A dipole of dipole moment `vecp=phati` lies along the x -axis in a non-uniform electric field `vecE=(c)/(x)hati` . The force acting on the dipole is

Options

A. zero
B. `-(pc)/(x^(2))hati`
C. `(pc)/(x^(2))hati`
D. `-(pc)/(2x^(2))hati`

Answer

To solve the problem of finding the force acting on a dipole in a non-uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Dipole moment: \(\vec{p} = p \hat{i}\)
- Electric field: \(\vec{E} = \frac{c}{x} \hat{i}\)

2. **Understand the Potential Energy of the Dipole:**
- The potential energy \(U\) of a dipole in an electric field is given by:
\[ U = -\vec{p} \cdot \vec{E} \]
- Since both \(\vec{p}\) and \(\vec{E}\) are along the x-axis, the angle between them is 0 degrees. Therefore, we can simplify the equation:
\[ U = -pE \cos(0) = -pE \]

3. **Substitute the Electric Field into the Potential Energy Equation:**
- Substitute \(\vec{E} = \frac{c}{x} \hat{i}\) into the potential energy formula:
\[ U = -p \left(\frac{c}{x}\right) = -\frac{pc}{x} \]

4. **Calculate the Force Acting on the Dipole:**
- The force \(F\) acting on the dipole can be found using the relation:
\[ F = -\frac{dU}{dx} \]
- Differentiate \(U\) with respect to \(x\):
\[ F = -\frac{d}{dx}\left(-\frac{pc}{x}\right) = \frac{pc}{x^2} \]

5. **Final Expression for the Force:**
- Therefore, the force acting on the dipole is:
\[ F = \frac{pc}{x^2} \]