Three point charges lie along a circle of radius r at angles of 30°, 150°, and 270° as shown in the figure below. Find a symbolic expression for the resultant electric field at the center of the circle.
If the charges are equal, the resultant field is zero, because the points are arranged as the corners of an equilateral triangle, and you are asking for the field at the center. Because of symmetry, the fields cancel.
E= −keq/r^2+keq/r^2 * cos60 +keq/r^2 * cos60
To determine the resultant electric field at the center of the circle, we need to find the individual electric fields created by each point charge and then sum them up.
Let's denote the magnitude of each point charge as Q.
The formula for the electric field due to a point charge is given by:
E = k * abs(Q) / r^2
where E is the electric field magnitude, k is the constant of electrostatics (approximately 9 x 10^9 N m^2/C^2), abs(Q) is the absolute value of the charge, and r is the distance from the charge to the point where we want to calculate the electric field.
Since the charges are located on a circle with radius r, the distance between each charge and the center of the circle is also r.
Now let's calculate the electric field created by each charge and find the symbolic expression for the resultant electric field at the center of the circle:
1. Charge at 30°:
The electric field created by this charge can be calculated as E1 = k * abs(Q) / r^2.
2. Charge at 150°:
The electric field created by this charge can be calculated as E2 = k * abs(Q) / r^2.
3. Charge at 270°:
The electric field created by this charge can be calculated as E3 = k * abs(Q) / r^2.
The resultant electric field at the center of the circle is the vector sum of the electric fields created by each charge.
Let's denote the resultant electric field at the center of the circle as ER. We can calculate it using the following equation:
ER = sqrt[(E1*cos(30°) + E2*cos(150°) + E3*cos(270°))^2 + (E1*sin(30°) + E2*sin(150°) + E3*sin(270°))^2]
Simplifying further, we get:
ER = sqrt[G^2 + H^2]
where G = E1*cos(30°) + E2*cos(150°) + E3*cos(270°) and H = E1*sin(30°) + E2*sin(150°) + E3*sin(270°).
Thus, the symbolic expression for the resultant electric field at the center of the circle is ER = sqrt[G^2 + H^2].
To find the resultant electric field at the center of the circle, we can use the principle of superposition. The electric field at the center of the circle due to each point charge can be calculated individually, and then we can find the vector sum of these individual electric fields.
The electric field at the center of the circle due to a point charge can be given by Coulomb's law:
E = k * (q / r^2)
Where:
- E is the electric field
- k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2)
- q is the charge
- r is the distance from the charge to the center of the circle
Let's denote the charges as q1, q2, and q3. Given that the charges lie at angles of 30°, 150°, and 270°, respectively, we can find the coordinates of each charge using basic trigonometry and the radius of the circle, r.
The coordinates of q1 are:
x1 = r * cos(30°)
y1 = r * sin(30°)
The coordinates of q2 are:
x2 = r * cos(150°)
y2 = r * sin(150°)
The coordinates of q3 are:
x3 = r * cos(270°)
y3 = r * sin(270°)
Now, we can find the distance from each charge to the center of the circle, which is simply the radius of the circle, r.
Using Coulomb's law, the electric field due to each charge can be calculated as follows:
E1 = k * (q1 / r^2)
E2 = k * (q2 / r^2)
E3 = k * (q3 / r^2)
Finally, to find the resultant electric field at the center of the circle, we need to find the vector sum of these individual electric fields:
E_resultant = E1 + E2 + E3
Note that the direction of the resultant electric field will depend on the angles at which the charges are located.