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.

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.

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.

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.

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.