two charged particles are located on an x axis: -q = -4.80 × 10-19 C at x = -3.00 m and q = 4.80 × 10-19 C at x = 3.00 m. What are the (a)x- and (b)y- components of the net electric field produced at point P at y = 5.00 m?
Well, well, well, we have some charged particles and an electric field puzzle here! Let's dive in, shall we?
(a) To find the x-component of the net electric field at point P, we need to calculate the electric field produced by each charged particle and then add them up. The formula for electric field due to a point charge is:
E = k * q / r^2
where E is the electric field, k is Coulomb's constant (8.99 × 10^9 Nm^2/C^2), q is the charge, and r is the distance between the charge and the point of interest.
For the negative charge at x = -3.00 m, the distance to point P is:
r1 = sqrt((-3.00)^2 + 5.00^2)
Using the formula, we find:
E1x = k * (-q) / r1^2
Now, for the positive charge at x = 3.00 m, the distance to point P is:
r2 = sqrt((3.00)^2 + 5.00^2)
Using the formula, we find:
E2x = k * q / r2^2
Finally, we add the x-components of the electric fields:
Net E_x = E1x + E2x
(b) As for the y-component of the net electric field at point P, it will be zero since both charges are on the x-axis and the y-components of their electric fields cancel each other out. So, E_y = 0.
There you have it! The x-component of the net electric field can be found by adding the electric fields produced by each charge, and the y-component is zero since the charges are located on the x-axis.
To find the net electric field at point P, we need to calculate the x and y components separately and then combine them at the end.
(a) To find the x-component of the net electric field at point P, we can use Coulomb's law. The electric field due to a point charge is given by:
E = k * (Q / r^2)
where E is the electric field, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge to the point where we want to find the electric field.
Let's calculate the x-component of the electric field due to the first charge (-q) at point P:
E1x = k * (-q) / (x1 - x)
where x1 is the position of the first charge (-3.00 m) and x is the x-coordinate of point P (which is 0, since it is located on the x-axis). Plugging in the values:
E1x = (9 × 10^9 Nm^2/C^2) * (-4.80 × 10^-19 C) / (-3.00 m - 0)
Similarly, let's calculate the x-component of the electric field due to the second charge (q) at point P:
E2x = k * q / (x2 - x)
where x2 is the position of the second charge (3.00 m), and plugging in the values:
E2x = (9 × 10^9 Nm^2/C^2) * (4.80 × 10^-19 C) / (3.00 m - 0)
Now, we can find the x-component of the net electric field at point P by summing up the x-components of the individual electric fields:
Net E_x = E1x + E2x
(b) To find the y-component of the net electric field at point P, we can use the same process as above, but considering the y component of each individual electric field.
The y-coordinate of point P is given as 5.00 m. Since both charged particles are on the x-axis, the y-component of the electric field due to each of them will be zero because the y-component of the position vector is zero. Therefore, the net electric field's y-component will also be zero.
So, to summarize:
(a) The x-component of the net electric field at point P is given by Net E_x = E1x + E2x.
(b) The y-component of the net electric field at point P is zero.
To find the net electric field at point P, we can break down the problem into two steps:
Step 1: Calculate the electric field due to each charged particle at point P.
Step 2: Add the electric fields vectorially to find the net electric field.
Before we proceed, let's define the variables:
-q = -4.80 × 10^-19 C (charge at x = -3.00 m)
q = 4.80 × 10^-19 C (charge at x = 3.00 m)
P is located at y = 5.00 m
Step 1: Calculate the electric field due to each charged particle at point P.
The electric field at point P due to a charged particle can be calculated using Coulomb's law:
Electric field (E) = (k * q) / r^2
where k is the electrostatic constant (k = 8.99 × 10^9 Nm^2/C^2), q is the charge producing the electric field, and r is the distance between the charge and the point at which the field is being measured.
For the charge at x = -3.00 m:
r1 = x1 - xp = -3.00 m (since xp = 0)
Electric field due to q1 at P: E1 = (k * (-q)) / r1^2
For the charge at x = 3.00 m:
r2 = x2 - xp = 3.00 m (since xp = 0)
Electric field due to q2 at P: E2 = (k * q) / r2^2
Step 2: Add the electric fields vectorially to find the net electric field.
Since the electric fields due to each charged particle are vectors, we have to consider their direction and magnitude.
Let's denote the electric field due to the negative charge as E1 (-q) and the electric field due to the positive charge as E2 (q).
The x-component of the net electric field (Ex) is given by:
Ex = E1 - E2
To find the y-component of the net electric field (Ey), we need to consider that the electric field due to a charged particle is always along the radial direction. In this case, the electric fields due to both charges are directed towards the positive y-direction, so their y-components will add up:
Ey = E1 + E2
Now we can calculate the electric fields and their components:
Step 1: Calculate the electric field due to each charged particle at point P.
Using Coulomb's law, we have:
E1 = (k * (-q)) / r1^2
E2 = (k * q) / r2^2
Substituting the values:
E1 = (8.99 × 10^9 Nm^2/C^2 * (-4.80 × 10^-19 C)) / (-3.00 m)^2
E2 = (8.99 × 10^9 Nm^2/C^2 * 4.80 × 10^-19 C) / (3.00 m)^2
Calculating E1 and E2 will give you their respective electric fields.
Step 2: Add the electric fields vectorially to find the net electric field.
Ex = E1 - E2
Ey = E1 + E2
Calculate Ex and Ey using the values obtained from Step 1. These values will give you the x- and y-components of the net electric field at point P.