Work Done Moving A Charge In An Electric Field

Hey guys! Ever wondered how much oomph it takes to move a tiny charged particle around when there's an electric field in the way? Well, buckle up because we're diving into a problem that's gonna make this crystal clear. We're tackling a classic physics question: How much work do we need to do to move a $5 \mu C$ charge from a spot that's 10 cm away to another spot that's 30 cm away, all while being pushed around by an electric field created by a $100 \mu C$ charge? Sounds like fun, right? Let's break it down step by step.

Understanding the Basics

Before we jump into the nitty-gritty calculations, let's get our heads around the key concepts. Work, in physics terms, is basically the energy you need to apply to move something against a force. In our case, the force is the electric force exerted by that $100 \mu C$ charge. The electric force, as you might remember, is described by Coulomb's Law. This law tells us that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it looks like this:

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

Where:

• $F$ is the electric force.
• $k$ is Coulomb's constant (approximately $8.99 × 10^9 N⋅m^2/C^2$).
• $q_1$ and $q_2$ are the magnitudes of the charges.
• $r$ is the distance between the charges.

Now, because the electric force isn't constant as we move the $5 \mu C$ charge (since the distance $r$ is changing), we need to use a little calculus magic to figure out the work done. The work done in moving a charge from point A to point B in an electric field is given by the integral of the electric force over the distance:

$W = \int_{r_A}^{r_B} F dr = \int_{r_A}^{r_B} k \frac{q_1 q_2}{r^2} dr$

Where:

• $W$ is the work done.
• $r_A$ is the initial distance (10 cm).
• $r_B$ is the final distance (30 cm).

Alright, now that we've got the theory down, let's get our hands dirty with some calculations!

Calculating the Work Done

So, we need to calculate the work done to move a $5 \mu C$ charge from 10 cm to 30 cm in the electric field of a $100 \mu C$ charge. Using the formula we just discussed, we can plug in the values and solve for the work ($W$).

First, let's convert the distances from centimeters to meters because, in physics, we like to keep things consistent with SI units. So, 10 cm becomes 0.1 m and 30 cm becomes 0.3 m.

Now, let's plug the values into our integral:

$W = \int_{0.1}^{0.3} k \frac{q_1 q_2}{r^2} dr = k q_1 q_2 \int_{0.1}^{0.3} \frac{1}{r^2} dr$

Where:

• $k = 8.99 × 10^9 N⋅m^2/C^2$
• $q_1 = 100 \mu C = 100 × 10^{-6} C$
• $q_2 = 5 \mu C = 5 × 10^{-6} C$

Now, let's calculate the integral:

$\int_{0.1}^{0.3} \frac{1}{r^2} dr = \left[ -\frac{1}{r} \right]_{0.1}^{0.3} = -\frac{1}{0.3} - \left( -\frac{1}{0.1} \right) = -\frac{1}{0.3} + \frac{1}{0.1} = -\frac{10}{3} + 10 = \frac{20}{3}$

Now, we plug this result back into our work equation:

$W = k q_1 q_2 \times \frac{20}{3} = (8.99 × 10^9) × (100 × 10^{-6}) × (5 × 10^{-6}) × \frac{20}{3}$

$W = (8.99 × 10^9) × (500 × 10^{-12}) × \frac{20}{3} = (8.99 × 500 × 20) × 10^{-3} / 3 = 89900 × 10^{-3} / 3 = 89.9 / 3 ≈ 29.97 J$

So, the work done is approximately 29.97 Joules.

The Final Answer

Alright, after all that calculating, we've arrived at the answer! The amount of work required to move a $5 \mu C$ charge from a point 10 cm away to a point 30 cm away, influenced by the electric field of a $100 \mu C$ charge, is approximately 29.97 Joules. Not too shabby, huh?

In summary, we used Coulomb's Law to understand the electric force, integrated that force over the distance to find the work done, and plugged in the given values to get our final answer. This problem showcases the relationship between electric fields, forces, and the work required to move charges around. It's a fundamental concept in electromagnetism, and mastering it will definitely give you a leg up in your physics studies. Keep up the great work, and happy calculating!

Practical Implications

Understanding the work done in moving charges isn't just an academic exercise; it has real-world applications in various fields. For example, in electronics, calculating the energy required to move electrons in a circuit is crucial for designing efficient and reliable devices. In medical equipment, such as MRI machines, precise control over electric and magnetic fields is necessary to generate accurate images. Even in everyday devices like touchscreens, the movement of charges is fundamental to how they operate. Therefore, grasping these concepts provides a strong foundation for tackling more complex problems in science and engineering.

Further Exploration

If you're interested in delving deeper into this topic, consider exploring related concepts such as electric potential energy, voltage, and capacitance. These concepts build upon the fundamentals we've discussed and offer a more comprehensive understanding of electromagnetism. Additionally, exploring numerical methods for solving more complex problems can be highly beneficial, as analytical solutions may not always be feasible. Remember, continuous learning and exploration are key to mastering physics and its applications.

Tips for Solving Similar Problems

When tackling similar problems, keep the following tips in mind:

1. Always start with a clear understanding of the problem: Identify the given quantities, what you need to find, and the relevant formulas.
2. Pay attention to units: Ensure all quantities are expressed in consistent units (SI units are generally preferred).
3. Draw a diagram: Visualizing the problem can help you understand the relationships between different quantities.
4. Break down the problem into smaller steps: Solve each step individually and then combine the results to get the final answer.
5. Check your answer: Make sure your answer is reasonable and has the correct units.

By following these tips, you can improve your problem-solving skills and gain a deeper understanding of physics concepts.

Conclusion

So, there you have it! We've successfully calculated the work done in moving a charge within an electric field. This type of problem helps illustrate the fundamental principles governing electromagnetism and has practical applications across various fields. Keep practicing and exploring, and you'll become a pro at solving these types of problems. Good luck, and happy learning!