A point electric dipole of moment `vec(p)_(1)=(sqrt(3)hat(i) +hat(j))` is held fixed at the origin. Another point electric dipole of moment `vec(p)_(2)` held at the point A (+a, 0, 0) will have minimum potential energy, if its orientation is given by the vector

Options

A. `sqrt(3) hat(i) + hat(j)`
B. `hat(i) - sqrt(3) hat(j)`
C. `2sqrt(3) hat(i) - hat(j)`
D. `hat(i) - 2sqrt(3) hat(j)`

Answer

To solve the problem of finding the orientation of the second electric dipole moment \(\vec{p}_2\) that minimizes the potential energy with respect to the first dipole moment \(\vec{p}_1\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electric Dipole Moment**:
An electric dipole moment \(\vec{p}\) is defined as \(\vec{p} = q \cdot \vec{d}\), where \(q\) is the charge and \(\vec{d}\) is the separation vector between the charges. The potential energy \(U\) of a dipole in an electric field \(\vec{E}\) is given by:
\[ U = -\vec{p} \cdot \vec{E} \]

2. **Identifying the Electric Field**:
The electric field \(\vec{E}_1\) due to the first dipole \(\vec{p}_1\) at the position of the second dipole (located at point A \((+a, 0, 0)\)) can be derived from the dipole field equations. For a dipole located at the origin, the electric field at a point along the axis of the dipole is given by:
\[ \vec{E}_1 = \frac{1}{4\pi \epsilon_0} \left( \frac{2\vec{p}_1}{r^3} \right) \]
where \(r\) is the distance from the dipole to the point A.

3. **Calculating the Electric Field**:
Given \(\vec{p}_1 = \sqrt{3}\hat{i} + \hat{j}\), we can find the electric field at point A. The distance \(r\) from the origin to point A is \(a\). Thus, the electric field \(\vec{E}_1\) at point A becomes:
\[ \vec{E}_1 = \frac{1}{4\pi \epsilon_0} \left( \frac{2(\sqrt{3}\hat{i} + \hat{j})}{a^3} \right) \]

4. **Finding the Orientation for Minimum Potential Energy**:
To minimize the potential energy \(U\), we need to align \(\vec{p}_2\) in the direction of \(\vec{E}_1\). The orientation of \(\vec{p}_2\) that minimizes the potential energy is given by:
\[ \vec{p}_2 = p_2 \hat{E}_1 \]
where \(p_2\) is the magnitude of the dipole moment \(\vec{p}_2\) and \(\hat{E}_1\) is the unit vector in the direction of \(\vec{E}_1\).

5. **Conclusion**:
The orientation of \(\vec{p}_2\) that will minimize the potential energy when placed at point A is in the direction of the electric field \(\vec{E}_1\) generated by \(\vec{p}_1\). Therefore, the orientation of \(\vec{p}_2\) should be:
\[ \vec{p}_2 = p_2 \hat{E}_1 \]
where \(\hat{E}_1\) is the normalized version of \(\vec{E}_1\).