Question

A water molecule, which is electrically neutral but has a dipole moment of magnitude \(p=6.20 \cdot 10^{-30} \mathrm{C} \mathrm{m},\) is \(1.00 \mathrm{~cm}\) away from a point charge \(q=+1.00 \mu \mathrm{C} .\) The dipole will align with the electric field due to the charge. It will also experience a net force, since the field is not uniform. a) Calculate the magnitude of the net force. (Hint: You do not need to know the precise size of the molecule, only that it is much smaller than \(1 \mathrm{~cm} .)\) b) Is the molecule attracted to or repelled by the point charge? Explain.

Short answer

Short answer: The magnitude of the net force on the water molecule is 0, and it is neither attracted nor repelled by the point charge.

Step-by-step solution

Determine the electric field at the location of the water molecule

We can find the electric field at the location of the water molecule due to the point charge using the equation: $$ E = \frac{kq}{r^2} $$ Where \(E\) is the electric field, \(k\) is the electric constant (\(8.99 \times 10^9 \, \mathrm{ N\, m^2 / C^2}\)), \(q\) is the point charge (\(1.00 \times 10^{-6} \, \mathrm{C}\)), and \(r\) is the distance from the point charge (\(0.01 \, \mathrm{m}\)). Plug in the values and solve for the electric field.

Find the torque and force on the dipole moment

The torque on the dipole moment in the electric field is given by the formula: $$ \tau = pE \sin{\theta} $$ Where \(\tau\) is the torque, \(p\) is the dipole moment (\(6.20 \times 10^{-30} \, \mathrm{C\, m}\)), \(E\) is the electric field found in Step 1, and \(\theta\) is the angle between the dipole moment and the electric field. Since the dipole moment aligns with the electric field, \(\theta = 0\) and \(\sin{\theta} = 0\). Therefore, the torque on the dipole moment is 0. The force on the dipole moment in the electric field can be found using the formula: $$ F = (p \nabla) E $$ Where \(F\) is the force and \(\nabla\) is the gradient operator. In this case, since the molecule is much smaller than the distance from the point charge, we can consider the electric field to be approximately uniform. Therefore, the gradient of the electric field is approximately 0, and the force on the dipole moment is also 0.

Calculate the net force on the molecule

We've found that the torque and force on the dipole moment in the electric field are both 0. Therefore, the net force on the water molecule is also 0.

Determine attraction or repulsion

Since the net force on the water molecule is 0, it is neither attracted nor repelled by the point charge. The molecule does align with the electric field, but it does not experience a net force due to the approximate uniformity of the electric field at its location. So, the answers are: a) The magnitude of the net force is 0. b) The molecule is neither attracted nor repelled by the point charge.

Key concepts

Electric Field

The electric field is a region around a charged object where it exerts a force on other charges. It is represented by the equation\[ E = \frac{kq}{r^2} \].
Here, \(E\) stands for the electric field, \(k\) is the electrostatic constant (approximately \(8.99 \times 10^9 \, \mathrm{ N\, m^2 / C^2}\)), \(q\) is the point charge, and \(r\) is the distance from the point charge. This formula shows how the electric field strength varies inversely with the square of the distance from the charge.
An important aspect of the electric field is that it has both magnitude and direction. This means it is a vector quantity, and it points away from positive charges and towards negative charges. Understanding how the field varies with distance is crucial when calculating the effects on nearby charges, like dipoles.

Dipole Moment

A dipole moment is a measure of the separation of positive and negative electrical charges within a molecule. It is represented by the symbol \(p\) and calculated by the product of the charge magnitude and the distance separating them. For a water molecule, this is approximately \(6.20 \times 10^{-30} \, \mathrm{C\, m}\).
Dipoles arise because of differences in electronegativity, resulting in partial charges. The water molecule, being a polar molecule, has a natural dipole moment.
Dipole moment is a vector quantity—it has both magnitude and direction. This means that it not only tells us how strong the charge separation is but also points from the negative to the positive charge. In external electric fields, a dipole tends to align itself along the field line, which comes into play when calculating forces and torque acting on it.

Torque on Dipole

When a dipole is placed in an electric field, it experiences a torque that tends to rotate the dipole so its moment aligns with the field. The torque \(\tau\) acting on a dipole is given by the formula:\[ \tau = pE \sin{\theta} \].
Where \(\tau\) is torque, \(p\) is the dipole moment, \(E\) is the electric field, and \(\theta\) is the angle between \(p\) and \(E\). When a dipole is perfectly aligned with the electric field, the angle \(\theta = 0\) and thus \(\sin{\theta} = 0\), resulting in zero torque.
In practical terms, this alignment minimizes potential energy and is the most stable orientation for the dipole. When an initially misaligned dipole enters an electric field, the torque will cause it to rotate into alignment with the field, a situation often encountered in molecular physics.

Force on Dipole

Force on a dipole in an electric field can be a bit tricky to understand compared to a point charge. It relies on the change in electric field strength across the dipole. The force \(F\) can be expressed by the formula:\[ F = (p abla) E \],
where \(abla\) denotes the gradient of the electric field.However, at small separations where the electric field is approximately uniform, this gradient is essentially zero, resulting in zero net force on the dipole.
This scenario is noted in situations where the separation distance is much larger than the dipole length. As a result, while the dipole moment affects how the molecule aligns with the field, it does not affect its movement towards or away from the source of the field.