In physics, dimensional analysis is often used to verify that the units in a formula produce the correct units for the answer. This process was first published by Frang¢ois. Daviet de Foncenex in 1761, and was later developed by other scientists, for example James Clerk Maxwell.
• a) In 1784, Charles-Augustin de Coulomb discovered that the force of attraction between two electric charges, 𝑞0 and 𝑞/ , can be calculated using the formula F = 5&% &$, where:
§ each electric charge is measured in coulombs (C),
§ r (the distance between the charges) is measured in meters (m),
§ the constant k = N𝑚/𝐶!/
Substitute these units into the formula and show that the formula produces the correct units of force, newtons (N) (Hint: Suppose 𝑞0 = 1, 𝑞/ = 2, r = 3):

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.

In physics, dimensional analysis is often used to verify that the units in a formula produce the correct units for the answer. This process was first published by François Daviet de Foncenex in 1761, and was later developed by other scientists, for example, James Clerk Maxwell. a) In 1784, Charles-Augustin de Coulomb discovered that the force of attraction between two electric charges, q0 and q2, can be calculated using the formula F = kq1q2/r^2, where: each electric charge is measured in coulombs (C), r (the distance between the charges) is measured in meters (m), and the constant k = Nm^2/C^2. Substitute these units into the formula and show that the formula produces the correct units of force, newtons (N) (Hint: Suppose q1 = 1, q2 = 2, r = 3).

Explore how dimensional analysis confirms that Coulomb's law formula for the force between two electric charges results in the correct units of force, newtons (N). This physics problem covers electrostatics and unit conversions, providing a step-by-step solution suitable for high school students in Grades 10-12.

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