To understand how the torque on an electric dipole changes when the magnitudes of the charges and their separation are altered, we need to start with the basic concepts of electric dipoles and torque in an electric field.
Electric Dipole Basics
An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance d. The dipole moment (p) is defined as:
p = q × d
This dipole moment is a vector quantity that points from the negative charge to the positive charge.
Torque on an Electric Dipole
The torque (τ) experienced by an electric dipole in a uniform electric field (E) is given by the formula:
τ = p × E
Here, τ is the torque, p is the dipole moment, and E is the electric field strength. The magnitude of the torque can also be expressed as:
τ = pE sin(θ)
where θ is the angle between the dipole moment and the electric field direction.
Changes to the Dipole
Now, let's analyze the changes made to the dipole:
- The magnitude of each charge is increased by a factor of 5, so the new charge becomes 5q.
- The separation distance between the charges is tripled, so the new distance is 3d.
Calculating the New Dipole Moment
With these changes, the new dipole moment (p') can be calculated as follows:
p' = (5q) × (3d) = 15qd
This shows that the new dipole moment is 15 times the original dipole moment (p = q × d).
Torque in the Electric Field
Assuming the electric field strength (E) remains constant, we can now find the new torque (τ') on the dipole:
τ' = p' × E = (15qd) × E = 15(qdE)
Since the original torque τ was given by τ = pE = (qd) × E, we can see that:
τ' = 15τ
Final Thoughts
In summary, when the magnitudes of the charges are increased by a factor of 5 and the separation is tripled, the torque on the dipole in a uniform electric field increases by a factor of 15. This illustrates how both the charge magnitude and the distance between charges significantly impact the behavior of electric dipoles in electric fields.
