Physics Q3 Notes:

• created by Desmond Gold (uWu)

Prerequisites

• protons and electrons are fundamental charge carriers which hold the smallest possible value of charge.
• two kinds of charge: positive and negative
• an object is electrically neutral when the number of protons are the same as the number of electrons.
• an object is electrically charged when the number of protons and the number of electrons are unequal
  • negative charge - there is excess of electrons
  • positive charge - there is lack of electrons
• $\overrightarrow{F}$ - force vector (has both magnitude and direction), can be both expressed in:
  • rectangular form (with unit vectors)
    • $\overrightarrow{F} = F_x \bold{\hat{i}} + F_y \bold{\hat{j}}$
      • $\bold{\hat{i}}$ - unit vector along x-axis = $<1, 0>$
      • $\bold{\hat{j}}$ - unit vector along y-axis = $<0, 1>$
      • $F_x$ - horizontal (x-axis) component of a vector
        • $F_x = F \ sin\theta$
      • $F_y$ - vertical (y-axis) component of a vector
        • $F_y = F \ cos\theta$
    • example: $\overrightarrow{F} = (5\bold{\hat{i}} - 3 \bold{\hat{j}}) N$
  • polar form (with magnitude and direction)
    • $\overrightarrow{F} = F, \theta_F$ (with reference angle)
      • $F$ - magnitude of a force or mathematically norm of a vector
        • $F = ||\overrightarrow{F} || = \sqrt{F_x^2 + F_y^2}$
      • $\theta$ - direction of the force, commonly expressed in degree form generally:
        • $\theta= tan^{-1}(\frac{F_y}{F_x})$
    • example: $\overrightarrow{F} = 5 N, \ 45 \degree$ North of East
• Net Force: $\overrightarrow{F_{net}} = \overrightarrow{F_{1}} + \overrightarrow{F_{2}} + ... + \overrightarrow{F_{n}}$
• Net Electric Field: $\overrightarrow{E_{net}} = \overrightarrow{E_{1}} + \overrightarrow{E_{2}} + ... + \overrightarrow{E_{n}}$

Fundamental Law of Charges

• like charges repel
• opposite charges attract

Coulomb’s Law

• formulated by Charles-Augustin de Coulomb
• states that:

electrical force is directly proportional to the product of two charges, and is inversely proportional to the square of center-to-center distance between these two charges.

• formula:
  $$F = \frac{k | q_1 \times q_2| }{r^2}$$
  • where:
    • $k$ (Coulomb’s constant) = $8.99 \times 10^9\ Nm^2/C^2$
    • $q_1$ and $q_2$ = signed magnitude (can be negative or positive) of charges
    • $r$ = head-to-head distance between two charged objects

Elementary Charged Particles

• $e$ (elementary charge) = $1.6 \times 10^-19 \ C$
• $q_e$ (charge value of electron) = $-e$
• $q_p$ (charge value of proton) = $+e$

Electric Field

• exists in a region around source charge
• contains electrical field lines
  • way to visualize by an arrow.
  • more electrical field lines implies higher charge, hence stronger electric field strength
• according to Michael Faraday:

Each charged object generates an electric field

• electric force per unit charge ($N/C$)
• electric field strength can be measured using test charge $q$
• formula:
  • $E = \frac{F}{|q|}$
  • or, $\overrightarrow{E} = \frac{\overrightarrow{F}}{q}$
    • electric field is directly proportional to electrical force exerted on $q$ by $Q$
    • electric field is inversely proportional to a test charge $q$
  • $E = \frac{k \ | Q |}{r^2}$
    • electric field is directly proportional to a source charge $Q$
    • electric field is inversely proportional to a square of distance between source and test charge

• electric field line arrows indicate the direction of electric field differing the location.
• positive source charge produces electrical field directed radially outward
• negative source charge produces electrical field directed radially inward.

• electric field of two positive charges:
• electric field of two negative charges:
  
• electric field of opposite charges:
  
• electric field of two parallel opposite charges
  

Source Charge and Test Charge

• In $+Q$

  • If $+q$ is a test charge:
    • $\overrightarrow{E}$ and $\overrightarrow{F}$ have the same directions
    • directed away from $+Q$
  • If $-q$ is a test charge:
    • $\overrightarrow{E}$ and $\overrightarrow{F}$ are of opposite directions
    • $\overrightarrow{E}$ is directed away from $+Q$
    • while, $\overrightarrow{F}$ is directed towards $+Q$

• In $-Q$

  • If $+q$ is a test charge:
    • $\overrightarrow{E}$ and $\overrightarrow{F}$ have the same directions
    • directed towards $-Q$
  • If $-q$ is a test charge:
    • $\overrightarrow{E}$ and $\overrightarrow{F}$ are of opposite directions
    • $\overrightarrow{F}$ is directed away from $-Q$
    • while, $\overrightarrow{E}$ is directed towards $-Q$

• Remarks:

  • $+Q$ is a source charge if and only if $\overrightarrow{E}$ is directed away from it.
  • $-Q$ is a source charge if and only if $\overrightarrow{E}$ is directed towards it.
  • $+q$ is a test charge if and only if both $\overrightarrow{E}$ and $\overrightarrow{F}$ have the same directions.
  • $-q$ is a test charge if and only if $\overrightarrow{E}$ and $\overrightarrow{F}$ are of opposite directions.

Challenge

Appendix

SI Units

• Distance: $m$ (meter)
• Mass: $kg$ (kilogram)
• Time: $s$ (second)
• Acceleration: $m/s^2$ (meter(s) per second squared)
• Force: $N$ (newton)
  • $1 \ N = 1 \ kg \ m / s^2$
• Charge: $C$ (coulomb)
• Electric Field Strength $N/C$ (newton(s) per coulomb)

Constants

• Elementary Charge Value: $e$ = $1.6 \times 10^{-19} \ C$
• Coulomb’s Constant: $k$ = $8.99 \times 10^9\ Nm^2/C^2$
• Gravitational Acceleration Constant: $g = 9.8 \ m/s^2$
• Mass of Electron: $m_e = 9.11 \times 10^{-31} kg$
• Mass of Proton: $m_p = 1.67 \times 10^{-27} kg$

Newton’s 2nd Law

• For force: $F = m \times a$
• For mass: $a = \frac{F}{m}$
• For acceleration: $m = \frac{F}{a}$

Coulomb’s Law

• For force:
  • $F = \frac{k | q_1 \times q_2 |}{r^2}$
• For distance:
  • $r = \sqrt{\frac{k | q_1 \times q_2 |}{F}}$
• For charge:
  • $q = \frac{F \times r^2}{k \times q_{other}}$

Electric Field

• For electric field:
  • $E = \frac{F}{|q|}$
  • $E = \frac{k \times |Q|}{r^2}$
• For test charge:
  • $|q| = \frac{F}{E}$
• For source charge:
  • $|Q| = \frac{E \times r^2}{k}$
• For distance:
  • $r = \sqrt{\frac{k \times |Q|}{E}}$

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