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The dipole moment is a key concept in understanding how electric dipoles interact with external electric fields. Essentially, it measures the separation of positive and negative charges within a system. For an electric dipole, which consists of a positive and a negative charge separated by a distance, the dipole moment (\textbf{p}) is calculated using the formula: \[ p = e \times d \]where \( e \) is the charge of an electron (\( 1.60 \times 10^{-19} \text{C} \)) and \( d \) is the distance between the charges. In our example, the dipole moment can be computed as:\[ p = (1.60 \times 10^{-19} \text{C}) \times (2.00 \times 10^{-9} \text{m}) = 3.20 \times 10^{-28} \text{Cm} \]This measure is crucial for further calculations involving potential energy in uniform electric fields.

Potential energy is the stored energy in a system due to its position in a force field, like an electric field. For an electric dipole in an electric field, the potential energy (\( U \)) depends on the alignment of the dipole with the field. When the dipole is aligned with the field, its potential energy (\( U_i \)) is given by:\[ U_i = - p E \]where \( p \) is the dipole moment and \( E \) is the electric field. In our exercise, the initial potential energy is:\[ U_i = - (3.20 \times 10^{-28} \text{Cm}) \times (3.00 \times 10^{6} \text{N/C}) = - 9.60 \times 10^{-22} \text{J} \]Conversely, when the dipole is aligned opposite the field, the potential energy (\( U_f \)) becomes:\[ U_f = + p E \]So:\[ U_f = (3.20 \times 10^{-28} \text{Cm}) \times (3.00 \times 10^{6} \text{N/C}) = 9.60 \times 10^{-22} \text{J} \]

A uniform electric field is one where the electric field strength (\( E \)) is constant in magnitude and direction throughout the region considered. This type of electric field simplifies calculations involving electric dipoles because the field strength remains the same. In our problem, we have a uniform electric field with a magnitude of:\[ E = 3.00 \times 10^6 \text{N/C} \]This means every point in this field experiences the same electric force per unit charge. A uniform field ensures that the calculations for potential energy and forces on the dipole are straightforward and consistent, making our analysis more manageable.

The change in potential energy (\( \triangle U \)) in a system gives insight into the work needed to move the system from one state to another. For an electric dipole flipping from alignment with a uniform electric field to an opposite alignment, the change in potential energy is given by:\[ \triangle U = U_f - U_i \]Given our calculated values for initial and final potential energies:\[ U_i = -9.60 \times 10^{-22} \text{J} \]\[ U_f = 9.60 \times 10^{-22} \text{J} \]We get:\[ \triangle U = 9.60 \times 10^{-22} \text{J} - (-9.60 \times 10^{-22} \text{J}) = 1.92 \times 10^{-21} \text{J} \]This result shows the energy needed to flip the dipole's orientation in the field, highlighting how potential energy changes in response to alignment in a uniform electric field.