Three charges are arranged in an equilateral triangle with sides of 4 cm. The coordinates and corresponding charge values are as follows: +4q at (-2,0); +2q at (2,0); -q at (0,3.464). The elementary charge of an electron is given by q = -1.602 x 10^-19 C.
Determine the torque that acts on the dipole and the work done to align the dipole at the center of the equilateral triangle. The dipole exhibits a dipole moment of ||p|| = 8 x 10^-12 Cm and an orientation of (-j, +i)
To determine the torque acting on the dipole, you first need to calculate the electric field at the position of the dipole due to each individual charge. Then, you can find the net electric field and the torque it exerts on the dipole.
Step 1: Calculate the electric field due to each charge:
The electric field due to a point charge at a distance r from the charge is given by:
$$E = \frac{k \cdot q}{r^2}$$
where E is the electric field, k is the Coulomb's constant (9 x 10^9 Nm^2/C^2), and q is the charge.
For the +4q charge at (-2,0):
Distance from the charge to the dipole position (0,0) is the magnitude of the position vector:
r = sqrt[(-2 - 0)^2 + (0 - 0)^2] = 2
Electric field due to this charge at the dipole position:
E1 = k * (4q) / r^2
For the +2q charge at (2,0):
Distance from the charge to the dipole position (0,0) is the magnitude of the position vector:
r = sqrt[(2 - 0)^2 + (0 - 0)^2] = 2
Electric field due to this charge at the dipole position:
E2 = k * (2q) / r^2
For the -q charge at (0,3.464):
Distance from the charge to the dipole position (0,0) is the magnitude of the position vector:
r = sqrt[(0 - 0)^2 + (3.464 - 0)^2] = 3.464
Electric field due to this charge at the dipole position:
E3 = k * (-q) / r^2
Step 2: Calculate the net electric field:
The net electric field at the position of the dipole is the vector sum of the electric fields due to each charge. Since the charges are arranged in an equilateral triangle, the magnitude of each electric field has the same value, but they have different directions.
To add the electric fields as vectors, you can calculate the x-component and y-component for each field, and then sum them up.
For the +4q charge at (-2,0):
x-component of E1 = -E1 * cos(60) (since the electric field points towards the negative x-direction)
y-component of E1 = E1 * sin(60)
For the +2q charge at (2,0):
x-component of E2 = E2 * cos(60) (since the electric field points towards the positive x-direction)
y-component of E2 = E2 * sin(60)
For the -q charge at (0,3.464):
x-component of E3 = 0 (since the electric field points along the y-axis)
y-component of E3 = -E3
Next, sum up the x-components and y-components:
Total x-component of the net electric field: E_net_x = E1 * (-cos(60)) + E2 * cos(60) + 0
Total y-component of the net electric field: E_net_y = E1 * sin(60) + E2 * sin(60) + (-E3)
Step 3: Calculate the torque acting on the dipole:
The torque on a dipole in an electric field is given by:
$$\tau = p \cdot E$$
where τ is the torque, p is the dipole moment, and E is the electric field.
The given dipole moment is p = 8 x 10^-12 Cm.
The torque vector τ is perpendicular to both p and E. Given that the orientation of the dipole is (-j, +i), we can represent the dipole moment as p = (0, -8 x 10^-12) Cm and the electric field as E = (E_net_x, E_net_y).
The torque acting on the dipole is then:
τ = p x E
τ = (0, -8 x 10^-12) Cm x (E_net_x, E_net_y)
Step 4: Calculate the work done to align the dipole at the center of the equilateral triangle:
The work done to align a dipole with a torque τ by an angle θ is given by:
$$W = -\tau \cdot \theta$$
where W is the work done and θ is the angle between the torque vector and the direction of alignment.
The given orientation of the dipole is (-j, +i), which has an angle of 90 degrees with the direction of alignment (perpendicular). Therefore, θ = 90 degrees.
Substituting the values, the work done to align the dipole is:
W = -τ · θ
Now, with the calculated torque vector τ and angle θ, you can substitute the values to find the torque acting on the dipole and the work done to align it at the center of the equilateral triangle.