Worksheet on Coulomb's Law with handwritten solutions and force diagrams.

Here are the solutions for the problems shown in your worksheet. I have broken them down by problem number so you can check your work or fill in what is missing.

Problem 22: Drawing Force Vectors

*Rule:* Opposite charges attract (pull toward each other). Like charges repel (push away from each other). The length of the arrow depends on the amount of charge ($F \propto q_1q_2$).

a. Two negative charges $(-)$ and $(-)$.
* Force: Repulsion.
* Vectors: Draw an arrow pointing left on the left charge, and an arrow pointing right on the right charge. They should be equal in length.

b. One positive $(+)$ and one negative $(-)$.
* Force: Attraction.
* Vectors: Draw an arrow pointing right (toward the negative) on the positive charge. Draw an arrow pointing left (toward the positive) on the negative charge. Equal length.

c. Four positives $(++++)$ vs Two negatives $(--)$.
* Force: Attraction (Opposites attract).
* Magnitude: The force depends on multiplying the charges ($4 \times 2 = 8$). This is a strong force.
* Vectors: Draw a long arrow pointing right on the positive group (pulling toward the negatives). Draw a long arrow pointing left on the negative group (pulling toward the positives). *Note: Your handwritten notes calculated this correctly as proportional to $8kq^2/r^2$.*

d. Three positives $(+++)$ vs One positive $(+)$.
* Force: Repulsion (Likes repel).
* Magnitude: $3 \times 1 = 3$. This is weaker than part (c).
* Vectors: Draw a medium-length arrow pointing left on the group of three. Draw a medium-length arrow pointing right on the single charge.

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Problem 23: Net Force on the Gray Charge

*Rule:* Use vector addition (tip-to-tail method) to find the net force ($\vec{F}_{net}$).

a. Two positive neighbors pushing a gray positive charge.
* The top-left neighbor pushes down-right. The bottom-left neighbor pushes up-right.
* $\vec{F}_{net}$: Add these two vectors. The result points straight to the right.

b. One negative (pulling) and one positive (pushing).
* The negative charge pulls the gray charge up-left. The positive charge pushes the gray charge up-right.
* $\vec{F}_{net}$: Since they are symmetric, the left/right parts cancel out. The result points straight up.

c. Two positives to the right.
* Both push the gray charge to the left.
* $\vec{F}_{net}$: Points straight to the left. (The vector from the closer charge should be longer).

d. Three positives surrounding the gray charge.
* Top pushes down. Bottom-left pushes up-right. Bottom-right pushes up-left.
* $\vec{F}_{net}$: The horizontal parts of the bottom two cancel each other out. The vertical parts add up against the top one. Assuming equal distances/charges, the result points straight down.

e. One negative (top) and two positives (bottom).
* Negative pulls Up. Positives push Up-Left and Up-Right.
* $\vec{F}_{net}$: All forces have an upward component. The sideways components cancel. The result points straight up.

f. One positive (left) and one negative (below).
* Positive pushes Right. Negative pulls Down.
* $\vec{F}_{net}$: Add a Right vector and a Down vector. The result points Down and to the Right (Southeast).

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Problem 24: Assigning Charges

*Rule:* Check if the arrows match attraction/repulsion rules.

a. Possible.
* Center is $(+)$.
* Top-Left must be $(+)$ to push away (repel).
* Bottom-Left must be $(-)$ to pull in (attract).
* Right must be $(+)$ to push away (repel).

b. Not Possible.
* Look at the top particle. It has an arrow pointing away from the center (Repulsion). This means it must have the same sign as the center.
* Look at the bottom particle. It has an arrow pointing toward the center (Attraction). This means it must have the opposite sign of the center.
* Therefore, Top and Bottom must have opposite signs. But the diagram shows the Top and Bottom particles interacting with each other via repulsion (arrows pointing away from each other), which implies they have the *same* sign. This is a contradiction.

c. Not Possible.
* Left particle is pushed Left (away from center). Right particle is pushed Right (away from center).
* This implies both outer particles have the same sign as the center.
* However, if they have the same sign as the center, they must also repel *each other*. The diagram does not show forces between the two outer particles, or implies a setup that doesn't fit standard pairwise Coulomb interactions simply. (Usually marked "Not possible" because you can't satisfy all vector directions simultaneously with simple point charges).

d. Possible.
* Let's assume the Center is Positive $(+)$.
* Left particle is pushed Left $\rightarrow$ Must be Positive $(+)$.
* Right particle is pulled Left $\rightarrow$ Must be Negative $(-)$.
* Check interaction between Left $(+)$ and Right $(-)$: They should attract. Left feels a pull to the Right; Right feels a pull to the Left.
* This fits! You just need to draw the charge signs: Left $(+)$, Center $(+)$, Right $(-)$.

Final Answer:
Problem 22:
a. Arrows point outward (away from each other).
b. Arrows point inward (toward each other).
c. Long arrows pointing inward (attraction between large charges).
d. Medium arrows pointing outward (repulsion).

Problem 23:
a. Net force vector points Right.
b. Net force vector points Up.
c. Net force vector points Left.
d. Net force vector points Down.
e. Net force vector points Up.
f. Net force vector points Down-Right.

Problem 24:
a. Yes. (Top-Left:+, Bottom-Left:-, Right:+)
b. No. (Contradiction in signs).
c. No.
d. Yes. (Left:+, Center:+, Right:-)