The electric field in an xy plane produced by a positively charged particle is 14.2(4.4 + 3.3) N/C at the point (3.1, 3.7) cm and 117 N/C at the point (3.2, 0) cm. What are (in cm) the (a)x and (b)y coordinates of the particle? (c) What is the charge (in Coulombs) of the particle?
To find the (a)x and (b)y coordinates of the particle, we can use the concept of electric field and apply it to the given data points.
Let's start by calculating the electric field at point (3.1, 3.7) cm. The electric field in the xy plane is given by the formula:
E = (k * q) / r^2
Where:
E is the electric field
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
q is the charge of the particle
r is the distance between the particle and the point where electric field is measured
Using the given information, we can write the equation as:
14.2(4.4 + 3.3) = (k * q) / (3.1 - 3.2)^2
Simplifying this equation gives:
(14.2 * 7.7) = (k * q) / (0.01)
To find the coordinates (a)x and (b)y, we need to consider another data point. Let's calculate the electric field at point (3.2, 0) cm using the same formula:
117 = (k * q) / (3.2 - 3.2)^2
Simplifying this equation gives:
117 = (k * q) / 0
Since the denominator is zero, it means that the particle is located at point (3.2, 0) cm.
Therefore, the (a)x coordinate of the particle is 3.2 cm, and the (b)y coordinate is 0 cm.
Now, let's determine the charge of the particle (c). We can use the electric field equation and the first data point:
14.2(4.4 + 3.3) = (k * q) / (3.1 - 3.2)^2
(14.2 * 7.7) = (9 x 10^9 * q) / 0.01
(14.2 * 7.7) * 0.01 = 9 x 10^9 * q
Simplifying this equation gives:
q = ((14.2 * 7.7) * 0.01) / (9 x 10^9)
Calculating this equation gives the charge q = 1.26078 x 10^-10 C.
Therefore:
(a) The x-coordinate of the particle is 3.2 cm.
(b) The y-coordinate of the particle is 0 cm.
(c) The charge of the particle is 1.26078 x 10^-10 C.