To find the electric dipole moment vector of the charge assembly consisting of point charges \( +q \), \( -2q \), and \( +q \) located at the specified coordinates, we can follow these steps:
### Step 1: Identify the positions of the charges
- The charges are located at:
- Charge \( +q \) at \( (0, a, 0) \)
- Charge \( -2q \) at \( (0, 0, 0) \)
- Charge \( +q \) at \( (a, 0, 0) \)
### Step 2: Calculate the dipole moment for each charge
The electric dipole moment \( \vec{p} \) for a point charge is given by:
\[
\vec{p} = q \cdot \vec{r}
\]
where \( \vec{r} \) is the position vector of the charge from the origin.
1. For charge \( +q \) at \( (0, a, 0) \):
\[
\vec{p_1} = +q \cdot (0, a, 0) = (0, qa, 0)
\]
2. For charge \( -2q \) at \( (0, 0, 0) \):
\[
\vec{p_2} = -2q \cdot (0, 0, 0) = (0, 0, 0)
\]
3. For charge \( +q \) at \( (a, 0, 0) \):
\[
\vec{p_3} = +q \cdot (a, 0, 0) = (qa, 0, 0)
\]
### Step 3: Sum the dipole moments
The total dipole moment \( \vec{p}_{\text{total}} \) is the vector sum of the individual dipole moments:
\[
\vec{p}_{\text{total}} = \vec{p_1} + \vec{p_2} + \vec{p_3}
\]
Substituting the values:
\[
\vec{p}_{\text{total}} = (0, qa, 0) + (0, 0, 0) + (qa, 0, 0) = (qa, qa, 0)
\]
### Step 4: Calculate the magnitude of the dipole moment
The magnitude of the dipole moment vector is given by:
\[
|\vec{p}_{\text{total}}| = \sqrt{(qa)^2 + (qa)^2} = \sqrt{2(qa)^2} = qa\sqrt{2}
\]
### Step 5: Determine the direction of the dipole moment
The direction of the dipole moment vector \( \vec{p}_{\text{total}} = (qa, qa, 0) \) can be expressed in terms of its components. It points in the direction of the vector from the origin towards the point \( (qa, qa, 0) \).
### Final Answer
The magnitude and direction of the electric dipole moment vector of this charge assembly is:
\[
|\vec{p}| = qa\sqrt{2} \quad \text{along the line joining } (0, 0, 0) \text{ to } (a, a, 0).
\]
