Calculating Distance With Coulomb's Law: A Worked Example

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Hey guys! Ever wondered how to figure out the distance between two charged objects based on the force they exert on each other? It's all thanks to Coulomb's Law! Let's break down a problem step-by-step so you can master this concept. We'll tackle a problem where we need to find the distance between two charged objects given the force between them and their charges. Get ready to dive in!

Understanding the Problem

Okay, so here's the scenario: We've got two objects. The first one has a charge of 12 microcoulombs (12 μC), and the second one has a charge of 15 microcoulombs (15 μC). These objects are attracting each other with a force of 2 Newtons (2 N). We also know Coulomb's constant, which is 9Imes109 Nm2/C29 Imes 10^9 \text{ Nm}^2/\text{C}^2. Our mission? To find the distance (r) separating these charged objects.

Key Concepts: Before we jump into the math, let's refresh some key ideas.

  • Charge (q): Measured in Coulombs (C). Microcoulombs (μC) are often used, and 1 μC = 10−610^{-6} C.
  • Force (F): Measured in Newtons (N). This is the attractive or repulsive force between the charges.
  • Distance (r): Measured in meters (m). This is what we're trying to find!
  • Coulomb's Constant (k): A constant of proportionality, approximately 9Imes109 Nm2/C29 Imes 10^9 \text{ Nm}^2/\text{C}^2 in a vacuum.
  • Coulomb's Law: This law states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it's expressed as: F=kImes(∣q1∣Imes∣q2∣)/r2F = k Imes (|q_1| Imes |q_2|) / r^2

In simpler terms, the bigger the charges, the stronger the force. The farther apart they are, the weaker the force. Make sense?

Applying Coulomb's Law

Now, let's use Coulomb's Law to solve for the distance 'r'. The formula, as we mentioned, is:

F=kImes(∣q1∣Imes∣q2∣)/r2F = k Imes (|q_1| Imes |q_2|) / r^2

Where:

  • F = 2 N (the force between the charges)
  • k = 9Imes109 Nm2/C29 Imes 10^9 \text{ Nm}^2/\text{C}^2 (Coulomb's constant)
  • q1q_1 = 12 μC = 12Imes10−612 Imes 10^{-6} C (charge of the first object)
  • q2q_2 = 15 μC = 15Imes10−615 Imes 10^{-6} C (charge of the second object)
  • r = distance (what we want to find)

Step 1: Rearrange the Formula

We need to isolate 'r' on one side of the equation. Here's how we do it:

  1. Multiply both sides by r2r^2: FImesr2=kImes∣q1∣Imes∣q2∣F Imes r^2 = k Imes |q_1| Imes |q_2|
  2. Divide both sides by F: r2=(kImes∣q1∣Imes∣q2∣)/Fr^2 = (k Imes |q_1| Imes |q_2|) / F

Now we have: r2=(kImes∣q1∣Imes∣q2∣)/Fr^2 = (k Imes |q_1| Imes |q_2|) / F

Step 2: Plug in the Values

Let's substitute the values we know into the rearranged formula:

r2=(9Imes109Imes12Imes10−6Imes15Imes10−6)/2r^2 = (9 Imes 10^9 Imes 12 Imes 10^{-6} Imes 15 Imes 10^{-6}) / 2

Step 3: Calculate

Now, let's do the math:

r2=(9Imes12Imes15Imes109−6−6)/2r^2 = (9 Imes 12 Imes 15 Imes 10^{9-6-6}) / 2 r2=(1620Imes10−3)/2r^2 = (1620 Imes 10^{-3}) / 2 r2=0.81r^2 = 0.81

Step 4: Solve for r

To find 'r', we need to take the square root of both sides:

r=Isqrt0.81r = Isqrt{0.81} r=0.9textmr = 0.9 text{ m}

So, the distance between the two charged objects is 0.9 meters.

Answer and Conclusion

Therefore, the distance separating the two objects is 0.9 m or 9Imes10−1textm9 Imes 10^{-1} text{ m}. The correct answer is (a).

In summary, to find the distance between two charged objects using Coulomb's Law, follow these steps:

  1. Write down Coulomb's Law: F=kImes(∣q1∣Imes∣q2∣)/r2F = k Imes (|q_1| Imes |q_2|) / r^2
  2. Rearrange the formula to solve for 'r': r=Isqrt(kImes∣q1∣Imes∣q2∣)/Fr = Isqrt{(k Imes |q_1| Imes |q_2|) / F}
  3. Plug in the values for the charges, force, and Coulomb's constant.
  4. Calculate the result. Remember to take the square root to find 'r'.

Practice makes perfect, so try solving similar problems to solidify your understanding of Coulomb's Law. You'll be a pro in no time!

Additional Tips and Tricks

  • Units are Key: Always make sure your units are consistent. If charge is given in microcoulombs, convert it to Coulombs before plugging it into the formula. The same applies to other units.
  • Signs Matter (Sometimes): Coulomb's Law gives the magnitude of the force. The problem stated an attractive force, meaning the charges have opposite signs. While this doesn't affect the distance calculation itself (we used absolute values), it's important to remember the sign convention when dealing with forces.
  • Estimating the Answer: Before doing the full calculation, try to estimate the answer. This can help you catch mistakes. For example, if you expect the distance to be around 1 meter and your calculation gives you 100 meters, you know something went wrong.
  • Double-Check Your Work: Carefully review your calculations to avoid errors. Pay attention to exponents and decimal places.

Further Exploration

Want to delve deeper into electrostatics? Here are some topics to explore:

  • Electric Fields: Learn about the electric field created by charged objects.
  • Electric Potential: Understand the concept of electric potential and how it relates to electric fields.
  • Capacitance: Explore how capacitors store electrical energy.
  • Electrostatic Potential Energy: Study the energy stored in a system of charges.

By understanding these related concepts, you'll gain a more complete understanding of electrostatics.

Keep practicing, and you'll become a master of electrostatics in no time! Good luck, and have fun learning!