Equilibrium Of Charged Spheres: A Physics Problem

Hey guys! Today, we're diving into a fascinating physics problem involving the equilibrium of charged spheres. This problem combines concepts from electrostatics and mechanics, making it a great exercise in applying fundamental physics principles. We'll break down the problem step-by-step, ensuring you grasp every detail. So, let's get started!

Problem Statement: Understanding the Setup

The problem presents a scenario with two small spheres hanging in equilibrium. These spheres are suspended by threads, each 40.0 cm long, and the threads are tied to a single fixed point. This setup immediately suggests we'll be dealing with forces acting at angles, requiring us to use vector components. One sphere has a mass of 2.40 g and carries a positive charge of +300 nC. The other sphere has the same mass, which simplifies our calculations somewhat. The key here is that the spheres are in equilibrium, meaning the net force acting on each sphere is zero. This is a crucial piece of information that allows us to apply Newton's First Law.

To fully understand the problem, let's visualize the scenario. Imagine two tiny balls hanging from strings, both tied to the same point on the ceiling. Because they both have a charge, they will repel each other. This repulsion, combined with gravity pulling them down and the tension in the strings holding them up, creates a balanced state where the spheres remain stationary. To solve this, we'll need to consider the forces acting on each sphere: the electrostatic force due to the other sphere, the gravitational force (weight), and the tension in the thread. We'll also need to use Coulomb's Law to calculate the electrostatic force and some trigonometry to deal with the angles involved.

Understanding the problem statement is the first and most important step in solving any physics problem. By carefully reading and visualizing the scenario, we can identify the key concepts and principles that apply. In this case, we know we'll be using concepts from electrostatics (Coulomb's Law), mechanics (Newton's Laws), and trigonometry (vector components). With a clear understanding of the setup, we're well-equipped to move on to the next step: identifying the unknowns and the relevant equations.

Identifying Unknowns and Relevant Equations

Now that we've visualized the problem, let's pinpoint what we need to find. The problem will likely ask us to determine things like the distance between the spheres, the tension in the threads, or the charge on the second sphere (if it's not given). Identifying these unknowns is crucial because it guides us in selecting the appropriate equations and solution strategy. In addition to the unknowns, we also need to list the known quantities. This includes the length of the threads (40.0 cm), the mass of the spheres (2.40 g), and the charge of one sphere (+300 nC). Making a clear list of knowns and unknowns is a great way to organize our thoughts and prevent confusion.

Next, we need to identify the relevant equations that govern the physics of this situation. Several key equations come into play here:

• Coulomb's Law: This law describes the electrostatic force between two charged objects. It states that the force (F) is directly proportional to the product of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them: F = k * |q1 * q2| / r^2, where k is Coulomb's constant (approximately 8.99 x 10^9 N m^2/C2).
• Newton's First Law: This law, also known as the law of inertia, states that an object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a net force. Since the spheres are in equilibrium, the net force on each sphere is zero. This means the vector sum of all forces acting on each sphere (electrostatic force, gravitational force, and tension) must equal zero.
• Gravitational Force Equation: The gravitational force (weight) acting on each sphere is given by W = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
• Trigonometric Functions: Since the forces are acting at angles, we'll need to use trigonometric functions (sine, cosine, tangent) to resolve the forces into their horizontal and vertical components. This is essential for applying Newton's First Law in both the x and y directions.

By identifying the unknowns and the relevant equations, we've laid the groundwork for solving the problem. The next step is to apply these equations, often involving a bit of algebraic manipulation, to find the values of the unknowns.

Solving for the Unknowns: A Step-by-Step Approach

Alright, guys, it's time to roll up our sleeves and get to the nitty-gritty of solving this problem. This is where we put our understanding of the physics principles and equations into action. We'll tackle this step-by-step to make it super clear.

First, let's draw a free-body diagram for one of the spheres. This is a crucial step because it helps us visualize all the forces acting on the sphere. The forces we need to include are:

• The electrostatic force (Fe) pushing the sphere away from the other sphere.
• The gravitational force (Fg or W) pulling the sphere downwards.
• The tension (T) in the thread, pulling the sphere upwards and inwards.

The tension force acts along the thread, so it has both vertical and horizontal components. We'll call these T_y (vertical component) and T_x (horizontal component). The angle (θ) between the thread and the vertical is important here, as it helps us relate T_x and T_y to the total tension T using trigonometric functions:

• T_x = T * sin(θ)
• T_y = T * cos(θ)

Now, because the sphere is in equilibrium, the net force in both the horizontal (x) and vertical (y) directions must be zero. This gives us two equations based on Newton's First Law:

• ΣFx = 0: Fe - T_x = 0 (The electrostatic force is balanced by the horizontal component of tension)
• ΣFy = 0: T_y - Fg = 0 (The vertical component of tension balances the gravitational force)

Let's substitute our expressions for T_x, T_y, Fe, and Fg into these equations:

• k * |q1 * q2| / r^2 - T * sin(θ) = 0
• T * cos(θ) - mg = 0

Here, 'r' is the distance between the spheres, 'q1' and 'q2' are the charges of the spheres, 'm' is the mass of the sphere, and 'g' is the acceleration due to gravity.

We now have two equations with several unknowns. Depending on what the problem asks us to find, we might need to solve for the distance 'r', the tension 'T', or perhaps the charge 'q2' on the other sphere. To solve these equations, we might need an additional relationship. This often comes from the geometry of the problem. In this case, the length of the thread (L) and the distance between the spheres (r) are related to the angle θ. Specifically:

• sin(θ) = (r/2) / L (This comes from considering a right triangle formed by the thread, the vertical line from the point of suspension, and half the distance between the spheres).

With this additional equation, we can now solve for the unknowns. The exact method for solving will depend on what the problem asks for, but it usually involves some algebraic manipulation and substitution. For example, you might solve the second equilibrium equation for T, and then substitute that expression into the first equilibrium equation. This often leads to an equation that can be solved for 'r', the distance between the spheres.

Remember, always keep track of your units and make sure they are consistent throughout the calculation. If you're working with grams, for instance, you might need to convert them to kilograms before using them in the equations. Also, double-check your algebra and make sure you're not making any sign errors.

Analyzing the Results: Does it Make Sense?

Once we've crunched the numbers and arrived at an answer, our work isn't quite done! The final, and arguably just as important, step is to analyze the results and ask ourselves: does this answer make sense in the context of the problem? This is where we engage our physical intuition and critical thinking skills.

For instance, let's say we calculated the distance between the spheres. We should consider: Is the distance a reasonable value given the length of the threads? If the threads are 40 cm long, and we calculated a distance of 1 meter between the spheres, that should raise a red flag! It's physically impossible for the spheres to be that far apart given the constraints of the problem.

Similarly, if we calculated the tension in the thread, we should ask: Is the tension a reasonable value compared to the weight of the sphere? The tension must be at least equal to the weight to support the sphere, and it will be higher due to the electrostatic repulsion. If we calculated a tension value much smaller than the weight, we know something went wrong.

Another important check is to consider the units of our answer. Are we getting a distance in meters or centimeters, a force in Newtons, and so on? If the units are incorrect, it's a clear indication of an error in our calculations.

Beyond these basic checks, it's also helpful to consider how the answer might change if we varied the parameters of the problem. For example:

• What would happen to the distance between the spheres if we increased the charge on one of the spheres? We'd expect the distance to increase due to stronger electrostatic repulsion.
• What if we increased the mass of the spheres? We'd expect the distance to decrease because the gravitational force would be stronger, pulling the spheres closer together.

By thinking about these qualitative relationships, we can gain confidence in our answer and catch potential errors. If our calculated answer doesn't align with our physical intuition, it's a sign to go back and review our work, looking for mistakes in our equations, substitutions, or calculations.

Analyzing the results is not just about finding errors; it's also about deepening our understanding of the physics involved. It helps us connect the mathematical solution to the physical reality of the problem, making the learning process more meaningful and lasting.

So, there you have it! We've walked through a comprehensive approach to solving a physics problem involving charged spheres in equilibrium. Remember, the key is to break the problem down into manageable steps: understand the problem, identify the unknowns and relevant equations, solve for the unknowns, and, most importantly, analyze the results to ensure they make sense. Keep practicing, and you'll become a pro at tackling these types of physics challenges! You got this! 🚀