prelecture_01_coulombs_law - KurtRudolph/phys212 GitHub Wiki

Coulomb's Law

• [ {\overrightarrow F_{ 12}} = k \frac{ {{q_1}{q_2}}}{ {r_{ 12}^2}} {\hat r_{ 12}}]
  • [ F_{ 12} \Rightarrow ] froce by [q_1] on [q_2]

Superposition Principle

• [ {\overrightarrow F_{ Net}} = {\sum\limits_i F_i}]
• The total electric force on a charge is just the vector sum of the electric forces on it exerted by all other charges.

SI Unis

• [Charge \Rightarrow \text{Coulombs} (\text{C})]
• [Distance \Rightarrow \text{meters} (\text{m})]
• [Force \Rightarrow \text{Newtons} (\text{N})]

Constants

• [ e = -1.6 \times 10^{-19} \text{C}]
  • Electric Charge
• [ k = \frac{ 1}{ 4 \pi \epsilon_o} = 9 \times 10^9 \frac{ \text{N} \cdot \text{m}^2}{ \text{C}^2}]
• [ G = 6.67 \times 10^{-11} \frac{ \text{N} \cdot \text{m}^2}{ \text{kg}}]
• [ m_e = 9.1 \times 10^{-31} \text{kg}]
  • Electron Mass
• [ m_p = 1.6 \times 10^{-27} \text{kg} ]
  • Proton Mass

Building Blocks of Matter

• [ \left| Q_{proton} \right| = \left| Q_{electron} \right| ]
• [ N_{protons} = N_{electrons} ]
• The magnitude of the charge of the proton is exactly equal to the magnitude of the charge on the electron. Since there are equal numbers of electrons and protons in an atom, atoms are electrically neutral.
• Types of material
  • Conductor
    • Charge carriers [the electrons] are totally free to move.
    • Materials such as metals
  • Insulator
    • Charges are fixed and cannot move.
    • Materials such as plastics

Questions

• The nucleus of a Helium atom has a charge equal to twice the proton's charge. Let [F_N] denote the magnitude of the force the Helium nucleus exerts on one of the electrons in a Helium atom, and let [F_e] denote the magnitude of the force one electron in the Helium atom exerts on the Helium nucleus.

  • [F_N = F_e ]
  • Even though the charges are not the same, the magnitude of the forces must be equal. The dominant force between the nucleus and the electron is the Coulomb force and this force is proportional to the product of the charges. Therefore, the magnitudes of [F_N] and [F_e] must be equal since the order in the multiplication doesn't matter. There is a more general reason that these forces must be equal, however: they comprise a Newton's third law interaction pair which means that their magnitudes are the same and their directions are opposite.

• Three charges are fixed in place as shown below. The +q charge is equidistant from the +2q and -q charges. Which of the vectors shown most closely represents the total force on the +q charge?

  • b
  • To find the total force on [+q] charge we need to add the force on it from the [+2q] charge ([F_2]) and the force on it from the [-q] charge ([F_1]) as shown in the diagram.
    • Note that [F_2] is twice as long as F1 since the distances from [+q] to [+2q] and to [-q] are the same, but the magnitude of [+2q] is twice that of [-q]. [F_2] points to the right since [+2q] exerts a repulsive force on [+q]. [F_1] points down since [-q] exerts an attractive force on [+q]. The total force ([F_2]) on [+q] is found by taking the vector sum as shown.