A charge distribution consists of q1=2q at (0,0,0), q2=-q at (0,0,a), and q3=q at (0,0,a). Compute the monopole moment, dipole moment, and approximate potential at large distance r.

The monopole moment $Q$ is the total charge of the distribution.

$$Q = \sum_{i} q_i = q_1 + q_2 + q_3$$ $$Q = 2q + (-q) + q$$ $$Q = 2q$$

The monopole moment is $\boxed{\mathbf{2q}}$.

The dipole moment $\vec{p}$ is given by $\vec{p} = \sum_{i} q_i \vec{r}_i$.

$$\vec{p} = q_1 \vec{r}_1 + q_2 \vec{r}_2 + q_3 \vec{r}_3$$ $$\vec{p} = (2q)(0\hat{k}) + (-q)(a\hat{k}) + (q)(a\hat{k})$$ $$\vec{p} = 0 - qa\hat{k} + qa\hat{k}$$ $$\vec{p} = 0$$

The dipole moment is $\boxed{\mathbf{0}}$.

The multipole expansion for the potential $V(\vec{r})$ at a large distance $r$ is given by:

$$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \left( \frac{Q}{r} + \frac{\vec{p} \cdot \hat{r}}{r^2} + \dots \right)$$

Since $Q = 2q$ and $\vec{p} = 0$, the dominant term in the expansion is the monopole term.

$$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{2q}{r}$$

The approximate potential at large distance $r$ is $\boxed{\mathbf{V(\vec{r}) = \frac{2q}{4\pi\epsilon_0 r}}}$.

The electric field $\vec{E}$ is related to the potential by $\vec{E} = -\nabla V$.
In spherical coordinates, for a potential that only depends on $r$:

$$\vec{E} = -\frac{\partial V}{\partial r} \hat{r}$$

Substitute the potential $V(\vec{r}) = \frac{2q}{4\pi\epsilon_0 r}$:

$$\vec{E} = -\frac{\partial}{\partial r} \left( \frac{2q}{4\pi\epsilon_0 r} \right) \hat{r}$$ $$\vec{E} = -\frac{2q}{4\pi\epsilon_0} \frac{\partial}{\partial r} (r^{-1}) \hat{r}$$ $$\vec{E} = -\frac{2q}{4\pi\epsilon_0} (-1 r^{-2}) \hat{r}$$ $$\vec{E} = \frac{2q}{4\pi\epsilon_0 r^2} \hat{r}$$

The electrical field is $\boxed{\mathbf{\vec{E} = \frac{2q}{4\pi\epsilon_0 r^2} \hat{r}}}$.