21: Electric Charges & Fields

21.1 Static Electricity; Electric Charge and its Conservation #

  • “Charged” objects posses a net electric charge
  • Unlike charges attract; like charges repel
    • Charges on glass are positive, charges on plastic is negative
  • Law of Conservation of Electric Charge:
    • Whenever a certain amount of charge is produced in one object, an equal amount of the opposite type of charge is produced in another object
      • Charges cannot be destroyed or created
    • E.x. a plastic ruler is rubbed with a paper towel. The plastic acquires a negative charge and the towel obtains an equal positive charge
    • In other words, the net amount of electric charge produced in any process is zero: .$\Sigma Q = 0$

21.2 Electric Charge in the Atom #

  • Atoms are made up of positive nucleus surrounded by at least one negatively charged electron.
    • Inside the nucleus are protons which are positively charged and neutrons which have no charge
    • The charges of electrons and protons are equal in magnitude
  • E.x. neutral atoms with no charge contain an equal number of protons and electrons
  • When an atom gains a charge (by losing/gaining electrons), it then has a net charge and is called an ion
  • Neutral objects have a net charge of zero
    • Over time, objects left alone with a charge tend to lose their charge
    • This is because over time, electrons are exchanged with water molecules in the air
      • Water molecules are polar: They are neutral, their charges aren’t equally distributed
    • Thus, on rainy days it’s harder for an object to maintain a charge for too long

21.3 Insulators and Conductors #

  • Conductor: Material that allow charge to flow between objects
    • Metals tend to be good conductors
    • Electrons (charges) are relatively lose: can move freely within metal, but can’t leave easily
      • Called free or conduction electrons
  • Insulator: Opposite of conductors; don’t easily allow a flow of charge
    • Most materials other than metals tend to be good insulators
      • Notably rubber and wood
    • Electrons are bound very tightly to the nuclei
    • Almost no free electrons
  • Semiconductors: Somewhere between the two former
    • Silicon, germanium
    • Less free electrons than a conductor, but more than an insulator

21.4 Induced Charge; Electroscope #

  • Conduction: Charge transfer by physical contact
    • E.x. a positively charged metal rod touches a neutral metal rod. Free electrons from the neutral rod will then flow (transfer) to the charged rod, leaving the formerly neutral rod now slightly positively charged
  • Induction: Charge distribution altered by bringing two objects close, but not touching
    • Unlike conduction, induction doesn’t alter the net charge of objects when the inducer is taken away
    • However, induction can redistribute the existing charges on the induced object
  • Grounded Objects
    • Objects can be ground to the earth with a conducting wire
    • The earth is very large and can conduct, so it easily accepts/gives up electrons
    • Therefore, when an object is induced by another charged object, the original objects will become charged
    • If the wire is ever cut when the object is under induction, the charge will stay in the object
  • Electroscope
    • .$\vec F \propto \text{angle of deflection}$
    • .$y$-axis: .$F_{T1} \sin \theta_1 = F_{21}$
    • .$x$-axis: .$F_{T1} \cos \theta_0 = m_1 g$
    • .$F_{21} = m_1 g \tan \theta_1 \approx m_1 g \theta_1$
    • .$F_{21} = - F_{12}$ ( Newton’s Third) .$ \Longrightarrow \theta_1/\theta_2 = m_2/m_1$
    • .$d = l (\theta_1 + \theta_2)$
    Forces Diagram Distance

21.5 Coulomb’s Law #

Coulomb’s Law: $$F = k \frac{Q_1 Q_2}{r^2}$$ where .$k$ is a constant equal to .$8.988 \cdot 10^9 \text{ N m$^2$/C$^2$}$
  • Very similar to universal gravitation equation
    • However…
      1. .$F_C$ can repel, whereas .$F_G$ is always attractive
      2. .$F_C$ only acts on charged objects, whereas .$F_G$ acts on neutral objects too
    • .$F_G/F_C \approx 10^{-40} \Longrightarrow F_C \gg F_G$
  • The coulomb (.$\text{C}$) is the SI unit for charge
  • Properties of Coulomb Force:
    1. It can be attractive and repulsive
    2. It is not a contact force
    3. Inversely proportional to .$r^2$
    4. Proportional to amount of charge .$Q$
  • The smallest charge we’ve observed is the elementary charge: .$e = 1.6022 \cdot 10^{-19} \text{ C}$
    • Electrons have a charge equal to .$-e$
    • Protons have a charge equal to .$+e = -Q_\text{electron}$
    • Charges are Quantized
      • That is, all charges are multiples of .$e$
      • Since electrons are elementary particles, by definition they can’t be divided.
  • .$k$ can also be written as .$\frac{1}{4\pi\varepsilon_0}$
    • .$\varepsilon_0$ is called the permittivity of free space
    • .$\varepsilon_0 = \frac{1}{4\pi k} = 8.85 \cdot 10^{-12} \text{C$^2$/N m$^2$}$

21.6 Electric Field #

  • Electric fields extend outward from every charge and permeates all of space $$\overrightarrow E = \lim_{q\to0}\frac{\overrightarrow F}{q} \Longrightarrow \overrightarrow F = q \overrightarrow E$$
    • .$q$ is a positive charge
    • .$\overrightarrow F$ is the forces the field exserts on .$q$
    • Has units newtons per coulomb (.$\text{N/C}$)
  • We can combine this with Coulomb’s law to get $$\overrightarrow E = \frac{kqQ/r^2}{q} = k \frac{Q}{r^2}$$
    • We see that .$\overrightarrow E$ is independent of the non-source particle .$q$
    • .$Q$ is the particle that is responsible for the field in the first place
  • An electric field at a given point is the sum of all other electric fields that act on that point $$\overrightarrow E = \overrightarrow E_1 + \overrightarrow E_2 + …$$

21.7 Electric Field Calculations for Continuous Charge Distributions #

  • We can extend our previous definition to calculus as $$\overrightarrow E = \int d \overrightarrow E = \int k\frac{1}{r^2}\ dq$$
    • .$dq = \lambda\ dl \text{ (line)} = \sigma\ dA \text{ (disk)} = \rho\ dV \text{ (sphere)}$
  • When solving problems, it’s a good idea to use symmetry, check charge direction, and (when applicable) use bounds of .$r \in [0, \infty]$
  • We can write equation for an infinite plane holding a uniform surface charge density .$\sigma$ $$\overrightarrow E = \frac{\sigma}{2\varepsilon_0}$$
    • This also applies in the case where a charge is close to an infinite surface (so that the distance to the surface is much greater than the distance to the edges)

21.8 Field Lines #

  • To visualize electric fields, we draw electric field lines or lines of force
  • Three properties of Electric Field Lines:
    1. Electric field lines indicate the direction of the electric field; the field points are in the direction tangent to the field line at any point – see point .$P$ in .$\text{(a)}$
    2. The lines are drawn so that the magnitude of the electric field, .$E$, is proportional to the number of lines crossing unit area perpendicular to the lines (i.e. a circle ‘hugging’ a point charge). The closer together the lines, the stronger the field.
    3. Electric field lines start on positive charges and end on negative charges; and the number starting or ending is proportional to the magnitude of the charge.
      • .$\text{Density} = \frac{\text{number of lines crossing surface}}{\text{area surface}}$
      • .$\text{1 Coulomb} = \frac{1}{\varepsilon_0} \cdot \text{ lines}$
      • .$\therefore \text{Density} = \frac{q}{\varepsilon_0 4\pi r^2} \Longrightarrow \vec E$
  • In the case of two oppositely charged parallel & equally spaces plates – such as case .$\text{(d)}$ – we can write the field as $$\overrightarrow E=\text{const.} = \frac{\sigma}{\varepsilon_0}=\frac{Q}{\varepsilon_0 A}$$
    • .$Q=\sigma A$ is the charge on one plate of area .$A$
  • Field lines never cross because it wouldn’t make sense for an electric field to have two directions at the same point.


Electric Dipole #

  • A combination of two equal but opposite charges next to one another – see .$(\text{a})$ above
  • Dipole Moment is when represented by vector .$\vec{p}$ of magnitude .$Ql$
    • Molecules that have dipole moments are called polar molecules
  • A dipole in a uniform electric field feels no net force, but does have a net torque (unless .$\vec p \parallel \vec E$)
  • If .$\vec p \not \parallel \vec E$, .$W=\int_{\theta_1}^{\theta_2} \tau d\theta$ where .$\tau = -\vec p\vec E\sin\theta = \vec p \times \vec E$
    • Simplifies to .$W=\vec p\vec E(\cos\theta_2 - \cos\theta_1)$
    • Thus, work/torque is most at .$\theta = 90^\circ$ or .$180^\circ$ depending on .$\vec E$ direction
    • Pay attention to right hand rule when solving
  • If .$r \gg l \Longrightarrow \overrightarrow E \propto 1/r^3$

21.9 Electric Fields and Conductors #

  • The static electric field inside a conductor is zero (in static situations where electrons have had time to stop moving)
    • For that reason, any net charge on a conductor distributes itself on the surface
    • Charges inside conductors act as if the conductor isn’t there
  • All the electric field lines just outside a charged conductor are perpendicular to the surface

21.10 Motion of Charged Particle #

  • Vector Form of Forces $$\overrightarrow F_{12} = k \frac{q_1 q_2}{r^2} \cdot \widehat r_{21}$$
    • Notation:
      • .$\overrightarrow F_{12}$ means force on .$q_1$ by .$q_2$ since .$q_2$ is the source charge
      • .$\widehat r_{21} = - \widehat r_{12} \Longrightarrow \overrightarrow F_{12} = -\overrightarrow F_{21}$
    • Direction
      • If .$q_1 q_2 > 0$ (same sign, repulse), then the force and unitary vectors both point away from the two charges
        Same Signs 12

        Same Signs 21

      • If .$q_1 q_2 < 0$ (opposite sign, attract), then the force vector points towards the two charges and the unitary direction vector still points away from the two charges
        Opposite Signs 12

        Opposite Signs 21

  • Superposition Principle
    • In a system considering multiple (3+) charges, forces acting on .$q_1$ by .$q_2$ (.$F_{12}$) is independent from whether other charges are present
    • Total forces acting on .$Q_1$ can be written as .$\overrightarrow F = \overrightarrow F_{12} + \overrightarrow F_{13} + …$
      • Remember to break down the vectors into .$x/y$ components when adding them
        • E.x. .$(F_{1})_x = (F_{12})_x + (F_{13})_x + …$
      • Realize that the axis are arbitrary
    • .$\theta = \tan^{-1}\Big(\frac{F_x}{F_y}\Big)$
  • Charges in Fields
    1. Charge moving with .$\vec v$ that is parallel to uniform field .$\overrightarrow E$
      • .$\overrightarrow F = q \overrightarrow E = m \vec a \Longrightarrow a_x = \frac{q}{m}\overrightarrow E = \text{const.}$
      • .$\vec v = \sqrt{2a_x \vec d} = \sqrt{\frac{2q}{m}\overrightarrow E_x \vec d}$
    2. Charge moving with .$\vec v$ that is orthogonal to uniform field .$\overrightarrow E$
      • Similar to projectile in gravitational field: .$\vec g \sim \overrightarrow E$
      • .$\overrightarrow F_x = 0 \Longrightarrow v_{x2} = v_{x1};\ \ a_x = 0$
      • .$\overrightarrow F_y = q \overrightarrow E = m a_y;\ \ a_y = \vec a = \frac{q}{m}\overrightarrow E = \text{const.}$
      • .$y(t) = \frac{1}{2} \frac{q\overrightarrow E}{m}t^2$

21.11 Electric Dipoles #