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Figure 22-21 shows two square arrays of charged particles. The squares with edges of 2d and d are centered at point P and are misaligned. The particles are separated by either d or d/2 along the perimeters of the squares.
What is the magnitude of the net electric field at P? (Note: The symbol used in the subscript of ε0 is a zero, not an "O".)?
Net electric field due to all charges exept the charges at the midpoints of vertical sides of the large square is zero. These two charges gives:
E=(1/4πε₀)•(2q)/(d/2)² - (1/4πε₀)•(q)/(d/2)²= (1/4πε₀)•(q)/(d/2)² (directed to the left)
To determine the magnitude of the net electric field at point P, we need to calculate the electric field due to each square array of charged particles and then combine them using vector addition.
Let's first consider the square array with edges of 2d centered at point P. The particles in this square are separated by d along the perimeter. Since the particles are all equally spaced, we can consider one side of the square. The electric field due to a point charge located at a distance d from point P can be calculated using Coulomb's Law.
The electric field due to one side of the square is given by:
E1 = (1/4πε₀) * (q/d²)
Next, we consider the smaller square array with edges of d centered at point P. The particles in this square are separated by d/2 along the perimeter. Again, we can consider one side of the square. The electric field due to a point charge located at a distance d/2 from point P can be calculated using Coulomb's Law.
The electric field due to one side of the smaller square is given by:
E2 = (1/4πε₀) * (q/(d/2)²)
Now, we need to find the net electric field at point P. Since the two square arrays are misaligned, their electric fields do not cancel out completely. We can find the net electric field by adding the two electric fields as vectors.
E_net = E1 + E2
Finally, we can calculate the magnitude of the net electric field at point P using the formula:
|E_net| = √(Ex² + Ey²)
Where Ex and Ey are the x and y components of the net electric field.
Please note that you will need the specific values of q, d, and ε₀ to calculate the actual numerical value of the magnitude of the net electric field at point P.
To find the magnitude of the net electric field at point P, we need to consider the individual electric fields generated by each of the charged particles in the two squares and then sum them up.
1. Let's start by determining the electric field generated by a single charged particle. The electric field at a point due to a point charge can be calculated using Coulomb's Law:
E = (k * q) / r^2
Where E is the electric field, k is Coulomb's constant (9 x 10^9 Nm^2/C^2), q is the charge of the particle, and r is the distance between the particle and the point where we want to find the electric field.
2. In this case, we have two different arrangements of charged particles:
- The first square has a side length of 2d and is centered at point P.
- The second square has a side length of d and is also centered at point P.
3. Both squares are misaligned, so the particles in the squares are separated by either d or d/2 along the perimeters.
4. Let's consider the first square with a side length of 2d. Each side of the square contains 4 particles.
- The particles on the top and bottom sides are separated by d/2.
- The particles on the left and right sides are separated by d.
The electric field generated by each of these particles will have a component along the x-axis and y-axis at point P. You can calculate the electric field components for each particle using Coulomb's Law and considering the distance between the particle and point P.
5. Now let's consider the second square with a side length of d. Each side of the square also contains 4 particles.
- The particles on the top and bottom sides are separated by d/2.
- The particles on the left and right sides are separated by d.
Again, calculate the electric field components for each particle using Coulomb's Law and considering the distance between the particle and point P.
6. Once you have calculated the electric field components for all the particles in both squares, sum up the x- and y-components separately. This will give you the net electric field components along the x-axis (Ex) and y-axis (Ey).
7. Finally, use Pythagorean's theorem to find the magnitude of the net electric field at point P:
E_net = sqrt(Ex^2 + Ey^2)
By following these steps, you can find the magnitude of the net electric field at point P.