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Electric Charges and Fields

Physics • Chapter 1 Flashcards
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Electric Charge

1. Basic Properties

  • Quantization: Total charge is always an integral multiple of basic quantum of charge $e$.
    $$ q = ne $$
  • Conservation: Total charge of an isolated system remains constant.
  • Additivity: Total charge is the algebraic sum of all individual charges in the system.

2. Key Facts

- Like charges repel, unlike charges attract.
- Value of $e = 1.6 \times 10^{-19} \text{ C}$
- Mass of electron $m_e = 9.1 \times 10^{-31} \text{ kg}$

Methods of Charging

  • Friction: Rubbing two insulators. e- transfer.
  • Conduction: Direct physical contact between conductors.
  • Induction: Charging without contact. Charges are redistributed due to presence of a nearby charged body.
Charging by Induction

Coulomb's Law

Scalar Form

$$ F = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2} $$
Coulomb's Law Vector Form

Vector Form

$$ \vec{F}_{12} = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{21} $$

Shows that $\vec{F}_{12} = -\vec{F}_{21}$ (Newton's 3rd Law).

Continuous Charge

When charge is distributed over a region, we use charge densities:

  • Linear ($\lambda$): Charge per unit length. (C/m)
  • Surface ($\sigma$): Charge per unit area. (C/m²)
  • Volume ($\rho$): Charge per unit volume. (C/m³)
Continuous Charge Distributions

Electric Field

Definition

Force experienced by a unit positive test charge.

$$ \vec{E} = \frac{\vec{F}}{q_0} $$

Field due to Point Charge

$$ E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} $$
Electric Field Visualization

Electric Field Lines

Key Properties

  • Originate from +ve, terminate at -ve.
  • Do not form closed loops.
  • Two field lines never intersect.
  • Relative closeness indicates field strength.
Electric Field Lines

Electric Dipole

Dipole Moment ($\vec{p}$)

$$ \vec{p} = q \times 2\vec{a} $$

Direction: -ve to +ve charge.
Unit: C·m.

Dipole in Uniform Field

  • Net Force: $F_{net} = 0$
  • Torque: $\vec{\tau} = \vec{p} \times \vec{E}$
  • Magnitude: $\tau = pE \sin\theta$
Dipole in Uniform Field

Dipole P.E. & Equilibrium

Potential Energy ($U$)

Work done in rotating the dipole in an external field.

$$ U = -\vec{p} \cdot \vec{E} = -pE \cos\theta $$

Stable vs Unstable

  • Stable Eq. ($\theta = 0^\circ$): $U = -pE$ (Minimum PE). Dipole is aligned with field.
  • Unstable Eq. ($\theta = 180^\circ$): $U = +pE$ (Maximum PE). Dipole is anti-aligned.

Dipole Field: Axial

For a short dipole ($r \gg a$):

$$ E_{axial} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3} $$

Direction: Same as $\vec{p}$

Dipole Axial Field

Dipole Field: Equat.

For a short dipole ($r \gg a$):

$$ E_{eq} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} $$

Direction: Opposite to $\vec{p}$

Dipole Equatorial Field

Relation

$$ E_{axial} = 2 \times E_{eq} $$

Electric Flux ($\Phi$)

Formula

$$ \Phi = \vec{E} \cdot \vec{A} = EA \cos\theta $$

Where $\theta$ is angle between E-field and Area Vector.

Electric Flux

For non-uniform field over closed surface:
$$ \Phi = \oint \vec{E} \cdot d\vec{S} $$

Gauss's Law

The total flux through a closed surface is $1/\varepsilon_0$ times the net charge enclosed.

$$ \oint \vec{E} \cdot d\vec{S} = \frac{q_{enclosed}}{\varepsilon_0} $$
Gauss's Law

Charges outside the surface do not contribute to total flux.

Gauss: Infinite Wire

Linear charge density $\lambda$:

$$ E = \frac{\lambda}{2\pi\varepsilon_0 r} $$

$E \propto 1/r$

Infinite Wire

Gauss: Infinite Sheet

Surface charge density $\sigma$:

$$ E = \frac{\sigma}{2\varepsilon_0} $$

(Independent of distance $r$!)

Infinite Sheet

Gauss: Spherical Shell

  • Outside ($r > R$): $E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}$
  • Surface ($r = R$): $E = \frac{1}{4\pi\varepsilon_0}\frac{q}{R^2}$
  • Inside ($r < R$): $E = 0$ (electrostatic shielding!)
Spherical Shell