Calculation of Electric Dipole Moment

To calculate torque acting on the dipole, we need to first calculate the electric dipole moment of the system.

The electric dipole moment is given by the product of the magnitude of the charge and the distance between the charges, multiplied by the unit vector in the direction from negative charge to positive charge.

μ = q × d

where, q is the magnitude of the charge and d is the distance between the charges.

Here, q = 4μC and d = AB = sqrt((2-1)^2+(-1-0)^2+(5-4)^2) = 1.73m

Thus, μ = 4μC × 1.73m = 6.92 × 10^-6 Cm

Calculation of Electric Field

Next, we need to calculate the electric field at the point where the charges are located.

The electric field is given by the formula:

E = k × q / r^2

where, k is Coulomb's constant, q is the magnitude of the charge, and r is the distance from the point charge to the point where the electric field is being measured.

Here, k = 9 × 10^9 Nm^2/C^2, q = -4μC for charge A and q = 4μC for charge B.

For point A, r = distance between A and point P (where the electric field is being measured) = sqrt((1-x)^2+(0-y)^2+(4-z)^2)

For point B, r = distance between B and point P = sqrt((2-x)^2+(-1-y)^2+(5-z)^2)

Thus, the electric field at point A is:

E = (9 × 10^9 Nm^2/C^2) × (-4μC) / [(x-1)^2+y^2+(z-4)^2]^1.5

And the electric field at point B is:

E = (9 × 10^9 Nm^2/C^2) × (4μC) / [(x-2)^2+(y+1)^2+(z-5)^2]^1.5

Calculation of Torque

The torque acting on the dipole is given by:

τ = μ × E × sinθ

where, θ is the angle between the electric dipole moment and the electric field.

Here, the electric dipole moment is in the direction from A to B, and the electric field is given by the vector sum of the electric fields at points A and B.

Thus, the electric field at point P is:

E = E1 + E2

where, E1 is the electric field due to charge A and E2 is the electric field due to charge B.

The angle between μ and E is given by:

sinθ = μ × E / (μ × E)

Thus, the torque acting on the dipole is:

τ = μ × E × sinθ

Plugging in the values, we get:

τ = (6.92 × 10^-6 Cm) × (0.20 î V/Cm) × sinθ

The angle θ can be calculated using vector addition of the electric fields at points A and B, along with the direction of the electric dipole moment.

The final answer is option A.