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Magazine Chapter 21: Problem 76
A point charge \(q_{1}=100 . \mathrm{nC}\) is at the origin of an \(x y\) -coordinate system, a point charge \(q_{2}=-80.0 \mathrm{nC}\) is on the \(x\) -axis at \(x=2.00 \mathrm{~m},\) and a point charge \(q_{3}=-60.0 \mathrm{nC}\) is on the \(y\) -axis at \(y=-2.00 \mathrm{~m} .\) Determine the net force (magnitude and direction) on \(q_{1}\).
Short Answer
Expert verified
Answer: The magnitude of the net force on \(q_1\) is \(22.5\:nN\), and the direction is \(36.9^\circ\) counterclockwise from the negative x-axis.
Step by step solution
01
Understanding the problem and setting up the variables
Let's first identify the given values and variables to set up the problem. The charges are: \(q_{1}=100\:nC\) (at the origin), \(q_{2}=-80.0\:nC\) (at \((2.00\:m,0)\)), and \(q_{3}=-60.0\:nC\) (at \((0,-2.00\:m)\)). We need to find the net force on \(q_1\), which is the sum of the individual forces acting on \(q_1\) due to \(q_2\) and \(q_3\).
02
Calculate the force between \(q_1\) and \(q_2\)
To calculate the force between \(q_1\) and \(q_2\), we can use Coulomb's Law which states: \(F=\frac{kq_1q_2}{r^2}\). In this case, the distance \(r_{12}\) between \(q_1\) (origin) and \(q_2\) (2.00 meters along the x-axis) is 2.00 meters. So, the force between \(q_1\) and \(q_2\) is given by:
\(F_{12}=\frac{(9*10^9)(100*10^{-9})((-80)*10^{-9})}{(2)^2}=-18*10^{-9}N\)
This force is acting in the negative x-direction.
03
Calculate the force between \(q_1\) and \(q_3\)
Now, we will calculate the force between \(q_1\) and \(q_3\) using Coulomb's Law, with the distance \(r_{13}\) being \(2.00\) meters. The force between \(q_1\) and \(q_3\) is given by:
\(F_{13}=\frac{(9*10^9)(100*10^{-9})((-60)*10^{-9})}{(2)^2}=-13.5*10^{-9}N\)
This force is acting in the positive y-direction.
04
Finding the net force on \(q_1\)
To find the net force on \(q_1\), we will sum the individual forces acting on \(q_1\) due to \(q_2\) and \(q_3\). The net force on \(q_1\) can be represented as a 2D vector:
\(\vec{F}_{net} = \vec{F}_{12}+\vec{F}_{13}\)
Given the direction of \(F_{12}\) (negative x-direction) and \(F_{13}\) (positive y-direction), we can rewrite \(\vec{F}_{net}\) as:
\(\vec{F}_{net} = (-18*10^{-9}\hat{i}) + (-13.5*10^{-9}\hat{j})\)
05
Compute the magnitude and direction of the net force
We can compute the magnitude of the net force using the Pythagorean theorem:
\(F_{net} = \sqrt{(-18*10^{-9})^2+(-13.5*10^{-9})^2}= 22.5*10^{-9}N\)
Next, we need to find the angle between the net force and the negative x-axis. We can use the arctangent function for this purpose:
\(\theta = \arctan\left(\frac{-13.5*10^{-9}}{-18*10^{-9}}\right)=36.9^\circ\)
So, the net force magnitude on \(q_1\) is \(22.5\:nN\), and the direction is \(36.9^\circ\) counterclockwise from the negative x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Point Charge
A point charge is a charge that is considered to be concentrated at a single point in space. In physics and electrostatics, we often use the concept of a point charge to simplify calculations and model electric interactions.
When we say point charge, we are describing a charge that has negligible size compared to the distance over which it interacts with other charges.
For example:
When we say point charge, we are describing a charge that has negligible size compared to the distance over which it interacts with other charges.
For example:
- If you have a tiny object carrying an electric charge, you might treat it as a point charge if the scale of interest is much larger than the object itself.
- This allows us to easily use formulas like Coulomb's Law to calculate the forces between charges.
Net Force Calculation
Calculating the net force on a point charge involves analyzing all the individual forces acting on it due to other charges in the system. This is crucial in understanding how electric charges interact in a field.
Here's how you can approach the calculation:
Here's how you can approach the calculation:
- **Identify each pair of interacting charges**: Determine which charges are affecting the point charge of interest. In our exercise, `q_1` is affected by `q_2` and `q_3`.
- **Calculate individual forces using Coulomb's Law**: For each pair, use the formula \[ F = \frac{k \, |q_1 q_2|}{r^2} \] where \( F \) is the force, \( k \) is the Coulomb's constant (\(9 \times 10^9 \, \text{Nm}^2/\text{C}^2\)), and \( r \) is the distance between the charges.
- **Consider the direction of the force**: Forces are vector quantities, meaning they have both magnitude and direction. Determine in what direction each force acts based on charge signs and positions.
- **Add the forces vectorially**: The net force is the vector sum of individual forces. In 2D, you add the x-components and y-components separately to find the resultant force vector.
Coordinate System
In physics, a coordinate system is employed to specify the positions of charges in space and help carry out calculations with clarity. It serves as a reference framework for analyzing interactions between charges.
In the exercise, we use a 2D Cartesian coordinate system:
In the exercise, we use a 2D Cartesian coordinate system:
- **The Origin:** The point charge `q_1` is located at the origin (0, 0), which acts as a central reference point with both x and y coordinates set to zero.
- **X-Axis and Y-Axis:** Other charges, such as `q_2` and `q_3`, are positioned on these axes, making it easier to compute distances and apply Coulomb's Law.
- **Vectors:** Using the coordinate system, forces can be expressed as vectors with both x and y components, which helps in calculating the net force.
Vector Addition
Vector addition is a fundamental concept in physics that helps in determining the net effect of multiple forces acting on a charge. It is crucial for calculating the resultant force direction and magnitude in electrostatics.
Here's how vector addition comes into play in the context of charges:
Here's how vector addition comes into play in the context of charges:
- **Representing Forces as Vectors:** Each force on a charge due to other charges is a vector, meaning it has both a magnitude (how strong the force is) and a direction (where the force is applied).
- **Adding Vectors:** To find the net force, combine these vectors using vector addition. You do this by summing up their x-components and y-components separately.
- **Calculating Magnitude and Direction:** Once vectors are added together, you can calculate the magnitude of the resultant force using the formula: \[ F_{\text{net}} = \sqrt{(F_x)^2 + (F_y)^2} \] The direction can be found using the arctangent function: \[ \theta = \arctan\left(\frac{F_y}{F_x}\right) \]
