To understand the electrostatic force acting on a tiny electric dipole, such as a water molecule (H2O), placed near an infinitely long charged wire, we need to consider a few key concepts in electrostatics. The dipole moment \( P \) of the molecule is a measure of its polarity, and when it is oriented perpendicular to the wire, it interacts with the electric field created by the wire. Let's break this down step by step.
Understanding the Electric Field of the Wire
An infinitely long wire with a linear charge density \( \lambda \) generates an electric field around it. The electric field \( E \) at a distance \( r \) from the wire can be calculated using Gauss's law. The formula for the electric field due to an infinitely long straight wire is given by:
E = \frac{\lambda}{2\pi \epsilon_0 r}
Here, \( \epsilon_0 \) is the permittivity of free space. This electric field points radially outward from the wire if the charge is positive and inward if the charge is negative.
Force on the Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a distance. The force \( F \) acting on a dipole in an electric field can be expressed as:
F = \nabla (\mathbf{p} \cdot \mathbf{E})
Where \( \mathbf{p} \) is the dipole moment vector and \( \mathbf{E} \) is the electric field vector. Since the dipole moment \( P \) is perpendicular to the wire, we can analyze the force acting on it due to the electric field generated by the wire.
Calculating the Force
For a dipole placed in a uniform electric field, the force can be simplified. However, in this case, the electric field is not uniform, as it varies with distance from the wire. The force on the dipole can be derived from the potential energy \( U \) of the dipole in the electric field:
U = -\mathbf{p} \cdot \mathbf{E}
The force can be found by taking the negative gradient of the potential energy:
F = -\nabla U
Since the dipole is oriented perpendicular to the wire, we can focus on the component of the electric field that affects the dipole. The force acting on the dipole can be expressed as:
F = \frac{dU}{dr} = \frac{d}{dr}(-\mathbf{p} \cdot \mathbf{E})
Final Expression for the Force
Substituting the expression for the electric field \( E \) into the equation, we can find the force acting on the dipole:
F = -\mathbf{p} \cdot \frac{dE}{dr}
Calculating \( \frac{dE}{dr} \) from the electric field equation gives:
\frac{dE}{dr} = -\frac{\lambda}{2\pi \epsilon_0 r^2}
Thus, the force acting on the dipole becomes:
F = \frac{p \lambda}{2\pi \epsilon_0 r^2}
This expression indicates that the force acting on the dipole is directly proportional to the dipole moment \( P \) and the linear charge density \( \lambda \), and inversely proportional to the square of the distance \( r \) from the wire.
Summary
In summary, the electrostatic force acting on a tiny electric dipole placed at a distance \( r \) from an infinitely long wire with linear charge density \( \lambda \), when the dipole moment is perpendicular to the wire, can be expressed as:
F = \frac{p \lambda}{2\pi \epsilon_0 r^2}
This relationship highlights the interaction between electric dipoles and electric fields, showcasing the fundamental principles of electrostatics in action.
