Calculating the Distance Between Two Point Charges in Space
Understanding the position and distance between charged particles is crucial in the field of electrostatics, which is a branch of physics. This article explains in detail how to determine the distance between two point charges using their vector positions. We will also discuss the mathematical concept behind the distance metric and provide a step-by-step approach to solve a specific problem.
Introduction to Point Charges and Positions
Point charges are hypothetical particles that carry an electric charge while being dimensionless. In practical scenarios, when the dimensions of the charges are much smaller compared to the distance between them, they can be modeled as point charges. This simplifies the problem and allows us to apply vector calculus.
Given Data
Two point charges are given:
Charge (Q_1 2 , mutext{C}) located at the point (r_1 4textbf{i} - 2textbf{j} 5textbf{k}) (measured in centimeters) Charge (Q_2 4 , mutext{C}) located at the point (r_2 8textbf{i} 5textbf{j} - 9textbf{k}) (measured in centimeters)To find the distance between the two charges, we will use the distance metric formula:
d sqrt{Delta x^2 Delta y^2 Delta z^2}
Where (Delta x), (Delta y), and (Delta z) represent the differences in the x, y, and z coordinates respectively.
Vector Subtraction: Finding (r_2 - r_1)
Let's begin by determining the vector difference between the positions of the two charges. We use the formula for vector subtraction:
(textbf{r}_2 - textbf{r}_1 (r_{2x} - r_{1x}) textbf{i} (r_{2y} - r_{1y}) textbf{j} (r_{2z} - r_{1z}) textbf{k})
Plugging in the given values, we get:
(8 textbf{i} 5 textbf{j} - 9 textbf{k}) - (4 textbf{i} - 2 textbf{j} 5 textbf{k}) 4 textbf{i} 7 textbf{j} - 14 textbf{k}
Calculating the Magnitude of the Vector
Using the distance metric formula, we calculate the magnitude of the vector obtained in the previous step. The formula is:
|r_2 - r_1| sqrt{(4)^2 (7)^2 (-14)^2}
First, we calculate the squared differences:
(4^2 16) (7^2 49) ((-14)^2 196)Adding these values together:
16 49 196 261
Now, we take the square root:
|r_2 - r_1| sqrt{261} approx 16.15 , text{cm}
Conclusion
The distance between the two point charges, measured from the position of charge (Q_1) to the position of charge (Q_2), is approximately 16.15 cm. This calculation is fundamental in electrostatics and is essential for understanding the behavior of electric fields and forces between charged particles.