulators can become conductors when wet (since liquids have dissolved ions that can conduct electricity)
- Semiconductors (silicon and germanium) can conduct under certain conditions only (e.g., external E)
- Resistance (R) is a measure of the opposition to current
flow (measured in Ohms, Ω)
- Friction inhibits relative motion between two substances; Resistance inhibits flow of electric charge
- Arises from collision of e- in current with atoms
- Causes metal to gain Internal Energy -- raises Temperature -- heats it (like kinetic friction)!
- Conductors have low resistance; Insulators have high resistance
- Resistance depends on (R = ρL/A):
- Type of substance (copper wire low resistance)
- Length: longer == more resistance
- Width: thinner == more resistance
- Temperature also affects it; think of water flowing through
a pipe; a resistor is a section of the pipe filled with wire
mesh:
Wire Mesh (Steel) __________________xxxxxxxxxxxxx_________________ / xxxxxxxxxxxxx \ | xxxxxxxxxxxxx | | /---------------xxxxxxxxxxxxx-------------\ | | | | | | | | | | | | | | | | | | | | | | \___________||||||||||____________________/ | | |||||||||| Filled with water | \_______________||||||||||_________________________/ Water PumpThe cylinder is filled with water (i.e., an incompressible fluid)- If the pipe filled with water was thin, resistance to flow of water increases (i.e., in this example, water has friction with the material of the pipe but not with itself); if it was longer, more mesh, higher resistance; friction from flowing water heats wire mesh; etc.
- Electric Circuits and Ohm's Law
- Batteries produce an Electric Field that forces e- to flow through a circuit → like inducing a Gravitational Field by changing the incline
- An electric circuit has:
- A power supply
- An electrical device (e.g., a resistor like a bulb)
- Wires (or conductors) to carry the current
- The power supply acts like a "charge pump" -- forces
charges to flow out of one terminal, through the circuit, and
into the other terminal
- Imagine a "sea of electrons" (like the "sea of water" in a water circuit)
- Pump pushes pre-existing water around; battery pushes pre-existing electrons
- Flow of water is water current; flow of electrons is electric current (or electricity)
- Just like in Mechanics, we went from considering Forces to
Energy:
- We can move from the idea of an electric field and the resulting force on the charges in a wire to the idea of Energy and Work to quantify the effect of a power supply
- Batteries cause e- to flow -- i.e., Force acts on e-, causing them to move a distance -- that's Work done by the battery!
- Therefore, batteries give e- energy (this energy is converted to light and heat in bulb)
- Voltage, V = Work/Q = Potential_Energy/Q = F•d/Q = E•d -- i.e., energy given to e- by the battery
- Units of V are Joules/Coulomb = J/C = Volt
- 9-Volt battery gives 9 joules of energy to each coulomb of electric charge that it moves through a circuit
- Voltage is also called electric potential and
ΔV is the Potential Difference
- Potential Difference between high and low PE
- Think of the + terminal of a battery as High PE
(like a higher height) and the - terminal of a battery
as the Low PE (the lower height) and a battery takes
charges from lower PE to higher PE (just like an
elevator takes masses from lower heights to higher
heights) -- see Fig. 17-15 on p. 407 of Beiser
Battery "raises" charge --- Elevator raises mass + |-----^ ___ooo___ -- High PE -- _____ _o_ | | | \ |m| | | | \ --- | | | \ | | | \ --- | ----o---- -- Low PE -- ----- |m|----- - --- - The Elevator gives a certain energy (PE) to each person (mass) → i.e., PE/m. The Battery also gives a certain energy (PE) to each charge (Q) → i.e., V = PE/Q.
- In Mechanics, PE/m = gh; in Electricity, PE/Q = V = E•d
- In the water analogy, Voltage Difference (and so
Intensity of Electric Field since V =
E•d) is analogous to Pressure Difference
Water Electricity Litre of water Coulomb of charge Water Flow is 1 litre/sec Electric Current is 1 coulomb/sec (ampere) Pump Battery Pipe Wire Pump pressure:
Increase Rate of Flow of water by
↓ Increasing height (F/m = g•h)
↓ Increases Pressure = F/A = mgh/A
↓ RESULTS in increased Force on each parcel of waterVoltage:
Increase Rate of Flow of Charge (the current) by
↓ Increasing Potential Difference, V (F•d/Q = E•d)
↓ Increases Electrical Field E = F/Q
↓ RESULTS in increased Force on each moving charge - Ohm's Law: I = V/R (only for "Ohmic" elements; but R = V/I always) → V = IR
- In-class Exercise 1: An electric fan has a
resistance of 30 Ω when the current
in it is 3 A. What is the voltage required for this current? (see p. 261)
Known Unknown R = 30 Ω V = ?V I = 3A - Series
- Electric elements connected one after the other. Each
wire is connected one after the other. E.g., 3 wires (and
2 bulbs) connected in series, one after the next:
___Wire 1___oBULBo___Wire 2___oBULBo___Wire 3____ / \ -ooo- + Terminal | | | | | | | | | - Terminal | --o-- | \_________________________________________________| - Only one path for the charges to follow
- So current is the same while voltage is divided among each element
- Individual voltages add up to total battery voltage → I.e., series circuits DIVIDE the energy provied by the battery ("on the slope" -- see analogy below)
- If one element "goes out", whole circuit is out
- Electric elements connected one after the other. Each
wire is connected one after the other. E.g., 3 wires (and
2 bulbs) connected in series, one after the next:
- Parallel
- Electric elements join at same "node" (more than one
path). Each wire connects to the same points on the
battery:
____________Wire 2_______oBULBo___ / \ /______Wire 1_______oBULBo_____ | // \ | -ooo- + Terminal | | | | | | | | | | | | - Terminal | | --o-- | | \\______________________________| | \___________________________________| - Like two friends approaching a fork in the road that meet up again (see Lab135b-lab3) when fork rejoins
- Just like the friends, the charges, and hence the current, is divided among the devices
- voltage is the same while the current is divided among each element → I.e., parallel circuits SHARE THE BATTERY ("on the plateau" -- see analogy below)
- If one element "goes out", the others are fine
- Electric elements join at same "node" (more than one
path). Each wire connects to the same points on the
battery:
- Here's one way to think about these circuit types: imagine that the
individual charges (remember, the flow of charges is
the current) are equivalent to individual hikers that hate
each other (since like charges repel each other). Then, the
Potential (the voltage, V) will be analogous to the height
that they start at. In a series circuit, they follow a
single, narrow path where they have to follow each other, one
after the next. However, they start off at the top of a
mountain and slowly descend.
Each slope can be thought of as a resistor that's in the circuit. The angle of the slope represents the strength of the resistor (with a shallow angle representing a weak resistor and a steep angle representing a strong resistor). The hikers, just like electrons, will take the path of least resistance and opt for the path with the shallowest slope (if at all possible).
In a series circuit, they start off at the top of a mountain and descend the mountain along a single path as follows:o o | | -ooo-____ | | \ --> 1st Slope/Resistor | | \ | | \ o o | | \ | | | | \_____ | | \ | | \ --> 2nd Slope/Resistor | | \ --o--__________________\So the flow of hikers (or the flow of charge, or current) stays the same because each hiker (or each electron) only has one single path to follow. So here, the current (or flow of hikers) stays the same whereas the voltage (or the height) changes across each resistor (slope) → i.e., the potential drops across the resistor (slope) while charge flows through it.
In a parallel circuit, however, you can imagine the hikers (that hate each other (since like charges repel each other)) going along a flat plateau (i.e, a flat plain with no hills or valleys) and hitting a fork in the road:____________________ o o / /\ | | / / \ -ooo-_______/ / \ | | \ / \---> Slope/Resistor | | \ / \ | | \______________/ \_________ | | \ / \ | | \ / \ | | \ / \________ | | \ / / | | | \ / / | | | \/______________/ | | | | --o--________________________________________________________|Now, the hikers will have the choice of TWO possible paths; one might take the top, while the other takes the bottom, path. Then, they'll each go down a slope (or resistor) after which theey'll go along the path and meet up when the fork re-converges at the end. Since the hikers represent electrons, and the motion of the hikers represents the flow of charges, we can see that the current divides in a parallel circuit. However, since the whole path is on a plateau, the voltage stays the same in a parallel circuit! - In-class Exercise 2: Three light bulbs are connected
in parallel with a 10-V battery. The resistance of the bulbs are
20 Ω, 10 Ω, and 20 Ω; what is the current produced
by the battery? (see p. 263)
Known Unknown V = 10V I = ?A R1 = 20Ω; R2 = 10Ω; R3 = 20Ω
- Power and Energy
- Power = energy per time = energy per coulomb x # coulombs
per time = V x I (units of J/C x C/sec = J/s = Watt)
- When a light bulb is rated at a certain wattage (e.g., 100 W), remember that its rating is only valid for the specified voltage (usually, 120-V AC in household circuits in North America). However, once you know its resistance and the current, you can always get the power rating, or wattage, via P = VI = (IR)I = I2R
- Energy = P • t (since P = E/t)
- In-class Exercise 3: If a 1,000 W hair dryer is used
for 2.5 minutes, then how much energy does it consume? (see
p. 268)
Known Unknown t = 2.5mins t = ?sec P = 1000W E = ?J
- Power = energy per time = energy per coulomb x # coulombs
per time = V x I (units of J/C x C/sec = J/s = Watt)
- AC and DC
- Two kinds of current: Alternating and Direct
- Direct Current flows in a fixed direction (+ to -)
- Alternating Current reverses polarity (the polarity of the
terminal reverses from + to - and back again -- the voltage
alternates)
- the e- flow, and thus the current direction, reverses
- Some devices can use either; others one or the other
- Transformers can "step up" or "step down" AC voltages