Question

Eight identical point charges of \(Q \mathrm{C}\) each are located at the corners of a cube of side length \(a\), with one charge at the origin, and with the three nearest charges at \((a, 0,0),(0, a, 0)\), and \((0,0, a)\). Find an expression for the total vector force on the charge at \(P(a, a, a)\), assuming free space.

Short answer

Question: Calculate the total vector force on a point charge located at P(a, a, a) in a cube of side length a, with equal point charges Q at each corner. Answer: To find the total vector force on the charge at P, follow these steps: 1. Visualize the cube and the charge positions. 2. Calculate the distance between the charge at P and each of the other charges. 3. Apply Coulomb's law to find the force between each pair of charges. 4. Calculate the force vector for each pair of charges. 5. Determine the total vector force on the charge at P by summing the force vectors from each pair of charges.

Step-by-step solution

Identify the given information and visualize the problem

We are given the point charges at each corner of the cube as \(Q\), the side length of the cube as \(a\), and the position vector of the charge we want to find the total force on: \(P(a, a, a)\). Visualize the cube and the charge positions to have a clear understanding of the problem.

Calculating the distance between charges

To apply Coulomb's law, we need the distance between each pair of charges. Using the position vectors, calculate the distance between the charge at P and each of the other charges, which will result in 7 distances.

Apply Coulomb's law to find the force between each pair of charges

Coulomb's law states that the force between two charged particles is: $$F = k \frac{Q_1 Q_2}{r^2}$$ where \(F\) is the force between two charges, \(Q_1\) and \(Q_2\) are the charges, \(r\) is the distance between the charges, and \(k\) is the Coulomb constant. Apply this formula to each pair of charges to determine the force between the charge at P and each of the other charges.

Calculate the force vector for each pair of charges

Now that we have the magnitudes of the forces between each pair of charges, we need to calculate the force vector for each pair. To do this, we can use the unit vector and multiply it by the magnitude of the force. The unit vector between two points is given by: $$\hat{r} = \frac{\vec{r_2} - \vec{r_1}}{|\vec{r_2} - \vec{r_1}|}$$ where \(\vec{r_1}\) and \(\vec{r_2}\) are the position vectors of the charges between which force is being calculated. Calculate the force vector for each pair of charges acting on the charge at P.

Determine the total vector force on the charge at P

To find the total vector force on the charge at P, sum the force vectors from each pair of charges. This will give you the total vector force acting on the charge at point P.

Key concepts

Electrostatic Force

The principle of electrostatic force is crucial to understanding how charged particles interact.

It is the basis of Coulomb's law, which states that the force (\( F \)) between two point charges (\( Q_1 \) and \( Q_2 \)) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (\( r \) ) between them. In mathematical terms, this is expressed as \[ F = k \frac{Q_1 Q_2}{r^2} \] where \( k \) is the Coulomb constant.

In the context of the exercise provided, where we are analyzing the forces between charges located at the corners of a cube, we use Coulomb's law to compute the force exerted by each of the other seven charges on the charge at point P. An important point for students to note is that the force is a vector quantity, meaning it has a direction as well as magnitude. The direction of this force is along the line joining the charges and follows the rule that like charges repel and unlike charges attract.

To ensure clarity, imagine two charged balloons. If they are both charged similarly, they try to move away from each other when brought close. This is due to the electrostatic force acting between them. The same concept is applied in the cube scenario, where each charge repels the charge at P.

Vector Superposition in Electromagnetism

The concept of vector superposition is fundamental when dealing with multiple forces in electromagnetism.

It implies that when several forces act on a charge simultaneously, the total force can be found by vectorially adding all the individual forces. This method is grounded in the superposition principle, which holds that for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.

In applying vector superposition to our exercise, calculations for the force between the charge at P and each of the seven other charges must be performed first. Each resulting force vector is then added together to ascertain the total force on the charge at P. Students must understand that each vector addition considers both magnitude and direction.

An everyday analogy would be carrying two bags, one in each hand. If the bags have different weights (magnitudes) and pull your hands in opposite directions (vectors), your body has to adapt to the combined effect (superposition) of carrying both bags simultaneously.

Electric Charge Interaction

Understanding electric charge interaction is key to solving problems related to electrostatics.

Charges can interact in complex environments, and these interactions obey specific rules. The interactions between charges are governed not only by the amount of charge but also by the distance separating the charges. Through this interaction, charges exert forces upon each other. For example, two charges of the same sign will exert a repulsive force on each other, while two charges of opposite signs will attract.

In our exercise, every corner of the cube has a charge that interacts with the charge at P. The interactions are dictated by the distances and charge values. Due to symmetry in geometrical arrangements, certain components of the forces might cancel out, simplifying the computation of the total force on charge P.

A useful tip for students is to draw a diagram illustrating the forces acting on the charge and look for symmetrical patterns. These patterns can provide insights into possible cancellations or reinforcements of forces, making the problem more manageable and the understanding of charge interactions clearer.