Giancoli Physics — Full Course (Lectures 1A through 10B)
Chapters 16–25 | Electrostatics · Circuits · Magnetism · EM Induction · Waves · Optics · Instruments
Electric charge comes in two kinds: positive (protons) and negative (electrons). Like charges repel, unlike attract. Charge is conserved (total charge in a closed system is constant) and quantized in units of e = 1.602 × 10⁻¹⁹ C.
An electroscope detects charge: gold leaves diverge when like charges are induced on them.
The force between two point charges Q₁ and Q₂ separated by distance r:
Equivalently, k = 1/(4πε₀) where ε₀ = 8.85 × 10⁻¹² C²/(N·m²) is the permittivity of free space.
The total force on a charge from multiple other charges is the vector sum of the individual Coulomb forces:
Strategy: compute each pairwise force magnitude with Coulomb's law, decompose into x and y components, sum components, then recombine.
The electric field at a point is the force per unit positive test charge placed there:
For a point charge Q at distance r:
For a continuous distribution, divide into infinitesimal charge elements dQ and integrate (or sum):
Common results to recognize (derived in lecture / textbook):
| Source | Field magnitude | Notes |
|---|---|---|
| Point charge | E = kQ/r² | Radial |
| Infinite line (linear density λ) | E = 2kλ/r = λ/(2πε₀r) | Perpendicular to line |
| Infinite plane (surface density σ) | E = σ/(2ε₀) | Uniform, ⊥ plane |
| Conducting plate (σ on surface) | E = σ/ε₀ | Just outside conductor |
| Ring (charge Q, radius R) on axis | E = kQx/(x²+R²)3/2 | x = distance along axis |
Field lines are a visual tool: tangent to E, density proportional to |E|.
In electrostatic equilibrium (no current flowing), free charges have rearranged so that:
Electric flux through a surface is the "flow" of E through it:
Gauss's Law: the total flux through any closed surface equals the enclosed charge divided by ε₀:
| Configuration | Gaussian surface | Result |
|---|---|---|
| Point charge Q | Sphere of radius r | E = kQ/r² (recovers Coulomb) |
| Spherical shell, charge Q | Sphere | E = 0 inside; E = kQ/r² outside |
| Infinite line, λ | Cylinder of radius r, length ℓ | E = λ/(2πε₀r) |
| Infinite plane, σ | Pillbox | E = σ/(2ε₀) |
| Conductor surface, σ | Pillbox half-inside | E = σ/ε₀ (outside only) |
Like gravity, the electric force is conservative. The electric potential energy change between points a and b:
The electric potential V is potential energy per unit charge:
For a uniform field between parallel plates separated by distance d:
In general: E points "downhill" — from high to low potential. Electric field strength can also be expressed in V/m, equivalent to N/C.
An equipotential is a surface where V is constant — no work is done moving a charge along it.
Choosing V = 0 at infinity:
Note: V is a scalar, sign included (positive Q → positive V; negative Q → negative V). Falls off as 1/r (slower than E ∝ 1/r²).
Superposition for V is easier than for E because V is a scalar — no vector decomposition needed:
For continuous distributions: V = ∫ k dQ / r.
For multiple charges, sum over all unique pairs.
A capacitor stores charge — typically two conductors separated by an insulator. When connected to a voltage V, it stores charge Q ∝ V:
Two plates of area A separated by distance d, vacuum between:
Larger area or smaller separation → more capacitance. Typical capacitors: pF to mF range.
Inserting an insulating material (a dielectric) between capacitor plates increases C by a factor K (the dielectric constant):
| Material | K | Dielectric strength (V/m) |
|---|---|---|
| Vacuum | 1.0000 | — |
| Air (1 atm) | 1.0006 | 3 × 10⁶ |
| Paper | 3.7 | 15 × 10⁶ |
| Mica | 7 | 150 × 10⁶ |
| Water | 80 | — |
Mechanism: the dielectric polarizes — dipoles align with E, partly canceling it inside the dielectric → smaller V for same Q → larger C.
Charging from 0 to Q requires work against the building voltage:
For a parallel-plate capacitor, U/Volume gives the energy density of the field itself — a result that turns out to be general:
A battery uses chemical energy to maintain a potential difference between its terminals — an "electrical pump" pushing positive charge from low to high potential internally.
Conventional current is in the direction positive charge would flow — opposite to actual electron flow in metals.
For many materials at fixed temperature, current is proportional to voltage applied:
Ohmic materials follow this linear V–I relationship; non-ohmic devices (diodes, transistors) do not.
For a uniform wire of length ℓ and cross-section A:
where ρ is the resistivity [Ω·m]. Resistivity depends on temperature:
| Material | ρ at 20°C (Ω·m) | α (1/°C) |
|---|---|---|
| Copper | 1.68 × 10⁻⁸ | 0.0068 |
| Aluminum | 2.65 × 10⁻⁸ | 0.00429 |
| Tungsten | 5.6 × 10⁻⁸ | 0.0045 |
| Carbon | 3.5 × 10⁻⁵ | −0.0005 |
| Glass | 10⁹–10¹² | — |
Free electrons move randomly at very high speeds (~10⁶ m/s thermal), but with an applied field, drift slowly in the field's direction:
where n = number density of charge carriers, e = electron charge, A = cross-section, vd = drift speed.
The EMF (ε, "electromotive force") is the voltage a source provides when no current flows. Real batteries have internal resistance r, so terminal voltage is less when current flows:
For a circuit with external load R: I = ε / (R + r). When R ≫ r: terminal voltage ≈ ε.
Note the formulas are opposite from resistors:
A capacitor in series with a resistor and battery. Charging:
Energy delivered: U = Pt. Electric companies bill in kilowatt-hours (1 kWh = 3.6 × 10⁶ J).
Household appliances are rated in watts at the household voltage (120 V US, 240 V EU). Current drawn: I = P/V.
AC voltage and current oscillate sinusoidally:
where f = 60 Hz (US) or 50 Hz (EU).
Average of V or I is zero, so we use rms (which gives the same average power as DC):
Danger comes primarily from current, not voltage. Effects on humans:
| Current (60 Hz AC, hand-to-hand) | Effect |
|---|---|
| 1 mA | Threshold of perception |
| 5 mA | Maximum harmless current |
| 10–20 mA | "Can't let go" — muscles contract |
| 50 mA | Pain, possible fainting |
| 100–300 mA | Ventricular fibrillation — can be fatal |
| > 1 A | Severe burns; possible cardiac arrest |
Body resistance: dry skin ~10⁵ Ω, wet skin/internal ~10³ Ω. At 120 V on wet skin: I ≈ 120 mA → potentially lethal.
Both built around a galvanometer (sensitive current meter, full-scale ~50 μA, internal R~30 Ω):
Below a critical temperature Tc, certain materials drop to zero resistance. Currents persist for years without dissipation.
Every magnet has two poles, North and South. Like poles repel, unlike attract. Unlike electric charges, no isolated magnetic monopole has ever been found; cutting a magnet in half gives two complete magnets.
Earth's geographic North is actually a magnetic south pole (so a compass N-pole points there). Field strength at surface ≈ 0.5 × 10⁻⁴ T.
Direction: right-hand rule #1 — point fingers in direction of I, curl toward B; thumb gives F. (Equivalently F = Iℓ×B.)
The tesla is defined by this equation: 1 T = 1 N/(A·m).
Right-hand rule: fingers point along v, curl into B, thumb gives F (for positive q; reverse for negative).
A charged particle moving perpendicular to B traces a circle (or helix if v has a parallel component). Setting magnetic force = centripetal force:
Field circles the wire. Direction: right-hand rule #2 — thumb along I, fingers curl in direction of B.
μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space.
Two parallel wires carrying currents I₁ and I₂ separated by distance d:
Same direction → attract; opposite directions → repel. (This is how the ampere was historically defined.)
When a current-carrying conductor sits in a B-field perpendicular to I, charge carriers are deflected to one edge — building up a transverse Hall voltage:
where w = width perpendicular to both I and B. Equivalently VH = IB / (n e t) where t = thickness, n = carrier density.
A rectangular loop of N turns, area A, carrying current I in a field B experiences a torque:
where θ is the angle between B and the area vector (normal to loop). Maximum when loop's plane is parallel to B (θ = 90°); zero when plane is perpendicular to B (θ = 0°).
A solenoid is a long, tightly-wound coil. Inside (away from ends) the field is nearly uniform and parallel to the axis:
Outside an ideal solenoid: B ≈ 0. With an iron core: replace μ₀ with μ = Kmμ₀ (Km can be 10²–10⁴) — the basis of electromagnets.
The magnetic version of Gauss's law for symmetric current distributions:
The line integral of B around a closed Amperian loop equals μ₀ × the current through any surface bounded by the loop.
| Source | Amperian loop | Result |
|---|---|---|
| Long straight wire, I | Circle of radius r around wire | B = μ₀I/(2πr) |
| Solenoid, n turns/m, I | Rectangle straddling wall | B = μ₀nI |
| Toroid, N turns, I | Circle inside torus | B = μ₀NI/(2πr) |
Faraday's seminal experiments (1831) showed that a changing magnetic environment near a coil induces an EMF — a current flows even with no battery in the circuit. Three ways to produce induced EMF:
What matters is the rate of change of magnetic flux:
where N is the number of turns. The minus sign carries Lenz's rule:
| Action | Flux change | Induced current direction |
|---|---|---|
| N-pole of magnet approaching coil | ΦB increasing into coil | Creates B out of coil → repels magnet |
| N-pole withdrawing | ΦB decreasing | Creates B into coil → attracts magnet |
| Loop area shrinking in B | ΦB decreasing | Current to maintain flux (same direction as B) |
When a conducting rod of length ℓ moves with velocity v perpendicular to a magnetic field B, free electrons in the rod experience a magnetic force F = qvB, driving them toward one end. This separation of charge creates a potential difference — an induced EMF.
This is valid when B, ℓ, and v are mutually perpendicular. If not perpendicular, use only the perpendicular components.
A changing magnetic field produces an electric field — not just in conductors, but in any region of space. This is one of Maxwell's key insights (Faraday's Law generalized).
This follows from F = qvB and E = F/q.
A generator converts mechanical energy → electrical energy. A coil of N turns and area A rotates at angular velocity ω in field B. By Faraday's law the induced EMF is:
The output is sinusoidal (alternating). The RMS output is:
At angle θ from vertical, velocity component perpendicular to B: v⊥ = v sin θ = (ωh/2) sin θ. Contribution from both arms: ε = 2NBlv⊥ = 2NBl·(ωh/2) sin θ = NBAω sin(ωt).
As a motor's armature spins, it acts like a generator and produces a back EMF (εback) opposing the applied voltage. Net current in the motor:
At start-up: εback = 0, so current is large (I = V/R). At full speed: εback ≈ V, current is small.
When a generator supplies current, Lenz's law produces a counter torque opposing rotation. More load → more counter torque → more mechanical energy input required. Energy is conserved.
Eddy currents are induced currents that circulate within bulk conductors in changing magnetic fields. They cause:
A transformer changes AC voltage using mutual inductance. Primary coil (NP turns, voltage VP) is linked to secondary coil (NS turns, voltage VS) through a laminated iron core.
| Type | NS vs NP | Voltage | Current | Example |
|---|---|---|---|---|
| Step-up | NS > NP | VS > VP | IS < IP | Power plant output |
| Step-down | NS < NP | VS < VP | IS > IP | Home supply, phone charger |
Power loss in transmission lines: Ploss = I²R. Transmitting at high voltage → low current → minimal loss.
Typical transmission: 240 kV → step down to 7200 V (substation) → step down to 120/240 V (home).
When current in coil 1 changes, it induces an EMF in nearby coil 2:
A changing current in a coil induces an EMF in that same coil opposing the change (self-inductance L):
where μ₀ = 4π × 10⁻⁷ T·m/A. For iron core: replace μ₀ with μ = Kmμ₀ where Km ≈ hundreds to thousands.
An inductor carrying current I stores energy in its magnetic field:
For a solenoid, using L = μ₀N²A/ℓ and B = μ₀NI/ℓ:
When a battery V₀ is connected to an inductor L and resistor R in series, current builds exponentially:
For AC sources with current I = I₀ cos(2πft):
| Element | Voltage–Current | Reactance/Resistance | Phase |
|---|---|---|---|
| Resistor R | V = IR | R (ohms) | In phase |
| Inductor L | V = IXL | XL = 2πfL = ωL | V leads I by 90° |
| Capacitor C | V = IXC | XC = 1/(2πfC) | I leads V by 90° |
In an LRC series circuit, current is maximum when the impedance Z is minimum. This occurs when XL = XC, i.e., at the resonant frequency:
At resonance, Z = R (purely resistive), and Irms = Vrms/R (maximum).
Energy oscillates between the capacitor (electric field) and the inductor (magnetic field), analogous to a mechanical pendulum:
A variable capacitor C is adjusted so that f₀ = 1/(2π√(LC)) matches the station frequency. Only that frequency produces significant current in the receiver circuit.
Maxwell unified all of electromagnetism into four equations. These are the foundation of classical electrodynamics and are consistent with special relativity.
| Equation | Named for | Physical meaning |
|---|---|---|
| 1. Electric Gauss's Law | Gauss / Coulomb | Electric charges are sources of E field |
| 2. Magnetic Gauss's Law | Gauss | No magnetic monopoles; B field lines close |
| 3. Faraday's Law | Faraday | Changing B → electric field |
| 4. Ampère–Maxwell Law | Ampère / Maxwell | Current or changing E → magnetic field |
Maxwell reasoned: if changing B → E (Faraday), and changing E → B (his extension of Ampère), then these fields can mutually sustain each other and propagate through space as a wave — even in vacuum.
Oscillating charges on an antenna (driven by an AC source) create oscillating E and B fields. These fields "detach" from the antenna and propagate outward as electromagnetic waves.
Maxwell calculated the speed of EM waves from his equations and found it exactly matched the measured speed of light. Therefore light is an electromagnetic wave.
where ε₀ = 8.85 × 10⁻¹² C²/(N·m²) and μ₀ = 4π × 10⁻⁷ N/A².
| Type | Wavelength | Frequency | Uses/Source |
|---|---|---|---|
| Radio waves | 10³ m – 0.1 m | 10⁴ – 10⁹ Hz | AM/FM radio, TV |
| Microwaves | 0.1 m – 1 mm | 10⁹ – 3×10¹¹ Hz | Radar, microwave ovens, WiFi |
| Infrared (IR) | 1 mm – 700 nm | 3×10¹¹ – 4×10¹⁴ Hz | Heat, remote controls, IR cameras |
| Visible light | 700 nm – 400 nm | 4–7.5 × 10¹⁴ Hz | Vision (red → violet) |
| Ultraviolet (UV) | 400 nm – 10 nm | 7.5×10¹⁴ – 10¹⁶ Hz | Sun, sterilization |
| X-rays | 10 nm – 0.01 nm | 10¹⁶ – 10¹⁸ Hz | Medical imaging, crystallography |
| Gamma rays | < 0.01 nm | > 10¹⁸ Hz | Nuclear decay, cancer therapy |
Historical methods: Rømer (1676, Jupiter's moons), Michelson (rotating mirror, 1926). Modern value:
In a medium with index of refraction n: v = c/n < c
EM waves carry energy. The energy density in fields E and B:
Since E = cB and c = 1/√(ε₀μ₀), the two contributions are equal: uE = uB.
Intensity = power per unit area (W/m²). For a plane EM wave with peak fields E₀ and B₀:
Radiation pressure is tiny but measurable (demonstrated by Crookes radiometer, solar sails concept).
E and B amplitudes fall as 1/r; intensity falls as 1/r².
Huygens' Principle: Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets after a time Δt.
This explains reflection and refraction using wave theory. It also explains why light bends around corners (diffraction) — prominent when λ is comparable to slit/obstacle size.
Two narrow slits separated by distance d are illuminated by coherent light (wavelength λ). Waves from the two slits interfere, producing alternating bright and dark fringes on a screen at distance L.
A diffraction grating has many closely spaced slits (N slits, spacing d). The condition for bright maxima is the same as double-slit, but the peaks are much sharper and brighter:
where d = 1/(number of lines per meter) = grating spacing.
Even a single slit of width a produces a diffraction pattern. Minima occur where:
Central maximum width: Δy = 2λL/a. Narrower slit → wider diffraction pattern (inverse relationship).
The actual double-slit pattern is the product of the two-slit interference pattern and the single-slit diffraction envelope. Some interference maxima may be "missing" if they coincide with single-slit minima.
When light reflects from a thin film (e.g., soap bubble, oil slick), rays reflecting from the top and bottom surfaces may interfere. The path difference is 2t (twice the film thickness for normal incidence).
| Condition | Bright (constructive) | Dark (destructive) |
|---|---|---|
| 0 or 2 phase shifts (both or neither surfaces shift) | 2t = mλn | 2t = (m+½)λn |
| 1 phase shift (one surface shifts) | 2t = (m+½)λn | 2t = mλn |
where λn = λ/n is the wavelength inside the film (n = index of film).
Circular thin-film interference rings formed between a convex lens and a flat glass surface. Central spot is dark (one phase shift, t→0).
Light travels in straight-line paths called rays when λ ≪ size of obstacles. This is the basis of geometric optics. Ray model successfully explains reflection, refraction, and image formation.
Plane mirror: Virtual, upright image. Image distance = object distance (di = do behind mirror). Image same size as object. Mirror need only be half person's height for full-body view.
When light crosses a boundary between media with different indices of refraction, it bends. Snell's Law:
n = index of refraction = c/v. Light bends toward the normal when entering a denser medium (n₂ > n₁), away from normal when entering less dense.
| Medium | Index n | Speed v = c/n |
|---|---|---|
| Vacuum | 1.000 | 3.00 × 10⁸ m/s |
| Air | 1.0003 ≈ 1.00 | ≈ c |
| Water | 1.33 | 2.26 × 10⁸ m/s |
| Glass (crown) | 1.52 | 1.97 × 10⁸ m/s |
| Diamond | 2.42 | 1.24 × 10⁸ m/s |
When light travels from denser (n₁) to less dense (n₂) medium, and θ₁ exceeds the critical angle θc, light is totally reflected (no refracted ray):
Glass has different n for different wavelengths: n is slightly larger for violet (shorter λ) than red (longer λ). This dispersion causes a prism to split white light into a spectrum. Rainbows are caused by dispersion in water droplets.
For a concave spherical mirror with radius of curvature r, parallel rays converge at the focal point F. The focal length:
Valid for paraxial rays (small angles only). Concave mirror: f > 0. Convex mirror: f < 0.
| Quantity | Positive (+) | Negative (−) |
|---|---|---|
| do | Object in front of mirror | Object behind mirror (virtual object) |
| di | Image in front of mirror (real) | Image behind mirror (virtual) |
| f | Concave mirror | Convex mirror |
| m | Image upright | Image inverted |
| Object location | Image type | Image location | Orientation | Size |
|---|---|---|---|---|
| do > 2f (beyond C) | Real | f < di < 2f | Inverted | Reduced |
| do = 2f (at C) | Real | di = 2f | Inverted | Same size |
| f < do < 2f | Real | di > 2f | Inverted | Enlarged |
| do = f | — | di = ∞ | — | — |
| do < f (inside F) | Virtual | Behind mirror | Upright | Enlarged |
A lens refracts light at two surfaces. A converging lens (thicker at center) has positive focal length; a diverging lens (thinner at center) has negative focal length.
| Shape | Name | f |
|---|---|---|
| Biconvex, plano-convex, converging meniscus | Converging | f > 0 |
| Biconcave, plano-concave, diverging meniscus | Diverging | f < 0 |
| Quantity | Positive (+) | Negative (−) |
|---|---|---|
| do | Object on incoming light side | Virtual object |
| di | Image on outgoing side (real) | Image on same side as object (virtual) |
| f | Converging lens | Diverging lens |
| m | Image upright | Image inverted |
| Object location | Image | Orientation | Size | Type |
|---|---|---|---|---|
| do > 2f | f < di < 2f | Inverted | Reduced | Real |
| do = 2f | di = 2f | Inverted | Same | Real |
| f < do < 2f | di > 2f | Inverted | Enlarged | Real |
| do < f | Same side as object | Upright | Enlarged | Virtual |
Diverging lens: always virtual, upright, reduced image.
For two thin lenses in contact or close together:
For separated lenses: treat each lens sequentially. The image from lens 1 becomes the object for lens 2.
where R₁, R₂ are radii of curvature of lens surfaces (positive if center of curvature is on the far side).
A camera uses a converging lens to form a real, inverted, reduced image on film/sensor. The image distance is adjusted by moving the lens (focusing).
Common f-stops: f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16. Each stop halves the light area → doubles exposure time needed. Lower f/# = wider aperture = more light = shallower depth of field.
The eye is a variable-power optical system. The cornea does most of the refracting (~2/3 of total power). The crystalline lens provides fine adjustment through accommodation (changing shape via ciliary muscles).
| Parameter | Value |
|---|---|
| Diameter of eye | ≈ 2.5 cm |
| Near point (young adult) | ≈ 25 cm (closest comfortable focus) |
| Far point (normal eye) | ∞ (can focus on very distant objects) |
| Lens power range | +18 to +24 D (accommodation range) |
Far point is closer than infinity — eye too powerful or too long. Cannot focus on distant objects. Corrected with diverging (negative) lens that moves the image of a distant object to the person's far point.
Near point is farther than 25 cm — eye too weak or too short. Cannot focus on close objects. Corrected with converging (positive) lens that moves a close object's image to the person's near point.
A converging lens held close to the eye allows an object to be placed closer than the near point (25 cm), producing a larger apparent (angular) size.
The angular magnification M compares the angle subtended with the lens to the angle subtended by the object at the near point (25 cm) without the lens:
A compound microscope uses two converging lenses to achieve high magnification of nearby objects:
where ℓ is the distance between the objective's back focal point and the front focal point of the eyepiece (the "tube length", typically ≈ 16 cm or 18 cm).
Minimum resolvable separation (Rayleigh criterion): δmin = 0.61λ / (n sin θ) where n sin θ is the numerical aperture (NA). Shorter wavelength → better resolution (electron microscopes use e⁻ waves with very short λ).
Telescopes magnify distant objects. Two main types:
Two converging lenses. Objective has long focal length; eyepiece has short focal length. Object at ∞ → objective forms real image at its focal point → eyepiece acts as magnifier.
The two lenses are separated by fobj + feye (sum of focal lengths) when viewing an object at infinity.
Uses a large concave mirror as the objective instead of a lens. Advantages:
Designs: Newtonian focus (flat secondary mirror at 45°), Cassegrain (convex secondary through hole in primary).
Proportional to area of objective: ∝ D2. Larger aperture → fainter objects visible. This, not magnification, is why astronomers want big telescopes.
Rays far from the optical axis focus at a different point than paraxial rays. Corrected by: compound lens systems, aspherical lenses, stopping down aperture.
Since n varies with λ (dispersion), different colors focus at different points. Corrected by an achromatic doublet: a converging and diverging lens of different glass types cemented together. Mirrors have no chromatic aberration.
Due to diffraction, even a perfect lens cannot form a perfect point image. Two point sources can just be resolved when the central maximum of one coincides with the first minimum of the other:
Smaller θmin (better resolution) requires: larger D or shorter λ.
| Instrument | Principle | Key formula |
|---|---|---|
| Spectrometer / spectroscope | Diffraction grating separates wavelengths | d sin θ = mλ |
| Interferometer (Michelson) | Path difference → interference fringes | ΔL = mλ/2 (bright) |
| Fiber optic scope (endoscope) | TIR keeps light in fiber | sin θc = n₂/n₁ |
The final covers everything from the term. Use the equation reference below as your master sheet.
| Constant | Symbol | Value |
|---|---|---|
| Speed of light | c | 3.00 × 10⁸ m/s |
| Permittivity of free space | ε₀ | 8.85 × 10⁻¹² C²/(N·m²) |
| Permeability of free space | μ₀ | 4π × 10⁻⁷ T·m/A = 1.257 × 10⁻⁶ T·m/A |
| Near point (standard) | N | 25 cm = 0.25 m |
| Euler's number | e | 2.718… |