Calculating Resultant Force: A Physics Guide

Oct 22, 2025 by Dimemap Team 45 views

Hey guys! Let's dive into a cool physics problem. We're going to figure out the resultant force on a positive charge, taking into account the forces exerted by two negative charges. It's like a tug-of-war, but with electrical forces. We'll break down the problem step-by-step, including a diagram and the use of angle decomposition, to make sure you get it. This is super helpful, whether you're studying for a test or just curious about how electric charges interact. So, grab your pencils and let's get started!

Understanding the Problem: Electric Forces and Coulomb's Law

Alright, first things first, let's understand what we're dealing with. We have three charges: one positive (+2 μC) and two negative (-4 μC each). The positive charge is the one feeling the force, while the negative charges are the ones exerting the force. The distance between the charges is also important: the positive charge is 60 mm away from each negative charge, and the negative charges are 80 mm apart. We're going to use Coulomb's Law, which describes the force between two charged objects. This law states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It’s a fundamental concept for understanding the interactions of electric charges, and will be key in our calculations.

Now, how do we use this law? Well, the formula is: F = k * |q1 * q2| / r², where:

F is the electrostatic force (what we want to find).
k is Coulomb's constant (approximately 8.9875 × 10⁹ N⋅m²/C²).
q1 and q2 are the magnitudes of the charges.
r is the distance between the charges.

Since we're calculating the resultant force, we'll need to figure out the force from each negative charge individually, and then combine them, taking direction into account, because force is a vector quantity (it has both magnitude and direction). That’s where the angle decomposition comes in handy. It helps us break down each force into components along the x and y axes, making it easier to add them up. The diagram is also key; it helps visualize how the forces are oriented, ensuring our calculations are accurate. So, let’s get into the specifics, shall we?

Setting Up the Diagram and the Approach

Let’s start with a neat diagram. Imagine the two negative charges are sitting on the x-axis, 80 mm apart. Place the positive charge somewhere above, equidistant from both negative charges. This creates a symmetrical setup. Now, draw arrows from each negative charge towards the positive charge. Those arrows represent the forces (F1 and F2) each negative charge exerts on the positive one. Remember, opposite charges attract, so the arrows should point towards the negative charges. The resultant force will be the sum of these two forces. To find the direction and magnitude of this resultant force, we'll use a combination of Coulomb's Law and trigonometry.

Here’s how we'll solve it, step-by-step:

Calculate the force magnitude: For each pair of charges using Coulomb's Law.
Determine the angles: Find the angles between the force vectors and the x-axis (we'll need some basic trigonometry, like cosine and sine).
Decompose the forces: Break down F1 and F2 into their x and y components.
Sum the components: Add the x-components together and the y-components together.
Find the resultant: Use the Pythagorean theorem to calculate the magnitude of the resultant force from the summed x and y components. Then, find the angle of the resultant vector to know its direction.

This methodical approach ensures we account for both the magnitude and direction of the forces involved. Before we proceed to the calculation, remember that we're dealing with very small numbers (microcoulombs and millimeters), so be super careful with your units! Keeping track of units is super important in physics, 'cause if you mess them up, your whole answer goes bonkers. Ready to crunch some numbers?

Calculating the Forces: Magnitude and Direction

Okay, let’s crunch those numbers! First, let's calculate the magnitude of the force (F1 and F2) between each negative charge and the positive charge. We know:

q1 = +2 μC = 2 x 10⁻⁶ C (convert microcoulombs to Coulombs)
q2 = -4 μC = -4 x 10⁻⁶ C
r = 60 mm = 0.06 m (convert millimeters to meters)
k = 8.9875 x 10⁹ N⋅m²/C²

Using Coulomb's Law, for the force from one of the negative charges (F1):

F1 = k * |q1 * q2| / r² F1 = (8.9875 x 10⁹ N⋅m²/C²) * |(2 x 10⁻⁶ C) * (-4 x 10⁻⁶ C)| / (0.06 m)² F1 ≈ 1.997 N

Since the situation is symmetrical, the force F2 will also be the same magnitude, approximately 1.997 N. Now, for the direction, notice that the force from each negative charge acts along a line connecting the positive and negative charges. So, the direction angle can be found using trigonometry. If we draw a line connecting the negative charges, and another line from the midpoint of that line to the positive charge (forming a right triangle), the angle we want is opposite to the side which is half the distance between the negative charges (40 mm = 0.04 m) and the adjacent is the distance from the midpoint of the negative charges to the positive charge. You can calculate the angles using the inverse sine or cosine function: arccos(0.04m/0.06m) = 48.19 degrees.

So both vectors F1 and F2 form 48.19 degree angles with the x-axis. Since the configuration is symmetrical, this simplifies our job. These force vectors are symmetrical, meaning they make the same angle with the vertical axis. The vertical components will add together, and the horizontal components will cancel each other out, because they are equal in magnitude and point in opposite directions. So, the resultant force will be straight up along the y-axis.

Decomposing Forces and Finding the Resultant

Time to break down the forces into their x and y components. Since we have equal forces and a symmetrical setup, this step becomes a little easier. The forces F1 and F2 each have x and y components. The x components will be equal in magnitude but opposite in direction. They’ll cancel each other out! The y-components, however, will add up because they point in the same direction (upwards).

Let’s calculate the y-component (Fy) for each force:

Fy = F * sin(θ), where θ is the angle from the x-axis (48.19 degrees) and F is the magnitude of the force (1.997 N).

So, Fy1 = 1.997 N * sin(48.19°) ≈ 1.49 N and since the configuration is symmetrical, Fy2 ≈ 1.49 N.

To find the resultant force, we add the y-components together: F_resultant_y = Fy1 + Fy2 = 1.49 N + 1.49 N = 2.98 N.

And since the x components cancel out, the resultant force only has a y-component. Its magnitude is approximately 2.98 N, and it points directly upwards, along the y-axis. The final step is to determine the direction. Since the only component is on the y-axis, the direction is straight up. The resultant force is the sum of all the forces acting on the positive charge, and it's a single force pointing upwards with a magnitude of 2.98 N. This makes sense; the two negative charges are both pulling the positive charge upwards towards them, and because they are placed symmetrically, the horizontal forces cancel each other out. This is all thanks to our diagram and angle decomposition, which helped us find both the magnitude and direction.

Conclusion and Diagram

So, there you have it, guys! We have successfully calculated the resultant force on the positive charge. The resultant force is approximately 2.98 N, and it points straight up. We've used Coulomb's Law, angle decomposition, and a clear diagram to solve the problem. Remember, these methods can be applied to many similar situations involving electrical forces. The key takeaway is to break down complex problems into manageable steps, use diagrams to visualize the interactions, and be careful with your units.

Here’s a basic diagram to visualize what we just calculated:

          +2 μC (Positive Charge)
            |      ^  Resultant Force (2.98 N)
            |     /|
            |    /  |
           F1 /    |    F2
          /   |    |
       /     |    |
 -4 μC ------|---- -4 μC
      (0, -0.04 m)  (0.08 m, 0)
       F1 and F2 are the forces acting on the positive charge
       The diagram shows that the resultant force is in the positive y direction.

Keep practicing these problems, and you’ll master them in no time! Physics can be super fun when you break it down step-by-step. Remember, practice makes perfect, so try out some more problems to solidify your understanding. Until next time, keep exploring the fascinating world of physics!