Apply Coulomb's Law - Physics

1. Find $q_2$ if $F=0.90\ \text{N}$, $q_1=+1\times 10^{-6}\ \text{C}$, and $r=0.10\ \text{m}$.

   $|q_2|\approx 1.0\times 10^{-6}\ \text{C}$. $|q_2| = \frac{Fr^2}{k|q_1|} = \frac{0.90 \times 0.01}{8.99\times10^9 \times 10^{-6}} \approx 1.0\times10^{-6}\ \text{C}$

2. What is the direction of the force between two charges with the same sign?

   Repulsive; along the line joining the charges. Like charges repel each other directly along their connecting line.

3. State the inverse-square relationship between force and separation distance in Coulomb's law.

   $F \propto \frac{1}{r^2}$. Force decreases with the square of the distance between charges.

4. Calculate $F$ for $q_1=+1\times 10^{-6}\ \text{C}$, $q_2=-1\times 10^{-6}\ \text{C}$, $r=0.10\ \text{m}$.

   $F\approx 0.90\ \text{N}$ (attractive). $F = 8.99\times10^9 \times \frac{1\times10^{-12}}{0.01} \approx 0.90\ \text{N}$, opposite signs attract.

5. Calculate $F_{\text{net}}$ on $q_2$ if $q_1=+1\times10^{-6}\ \text{C}$ at $0.10\ \text{m}$ left and $q_3=+1\times10^{-6}\ \text{C}$ at $0.10\ \text{m}$ right.

   $F_{\text{net}}=0\ \text{N}$. Equal charges at equal distances create equal but opposite forces that cancel.

6. Find the force ratio $\frac{F_2}{F_1}$ if the distance changes from $r_1$ to $r_2=3r_1$.

   $\frac{F_2}{F_1}=\frac{1}{9}$. Since $F \propto 1/r^2$, tripling $r$ reduces force by factor of 9.

7. Find $r$ if $q_1=q_2=+1\times 10^{-6}\ \text{C}$ and the force magnitude is $F=1.0\ \text{N}$.

   $r\approx 0.095\ \text{m}$. $r = \sqrt{\frac{k|q_1q_2|}{F}} = \sqrt{\frac{8.99\times10^{-3}}{1.0}} \approx 0.095\ \text{m}$

8. Identify whether the force is attractive or repulsive for $q_1=+2\ \mu\text{C}$ and $q_2=-5\ \mu\text{C}$.

   Attractive. Opposite signs ($+$ and $-$) mean attractive force.

9. What is the direction of the Coulomb force on each charge relative to the line joining them?

   Along the line connecting the charges. Electric forces act along the line between charges.

10. What is the SI value of Coulomb's constant $k$ used in Coulomb's law?

    $k = 8.99\times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2$. This constant relates charge, distance, and force in SI units.

11. What is the net force magnitude if two forces of $5\ \text{N}$ and $3\ \text{N}$ act on a charge in opposite directions?

    $2\ \text{N}$. Forces in opposite directions subtract: $5 - 3 = 2$ N.

12. Identify the vector direction of the force on $q_1$ due to $q_2$ in Coulomb's law problems.

    Along the line from $q_1$ to $q_2$ (toward or away). Force acts along the line connecting the charges.

13. What is the net force magnitude if two forces of $3\ \text{N}$ and $5\ \text{N}$ act on a charge in the same direction?

    $8\ \text{N}$. Forces in same direction add: $3 + 5 = 8$ N.

14. Calculate $F$ for $q_1=+4\times 10^{-6}\ \text{C}$, $q_2=+2\times 10^{-6}\ \text{C}$, $r=0.20\ \text{m}$.

    $F\approx 1.8\ \text{N}$. $F = 8.99\times10^9 \times \frac{8\times10^{-12}}{0.04} \approx 1.8\ \text{N}$

15. What is the sign of the force between like charges (both $+$ or both $-$)?

    Repulsive. Like charges repel each other.

16. What happens to $F$ if both charges change from $(q_1,q_2)$ to $(2q_1,3q_2)$ with $r$ unchanged?

    $F\to 6F$. Force scales with product of charges: $2 \times 3 = 6$.

17. State the formula for the magnitude of the electric force between two point charges.

    $F = k\frac{|q_1 q_2|}{r^2}$. Force is proportional to charge product and inversely proportional to distance squared.

18. What is the direction of the force between two charges with opposite signs?

    Attractive; along the line joining the charges. Opposite charges attract each other directly along their connecting line.

19. What happens to $F$ if the separation distance changes from $r$ to $2r$ (charges unchanged)?

    $F\to \frac{F}{4}$. Doubling distance reduces force by factor of $2^2 = 4$.

20. What is the value of Coulomb's constant $k$ in $\text{N}\cdot\text{m}^2/\text{C}^2$?

    $k = 8.99\times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2$. This fundamental constant relates charge, distance, and force in SI units.

21. For three collinear charges, what principle tells you the net force on one charge is the vector sum of pairwise forces?

    Superposition: $\vec{F}_{\text{net}}=\sum \vec{F}_i$. Vector sum of individual forces gives the net force on a charge.

22. What is the net force magnitude on $q_3$ if two equal forces of $5\ \text{N}$ act on it in opposite directions?

    $0\ \text{N}$. Equal forces in opposite directions cancel out.

23. State Newton's third law relationship for the forces two charges exert on each other.

    $\vec{F}_{12}=-\vec{F}_{21}$. Forces form an action-reaction pair with equal magnitude, opposite direction.

24. What happens to $F$ if the separation distance changes from $r$ to $2r$?

    $F\rightarrow\frac{F}{4}$. Force decreases by factor of $r^2$, so doubling $r$ gives $F/4$.

25. Identify the SI unit of electric charge used in Coulomb's law calculations.

    $\text{coulomb (C)}$. The standard unit for measuring electric charge in the SI system.

26. What is the sign of the force between opposite charges (one $+$ and one $-$)?

    Attractive. Opposite charges attract each other.

27. State the formula for the magnitude of the electrostatic force between two point charges.

    $F = k\frac{|q_1 q_2|}{r^2}$. Force is proportional to charge product and inversely proportional to distance squared.

28. Identify the SI unit of electric charge used in Coulomb's law.

    $\text{coulomb (C)}$. Named after Charles-Augustin de Coulomb who discovered the law.

29. What happens to $F$ if the separation distance changes from $r$ to $\frac{r}{3}$ (charges unchanged)?

    $F\to 9F$. Reducing distance by 3 increases force by factor of $3^2 = 9$.

30. Find the force ratio $\frac{F_2}{F_1}$ if $q_1$ is tripled and $q_2$ is halved while $r$ stays constant.

    $\frac{F_2}{F_1}=\frac{3}{2}$. Force ratio equals charge product ratio: $(3)(1/2) = 3/2$.

31. What happens to $F$ if the separation distance changes from $r$ to $\frac{r}{3}$?

    $F\rightarrow 9F$. Force increases by factor of $r^2$, so $r/3$ gives $9F$.

32. Identify the SI unit of electric force in Coulomb's law problems.

    $\text{newton (N)}$. Force is measured in newtons in the SI system.

33. What happens to $F$ if $q_1$ is doubled while $q_2$ and $r$ stay the same?

    $F\rightarrow 2F$. Force is directly proportional to each charge.

34. State Newton's third-law relationship for the forces between two point charges.

    $\vec{F}_{12} = -\vec{F}_{21}$. Forces form an action-reaction pair with equal magnitude, opposite direction.

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