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An electric dipole consists of two equal and opposite charges separated by a distance. The electric dipole moment, denoted by \(\mathbf{p}\), is a vector quantity. It points from the negative charge to the positive charge, and its magnitude is given by:

• \( p = q \cdot d \)

where \( q \) is the magnitude of either charge, and \( d \) is the separation distance between the charges. In our example, the charges are \(+2 e\) and \(-2 e\), and they are separated by \(0.78 \text{ nm}\). We convert this to meters for consistency in SI units: \(0.78 \text{ nm} = 0.78 \cdot 10^{-9} \text{ m} \). With \(q\) being \(2 \cdot 1.6 \cdot 10^{-19} \text{ C} \), we get the dipole moment to be \( p = 2.496 \cdot 10^{-28} \text{ Cm} \). This dipole moment is fundamental for understanding the interaction between the dipole and an external electric field.

Torque (\textbf{\tau\}) on a dipole in an electric field describes the rotational effect of the field on the dipole. It depends on the electric field strength (\textbf{E}), the dipole moment (\textbf{p}), and the angle (\textbf{\theta\}) between them. The formula to calculate this is:

• \( \tau = pE \cdot \sin(\theta) \)

This equation shows that \(\tau\) achieves its maximum value when the dipole is perpendicular to the electric field (\textbf{\theta = 90^\text{o}\}) and drops to zero when the dipole is parallel (\textbf{\theta = 0^\text{o}\}) or antiparallel (\textbf{\theta = 180^\text{o}\}) to the field. Calculating torque helps us understand how a dipole placed in an electric field will rotate, aligning itself with the field direction.

The electric field strength (\textbf{E}) is defined as the force per unit charge exerted by a field on a test charge. It is measured in \( \text{N/C} \) (Newtons per Coulomb). In this exercise, the electric field strength is given as \(3.4 \cdot 10^6 \text{ N/C}\). This strong field will exert significant forces on any charged particles or dipoles within it. Understanding the electric field strength is crucial because it directly influences the magnitude of the torque on the dipole through the equation \( \tau = pE \cdot \sin(\theta) \). A higher electric field strength will yield a larger torque, affecting the rotational dynamics of the dipole.

The angle (\textbf{\theta\}) between the dipole moment (\textbf{p}) and the electric field (\textbf{E}) is crucial in determining the magnitude of the torque. This relationship is given by the sine function in the torque formula: \(\tau = pE \cdot \sin(\theta) \). Here's how different angles affect the torque:

• Parallel (\theta = 0^\text{o}): \sin(0^\text{o}) = 0\, thus, \(\tau = 0\).
• Perpendicular (\theta = 90^\text{o}): \sin(90^\text{o}) = 1\, thus, \(\tau = pE\).
• Antiparallel (\theta = 180^\text{o}): \sin(180^\text{o}) = 0\, thus, \(\tau = 0\).

The torque is greatest when the dipole is perpendicular to the field because \(\sin(90^\text{o}) = 1\). It is zero when the dipole is either parallel or antiparallel to the field because \(\sin(0^\text{o}) \text{and} \sin(180^\text{o}) = 0\). This demonstrates how torque varies with the dipole's orientation relative to the electric field.