Charge q1 = -6.5 nC is located at the coordinate system origin, while charge q2 = -1.5 nC is located at (a, 0), where a = 1.2 m. The point P has coordinates (a, b), where b = 2.1 m.
(a) At the point P, find the x-component of the electric �field Ex in units of N/C.
(b) At the point P, find the y-component of the electric �field Ey in units of N/C.
a) Ex = -(1.5*10^-9*6.5*10^-9)/(1.2^2) = -2.8125*10^-17 N/C
b) Ey = -(1.5*10^-9*6.5*10^-9)/(2.1^2) = -1.7143*10^-17 N/C
To find the electric field components at point P, we can use Coulomb's Law and the principle of superposition.
(a) To find the x-component of the electric field (Ex) at point P, we need to calculate the x-component of the electric field contribution from both charges.
The electric field (E1x) at P due to charge q1 can be calculated using the formula:
E1x = (k * q1 * dx) / r^3
where k is the electrostatic constant (9 * 10^9 Nm^2/C^2), q1 is the charge at the origin (-6.5 nC), dx is the x-component of the distance between point P and the origin (a - 0 = a), and r is the total distance between charge q1 and point P.
Substituting the given values, we get:
E1x = (9 * 10^9 Nm^2/C^2) * (-6.5 * 10^-9 C) * (1.2 m) / [(a^2 + b^2)^(3/2)]
Now let's calculate the electric field (E2x) at P due to charge q2:
E2x = (k * q2 * dx) / r^3
where q2 is the charge at (a, 0) (-1.5 nC).
Substituting the values, we get:
E2x = (9 * 10^9 Nm^2/C^2) * (-1.5 * 10^-9 C) * (a) / [(a^2 + b^2)^(3/2)]
Now we can add the x-components of the electric field contributions from both charges:
Ex = E1x + E2x
(b) To find the y-component of the electric field (Ey) at point P, we can follow the same steps as above, but calculate the y-component of the electric field contributions from both charges.
The electric field (E1y) at P due to charge q1 can be calculated using the formula:
E1y = (k * q1 * dy) / r^3
where dy is the y-component of the distance between point P and the origin (b), and r is the total distance between charge q1 and point P.
Substituting the given values, we get:
E1y = (9 * 10^9 Nm^2/C^2) * (-6.5 * 10^-9 C) * (b) / [(a^2 + b^2)^(3/2)]
Now let's calculate the electric field (E2y) at P due to charge q2:
E2y = (k * q2 * dy) / r^3
where dy is the y-component of the distance between charge q2 and point P (b), and q2 is the charge at (a, 0) (-1.5 nC).
Substituting the values, we get:
E2y = (9 * 10^9 Nm^2/C^2) * (-1.5 * 10^-9 C) * (0) / [(a^2 + b^2)^(3/2)]
Since the y-component of the electric field due to charge q2 is zero (charge q2 is located on the x-axis), the total y-component of the electric field at point P is only due to the contribution from charge q1:
Ey = E1y
Now we have the x-component of the electric field (Ex) and the y-component of the electric field (Ey) at point P in units of N/C.
To find the electric field components at point P, let's use Coulomb's law. The equation for the 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 x 10^9 N m^2/C^2)
|q| is the magnitude of the charge
r is the distance between the charge and the point where the electric field is being calculated
First, let's calculate the x-component of the electric field (Ex) at point P:
(a) To calculate Ex, we need to find the distance between charge q1 and point P, which is the x-coordinate of point P, a. Since q1 is located at the origin (0, 0), the distance between q1 and P is simply a. We can then use Coulomb's law to calculate the electric field due to q1 at point P:
Ex1 = k * |q1| / a^2
Where Ex1 is the x-component of the electric field due to q1.
Substituting the values:
Ex1 = (8.99 x 10^9 N m^2/C^2) * (6.5 x 10^-9 C) / (a^2)
Next, let's calculate the x-component of the electric field (Ex2) due to q2 at point P:
Ex2 = k * |q2| / (a - a)^2
Where Ex2 is the x-component of the electric field due to q2. Since q2 is along the x-axis, the distance between q2 and point P is zero, so we divide by (a - a)^2, which simplifies to 0.
Now, to find the total x-component of the electric field (Ex) at point P, we simply sum up the contributions from both charges:
Ex = Ex1 + Ex2
(b) Similarly, let's calculate the y-component of the electric field (Ey) at point P:
To calculate Ey, we need to find the distance between charge q1 and point P, which is the y-coordinate of point P, b. Since q1 is located at the origin (0, 0), the distance between q1 and P is simply b. We can then use Coulomb's law to calculate the electric field due to q1 at point P:
Ey1 = k * |q1| / b^2
Where Ey1 is the y-component of the electric field due to q1.
Next, let's calculate the y-component of the electric field (Ey2) due to q2 at point P:
Ey2 = k * |q2| / (a^2 + b^2)
Where Ey2 is the y-component of the electric field due to q2.
To find the total y-component of the electric field (Ey) at point P, we sum up the contributions from both charges:
Ey = Ey1 + Ey2
With these formulas, you can substitute the values of the charges and distances to calculate the x-component (Ex) and y-component (Ey) of the electric field at point P.