Physics Chapter 16 – Electric Charge and Electric Field

Static Electricity, Electric Charge, and Its Conservation

• Everyday observation: rubbing certain materials (plastic, glass, wool, silk) produces noticeable forces → the phenomenon of static electricity.
  • Experiment 1 (undisturbed plastic rods):
  • No interaction → objects are electrically neutral.
  • Experiment 2 (both plastic or both glass rods rubbed):
  • Like–like interactions (plastic-plastic, glass-glass) → long-range repulsion.
  • Rubbing = charging; charged objects exert new forces.
  • Experiment 3 (rubbed glass vs. rubbed plastic):
  • Opposite materials attract → existence of two distinct charge types.
  • Convention: glass → positive charge, plastic → negative charge.
  • Experiment 4 (vary distance & vigor of rubbing):
  • Force decreases with increasing separation.
  • Force increases with greater amount of charge transferred.
• Fundamental qualitative rules:
  • Like charges repel; opposite charges attract.
  • Electric force can be attractive or repulsive (contrast with always-attractive gravity).
• Conservation of charge:
  • In any process the algebraic sum of all charge is constant; charge can move or redistribute but cannot be created or destroyed.

Atomic Structure of Charge

• Atom model:
  • Dense nucleus: positive protons (+) & neutral neutrons (0).
  • Light electrons (−) form surrounding cloud.
  • Charge is an intrinsic property of electrons and protons; magnitude of elementary charge $e = 1.602\times10^{−19}\ \text{C}$.
• Charge quantization:
  • Net charge $Q$ is an integer multiple of $e$.
• Charged matter:
  • Positive objects have lost electrons (not gained protons).
  • Ionization: removing an electron → positive ion; adding an electron → negative ion.
• Polar molecules:
  • Overall neutral but internal charge distribution is asymmetric → permanent electric dipole.
  • Example: $\text{H}_2\text{O}$ — slight $+$ on H side, $−$ on O side.

Conductors, Insulators, and Semiconductors

• Conductor:
  • Electrons are free to move throughout bulk (e.g., metals, electrolytes).
  • Charge placed on a conductor redistributes until electrostatic equilibrium.
• Insulator (non-conductor):
  • Electrons bound; almost no macroscopic charge flow (e.g., glass, rubber, wood).
• Semiconductor: intermediate behavior (e.g., Si, Ge); conductivity can be engineered.

Charging by Conduction and Induction; The Electroscope

• Conduction (contact):
  • Touch a neutral conductor (A) with a charged object (B) → electrons move until potentials equalize → A acquires same-sign charge as B.
• Induction (no contact):
  • Bring charged body near neutral conductor → internal electrons shift (charge separation).
  • Ground the far side → electrons leave/enter → remove ground & charged body → conductor attains opposite sign to inducing charge.
• Electroscope:
  • Metal rod plus gold leaves inside insulating container.
  • Leaves diverge when charged (either by conduction or induction).
  • Using a known-sign charge, can determine sign of an unknown charge by observing attraction/repulsion during test.
• Non-conductors under induction:
  • Cannot transfer charge, but their molecular dipoles re-orient → polarization leads to attractive force toward charged object.

Coulomb’s Law

• Empirical law relating point charges:
  • $F = k \frac{|Q<em>1 Q</em>2|}{r^2}$
  • $F$: magnitude of electric force between two point charges.
  • $r$: center-to-center separation.
  • $k = 8.988\times10^{9}\ \text{N·m}^2/\text{C}^2 = \frac{1}{4\pi\varepsilon_0}$.
  • Direction: along line joining charges; attractive if $Q<em>1 Q</em>2 < 0$, repulsive if $Q<em>1 Q</em>2 > 0$.
• Typical laboratory charges: $1\ \mu\text{C} = 10^{−6}\ \text{C}$.

Problem-Solving with Coulomb’s Law and Vectors

• Superposition principle:
  • Net force on any charge = vector sum of individual Coulomb forces from every other charge.
• Vector methods:
  • Tail-to-tip, parallelogram, or component addition $F<em>x = \sum F</em>i \cos\theta<em>i,\; F</em>y = \sum F<em>i \sin\theta</em>i$.
• Example (charges −1.0 nC & +4.0 nC, $r = 1.0\ \text{cm}$):
  • $F = k \frac{(1.0\times10^{−9})(4.0\times10^{−9})}{(1.0\times10^{−2})^2} \approx 3.6\times10^{−4}\ \text{N}$, direction: attractive (toward each other).

The Electric Field

• Definition: field at point in space equals force per unit positive test charge.
  • $\mathbf{E} = \frac{\mathbf{F}}{q_{\text{test}}}$
• Field of a single point charge $Q$:
  • $\mathbf{E}(\mathbf{r}) = k \frac{Q}{r^2}\,\hat{r}$ (radial; outward if Q>0, inward if Q<0).
• Force on charge in known field:
  • $\mathbf{F} = q\,\mathbf{E}$.
• Superposition for fields:
  • $\mathbf{E}<em>{\text{net}} = \sum</em>i \mathbf{E}_i$.
• Example 20.7 (field of proton at electron orbit, $r=0.053\ \text{nm}$):
  • $E = k\,\frac{e}{r^2} = 5.1\times10^{11}\ \text{N/C}$, radially outward.

Electric Field Lines

• Graphical representation rules:
  1. Lines are tangent to $\mathbf{E}$ direction at every point.
  2. Density of lines ∝ magnitude of field.
  3. Begin on $+$ charges, end on $−$ charges; number of originating/terminating lines ∝ |charge|.
• Special cases:
  • Electric dipole: equal & opposite charges separated by small distance; characteristic pattern curves from positive to negative.
  • Parallel-plate capacitor: between two large, closely spaced, oppositely charged plates lines are parallel & equally spaced → uniform field.

Electric Fields and Conductors

• Electrostatic equilibrium:
  • Inside conductor: $\mathbf{E}=0$ → no net motion of free electrons.
  • Excess charge resides on outer surface.
  • Field at surface is perpendicular; any tangential component would drive surface currents.
• Shielding (Faraday cage):
  • Enclosing region with conducting shell screens internal volume from external static fields (and vice-versa).
• Charge density non-uniformity:
  • Accumulates at sharp points → large local $E$; used deliberately (lightning rods) to initiate air ionization and safely bleed charge.

Applications and Biological Connections

• Heart dipole & electrocardiogram (ECG):
  • Depolarization/repolarization waves create a time-varying electric dipole; field extends through torso, measured by surface electrodes to diagnose cardiac function.
• Molecular biology – DNA:
  • Double helix stabilized by electrostatic attraction between complementary nucleotide bases (A–T, G–C).
  • During replication, random thermal motion brings bases together; correct pairs attract electrostatically, guiding accurate copying.
• Photocopy machines & laser printers:
  • Steps:
  1. Drum given uniform positive charge via charging roller.
  2. Optical image (or laser-written pattern) discharges selected areas.
  3. Negatively charged toner particles adhere to remaining $+$ regions.
  4. Paper contacts drum; toner transfers.
  5. Heat/pressure rollers fuse toner permanently.

Gauss’s Law and Electric Flux

• Electric flux through surface element $\Delta A$ at angle $\theta$ from $\mathbf{E}$:
  • $\Delta\Phi_E = \mathbf{E}\cdot\Delta\mathbf{A} = E\,\Delta A \cos\theta$.
• Total flux through closed surface:
  • $\Phi<em>E = \oint</em>S \mathbf{E}\cdot d\mathbf{A}$.
• Gauss’s law:
  • $\Phi<em>E = \frac{Q</em>{\text{enc}}}{\varepsilon_0}$.
• Useful for highly symmetric charge distributions (spherical, cylindrical, planar) to obtain $\mathbf{E}$ without direct integration.
• Sample numeric exercise: given $\Phi<em>E = 5.0\times10^{4}\ \text{N·m}^2/\text{C}$ through sphere of radius 20 cm → $Q</em>{\text{enc}} = \Phi<em>E \varepsilon</em>0 = (5.0\times10^{4})(8.85\times10^{−12}) \approx 4.4\times10^{−7}\ \text{C}$.

Chapter-End Key Facts & Equations

• Charge properties:
  • Two kinds (( +, − )), conserved, quantified in units of $e$.
• Material categories: conductors (mobile e⁻), insulators, semiconductors.
• Charging mechanisms: conduction, induction, polarization.
• Coulomb’s law: $F = k \dfrac{Q<em>1 Q</em>2}{r^2}$.
• Electric field: $\mathbf{E} = \dfrac{\mathbf{F}}{q} = k \dfrac{Q}{r^2}\,\hat{r}$.
• Field lines visualization rules (direction, density, origin/termination).
• Conductor equilibrium: $\mathbf{E}_{\text{inside}} = 0$; excess charge on surface; surface $\mathbf{E}$ perpendicular.
• Electric flux: $\Phi_E = \oint\mathbf{E}\cdot d\mathbf{A}$.
• Gauss’s law: $\Phi<em>E = Q</em>{\text{enc}}/\varepsilon_0$.
• Constants:
  • $e = 1.602\times10^{−19}\ \text{C}$, $k = 8.988\times10^{9}\ \text{N·m}^2/\text{C}^2$, $\varepsilon_0 = 8.854\times10^{−12}\ \text{C}^2/\text{N·m}^2$.
• Practical scales: rubbing produces micro-coulomb charges; atomic fields ≈ $10^{11}\ \text{N/C}$; everyday copier fields cause toner attraction; biological dipoles drive ECG signals.