91Ó°ÊÓ

An electric dipole consists of two equal but opposite charges separated by a specific distance. The dipole moment (\( p \)) is a vector quantity that characterizes the dipole. It provides both the magnitude and the direction of the dipole's ability to produce electric fields. The dipole moment is calculated as the product of the charge (\( q \)) and the separation distance (\( d \)) between charges, given by \( p = q \cdot d \).
For example, let's consider a dipole composed of charges \( +2e \) and \( -2e \), where \( e \) is the elementary charge (\( 1.6 \times 10^{-19} \text{ C} \)) and the distance separating them is \( 0.78 \times 10^{-9} \text{ m} \).

• Charge, \( q = 2 \times (1.6 \times 10^{-19}) \text{ C} \)
• Distance, \( d = 0.78 \times 10^{-9} \text{ m} \)

The dipole moment can be calculated as \( p = 2 \times (1.6 \times 10^{-19}) \times 0.78 \times 10^{-9} = 2.496 \times 10^{-29} \text{ C} \cdot \text{m} \).
The dipole moment points from the negative to the positive charge and its magnitude can help us understand the interaction of the dipole with external electric fields.

Torque measures the tendency of a force to rotate an object about an axis. In the context of a dipole placed in an external electric field, the torque (\( \tau \)) depends on both the dipole moment and its orientation with respect to the field.
The torque is given by the formula:\[\tau = pE\sin\theta\]where \( p \) is the dipole moment, \( E \) is the strength of the electric field, and \( \theta \) is the angle between the dipole moment and the electric field.

• **Parallel orientation (\( \theta = 0^{\circ} \))**: When the dipole is parallel to the electric field, \( \sin 0^{\circ} = 0 \), resulting in zero torque (\( \tau = 0 \)).
• **Perpendicular orientation (\( \theta = 90^{\circ} \))**: When the dipole is perpendicular to the electric field, \( \sin 90^{\circ} = 1 \), the torque is maximized and given by \( \tau = pE \). For our example, this results in \( \tau = 2.496 \times 10^{-29} \times 3.4 \times 10^{6} = 8.4864 \times 10^{-23} \text{ Nm} \).
• **Antiparallel orientation (\( \theta = 180^{\circ} \))**: When the dipole is antiparallel, \( \sin 180^{\circ} = 0 \), so the torque is again zero (\( \tau = 0 \)).

Thus, the orientation of the dipole relative to the electric field is crucial in determining the torque experienced by the dipole.

The electric field (\( E \)) is a vector field that surrounds electric charges and exerts force on other charges within the field. It is usually described as the force per unit charge, implying that any charged particle placed in the field would experience a force proportional to its charge.
The strength of the electric field is measured in units of newtons per coulomb (\( \text{N/C} \)) and is often a crucial factor in determining how charged objects like dipoles behave when exposed to it. The uniform electric field can affect the orientation and stability of an electric dipole, and this is characterized by the torque experienced by the dipole.
In our scenario, the given electric field strength is \( 3.4 \times 10^{6} \text{ N/C} \). This strong field can significantly influence the dipole's behavior:

• When aligning with the electric field (\( \theta = 0^{\circ} \) or \( \theta = 180^{\circ} \)), the dipole experiences no torque.
• When perpendicular (\( \theta = 90^{\circ} \)), the electric field induces a maximum torque, rotating the dipole to align with the field.

Understanding electric fields is important, as they can determine the dynamics and equilibrium states of electric dipoles in space.