Two positive charges q1 and q2 fixed to a circle. At the center of the circle they produce a net electric field that is directed upward along the vertical axis. Determine the ratio q2/q1.
Angle for q1 is 30 degrees from diameter.
Angle for q2 is 60 degrees from diameter.
Are these angles above or below the diameter?
Also, is that diameter vertical or horizontal?
If vertical and charges below:
horizontal components equal and opposite
(kQ2/r^2) cos 60 = (k Q1/r^2) cos 30
Q2 cos 60 = Q1 cos 30
Q2/Q1 = cos 30/cos 60
To determine the ratio q2/q1, we can use the principle that the net electric field at the center of the circle is the vector sum of the electric fields produced by each charge.
Let's denote the magnitudes of the charges as q1 and q2, and let's assume q1 is located at an angle of 30 degrees from the diameter, and q2 is located at an angle of 60 degrees from the diameter.
The electric field produced by a point charge at a distance r is given by the equation:
E = k * (q / r^2)
Where k is the electrostatic constant.
Since q2 and q1 produce net electric field along the vertical axis, the vertical components of their electric fields must cancel each other out. Therefore, the magnitudes of their vertical components should be equal.
To find the ratio q2/q1, we need to find the ratio of their horizontal components. Let's calculate the horizontal and vertical components of their electric fields:
For q1:
The horizontal component of the electric field produced by q1 can be calculated as:
Ex1 = E1 * cos(theta1)
Where theta1 is the angle of q1 from the x-axis (diameter) and E1 is the magnitude of the electric field produced by q1.
For q2:
The horizontal component of the electric field produced by q2 can be calculated as:
Ex2 = E2 * cos(theta2)
Where theta2 is the angle of q2 from the x-axis (diameter) and E2 is the magnitude of the electric field produced by q2.
Since the vertical components cancel each other out, we have:
E1 * sin(theta1) = E2 * sin(theta2)
Taking the ratio of the horizontal components, we get:
Ex2 / Ex1 = (E2 * cos(theta2)) / (E1 * cos(theta1))
Simplifying further:
Ex2 / Ex1 = (E2 / E1) * (cos(theta2) / cos(theta1))
Since we are given that the net electric field is directed upward along the vertical axis, we can assume that the vertical components are equal:
E1 * sin(theta1) = E2 * sin(theta2)
E1 / E2 = sin(theta2) / sin(theta1)
Substituting this expression back into our previous equation, we have:
Ex2 / Ex1 = (sin(theta2) / sin(theta1)) * (cos(theta2) / cos(theta1))
Now we need to plug in the actual values for theta1 and theta2:
Theta1 = 30 degrees
Theta2 = 60 degrees
Plugging these values into the equation above, we get:
Ex2 / Ex1 = (sin(60) / sin(30)) * (cos(60) / cos(30))
= (sqrt(3) / 0.5) * (0.5 / sqrt(3))
= 1
Therefore, the ratio q2/q1 is 1.
To determine the ratio q2/q1, we need to analyze the electric field contributions from each charge (q1 and q2) and consider their directions.
First, let's break down the problem into two components: the horizontal and vertical components of the electric field.
1. Horizontal Component:
Since both charges are located on a circle, the horizontal components of their electric fields will cancel each other out, as the horizontal components for both charges are equal in magnitude but opposite in direction. Therefore, there is no net horizontal electric field.
2. Vertical Component:
The vertical components of the electric fields produced by q1 and q2 will add up.
Let's assume q1 produces an electric field vector E1 and q2 produces an electric field vector E2. The angle for q1 is given as 30 degrees from the diameter, and the angle for q2 is given as 60 degrees from the diameter.
The vertical component of E1 can be calculated as E1 * sin(30°).
Similarly, the vertical component of E2 can be calculated as E2 * sin(60°).
Since the net electric field is directed upward along the vertical axis, we can equate the sum of the vertical components of the electric fields:
E1 * sin(30°) + E2 * sin(60°) = net electric field
Since we know that the charges are positive and produce a net electric field directed upward, the magnitudes of the electric fields E1 and E2 will be equal. Let's assume the magnitude is E.
Therefore, the equation becomes:
E * sin(30°) + E * sin(60°) = net electric field
To simplify further, we can use the trigonometric identities:
sin(30°) = 1/2 and sin(60°) = √3/2
E * (1/2) + E * (√3/2) = net electric field
Simplifying the expression:
E/2 + (√3 * E)/2 = net electric field
Combining like terms:
(E + √3 * E)/2 = net electric field
Simplifying:
(1 + √3)/2 * E = net electric field
Since we are given that the charges q1 and q2 produce the net electric field, we can write:
q1 * E = net electric field
q2 * E = net electric field
Dividing these two equations, we get:
q2/q1 = (net electric field produced by q2) / (net electric field produced by q1) = E/E = 1
Hence, the ratio q2/q1 is 1.