Chapter Review

Electrostatics

Coulomb Law and Electric Field · Electric Field Lines and Infinite Sheet Field · Electric Potential and Capacitor Transients

Coulomb's Law

Quantifies the electrostatic force between two stationary point charges — proportional to the product of charges and inversely proportional to the square of their separation.

Key Points

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Force is repulsive for like charges, attractive for unlike charges
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$F \propto q_1 q_2$ — doubling one charge doubles the force
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$F \propto 1/r^2$ — doubling distance reduces force to 1/4
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In a dielectric medium, force is reduced by factor $\epsilon_r$: $F_{medium} = F_{vacuum}/\epsilon_r$
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Key values: vacuum $\epsilon_r = 1$, air ≈ 1, glass ≈ 5–10, water ≈ 78.5
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Newton's Third Law applies: $\vec{F}_{12} = -\vec{F}_{21}$

Formula

$$F = k \frac{|q_1 q_2|}{r^2}$$

Electric Field Intensity

Force per unit positive test charge at a point — a vector quantity describing the strength and direction of the field created by source charges, independent of any test charge.

Key Points

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Defined as $\vec{E} = \vec{F}/q_0$ — units N/C or V/m (equivalent)
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Positive charge in field: force parallel to $\vec{E}$; negative charge: force antiparallel
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Field exists independently of the test charge — the test charge merely probes it
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Test charge must be small enough to not disturb the source charge distribution
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Force on any charge in a known field: $\vec{F} = q\vec{E}$

Formula

$$E = k \frac{|Q|}{r^2}$$

Superposition of Electric Fields

The net electric field at any point is the vector sum of individual fields from each source charge. Requires component-wise addition — magnitudes cannot be simply added.

Key Points

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Break each $\vec{E}_i$ into x and y components, sum separately, then recombine
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Two equal like charges at midpoint: fields cancel ($\vec{E}_{net} = 0$)
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Two equal unlike charges at midpoint: fields add (both toward negative charge)
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For two unlike charges, null point lies outside the pair on the side of the smaller charge
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For two like charges, null point lies between them, closer to the smaller charge

Formula

$$\vec{E}_{net} = \sum_i \vec{E}_i$$

Electric Field Lines

Visual representation of the electric field — tangent gives direction, spacing indicates strength. Lines originate on positive charges and terminate on negative charges.

Key Points

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Lines never intersect (field has unique direction at each point)
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Closely packed lines = strong field; widely spaced = weak field
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Single positive charge: radial outward; single negative charge: radial inward
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Dipole: lines curve from + to −; parallel plates: uniform, equally spaced lines
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Number of lines is proportional to the magnitude of the source charge

Electric Flux and Gauss's Law

Electric flux measures the total field passing through a surface. Gauss's Law relates the net flux through any closed surface to the enclosed charge alone.

Key Points

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Flux $\Phi_e = EA\cos\theta$ — $\theta$ is angle between field and the surface normal, not the plane
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$\theta = 0°$: maximum flux (field perpendicular to surface); $\theta = 90°$: zero flux
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Gauss's Law: $\Phi_e = Q_{enclosed}/\epsilon_0$ — flux depends only on enclosed charge, not shape or size
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External charges contribute zero net flux (lines entering must also exit)
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Zero net flux does NOT mean zero field (e.g., a dipole enclosed has $\Phi = 0$ but strong local fields)

Formula

$$\Phi_e = \frac{Q_{enclosed}}{\epsilon_0}$$

Infinite Sheet and Parallel Plates

An infinite non-conducting sheet produces a uniform field independent of distance. Two oppositely charged parallel plates produce a uniform field between them and zero field outside.

Key Points

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Single infinite sheet: $E = \sigma/2\epsilon_0$ — constant at all distances (no distance dependence)
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Gaussian surface: cylinder with flat faces on either side; curved surface contributes zero flux
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Two parallel plates ($+\sigma$ and $-\sigma$): fields add between plates, cancel outside
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Between parallel plates: $E = \sigma/\epsilon_0$ (double the single-sheet value)
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Outside parallel plates: $E = 0$ — fields from both plates point in opposite directions and cancel

Formula

$$E = \frac{\sigma}{\epsilon_0}$$

Hollow Conducting Sphere (Electrostatic Shielding)

Inside a hollow charged conductor the electric field is zero because all excess charge resides on the outer surface. Outside, the field behaves as if all charge were a point charge at the center.

Key Points

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$E = 0$ for $r < R$ — Gaussian surface inside encloses no charge
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$E = kq/R^2$ at the surface ($r = R$)
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$E = kq/r^2$ for $r > R$ — identical to a point charge at the center
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Faraday cage effect: metal enclosures block external electric fields
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Flux depends only on enclosed charge, not on the sphere's radius

Electric Potential and Potential Difference

Electric potential is work done per unit charge from infinity to a point — a scalar quantity. Potential difference between two points drives current and determines energy transfer.

Key Points

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$V = W/q_0$ — scalar with units volts (V = J/C)
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Potential due to point charge: $V = kq/r$ — inverse distance (not $1/r^2$)
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Potential is algebraic: $V_{net} = V_1 + V_2 + \cdots$ — simply add signs, no components
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Midpoint between opposite charges: $V = 0$ but $E \neq 0$ (non-zero field at zero potential)
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Midpoint between like charges: $E = 0$ but $V \neq 0$ (non-zero potential at zero field)
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Field points from high to low potential: $E = -\Delta V/\Delta r$

Formula

$$V_r = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$

RC Time Constant and Capacitor Transients

A capacitor charges and discharges exponentially through a resistor, governed by the time constant $\tau = RC$. Charging uses $(1 - e^{-t/RC})$, discharging uses $e^{-t/RC}$.

Key Points

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Time constant $\tau = RC$ — units of seconds ($\Omega \times \text{F} = \text{s}$)
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Charging: $V_C$ rises from 0 → $V_0$; current starts at $I_0 = V_0/R$ and decays to 0
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Discharging: $V_C$ falls from $V_0$ → 0; current flows opposite to charging direction
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At $t = \tau$: charging reaches 63.2%, discharging drops to 36.8%
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At $t = 5\tau$: process is 99.3% complete (effectively steady state)
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To find time for a specific voltage during discharge: $t = -RC \ln(V_C/V_0)$

Formula

$$V_C = V_0 \left(1 - e^{-t/RC}\right)$$

Formulas

Coulomb's Law

Force between two point charges in vacuum.

Formula

$$F = k \frac{|q_1 q_2|}{r^2}$$

Point Charge Electric Field

Field magnitude at distance r from source charge Q.

Formula

$$E = k \frac{|Q|}{r^2}$$

Gauss's Law

Net flux through closed surface equals enclosed charge over permittivity.

Formula

$$\Phi_e = \frac{Q_{enclosed}}{\epsilon_0}$$

Infinite Sheet Field

Field of a single non-conducting sheet — independent of distance.

Formula

$$E = \frac{\sigma}{2\epsilon_0}$$

Parallel Plates Field

Field between two oppositely charged plates.

Formula

$$E = \frac{\sigma}{\epsilon_0}$$

Point Charge Potential

Potential at distance r from point charge q.

Formula

$$V_r = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$

Potential Gradient

Field equals negative rate of change of potential with distance.

Formula

$$E = -\frac{\Delta V}{\Delta r}$$

Capacitor Charging

Voltage across charging capacitor rises toward V₀.

Formula

$$V_C = V_0 \left(1 - e^{-t/RC}\right)$$

Capacitor Discharging

Voltage across discharging capacitor decays toward zero.

Formula

$$V_C = V_0 \, e^{-t/RC}$$