We are tasked with verifying that the units of Coulomb's law yield the correct units of force, which are newtons (N), using dimensional analysis.

Coulomb's Law Formula:

$F = \frac{k \cdot q_1 \cdot q_2}{r^2}$

Where:

• $F$ is the force (in newtons, $N$),
• $q_1$ and $q_2$ are the charges (in coulombs, $C$),
• $r$ is the distance between the charges (in meters, $m$),
• $k$ is the electrostatic constant (with units $\text{Nm}^2/\text{C}^2$).

Now, let's perform dimensional analysis by substituting the units into this formula.

Step 1: Substitute the units

We know the following:

• $q_1$ and $q_2$ are in coulombs (C),
• $r$ is in meters (m),
• $k$ has units of $\text{Nm}^2/\text{C}^2$.

Thus, the formula for $F$ with units looks like this:

$F = \frac{\left( \frac{\text{Nm}^2}{\text{C}^2} \right) \cdot C \cdot C}{m^2}$

Step 2: Simplify the expression

First, simplify the units inside the fraction:

$F = \frac{\text{Nm}^2 \cdot C^2}{\text{C}^2 \cdot m^2}$

Next, cancel the $C^2$ terms:

$F = \frac{\text{Nm}^2}{m^2}$

Finally, cancel the $m^2$ terms:

$F = N$

Conclusion:

The units simplify to newtons (N), which is the correct unit of force. This confirms that Coulomb's law has the correct dimensional consistency.

Would you like more details on any part of this process or have any other questions?

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Here are 5 related questions you might find interesting:

1. How does Coulomb’s law compare to Newton's law of gravitation in terms of dimensional analysis?
2. What is the physical significance of the constant $k$ in Coulomb's law?
3. How can we apply dimensional analysis to check the correctness of more complex physical formulas?
4. How do the units for electric field strength ($E$) relate to Coulomb's law?
5. What are the SI base units of the newton, and how can dimensional analysis be used to derive them?

Tip: Dimensional analysis is a powerful tool not just for verifying units but also for deriving relationships between physical quantities without detailed calculations.