1. Given data

• Charge at origin: $q = 10^{-3}\,\mu C = 10^{-3}\times 10^{-6} C = 10^{-9} C$
• Point $A = (\sqrt{2},\sqrt{2})$
• Point $B = (2,0)$

The electric potential due to a point charge is $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r} = k\frac{q}{r}$ where $k = 9\times 10^9\,\text{SI units}$.

2. Find distance of point $A$ from origin

$r_A = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2}$ $r_A = \sqrt{2+2} = \sqrt{4} = 2$

3. Find distance of point $B$ from origin

$r_B = \sqrt{(2)^2 + 0^2} = 2$

4. Find potentials at $A$ and $B$

Since both points are at the same distance from the charge, $V_A = k\frac{q}{r_A} = 9\times 10^9 \cdot \frac{10^{-9}}{2} = \frac{9}{2} = 4.5\,\text{V}$

Similarly, $V_B = k\frac{q}{r_B} = 9\times 10^9 \cdot \frac{10^{-9}}{2} = 4.5\,\text{V}$

5. Potential difference between $A$ and $B$

$V_A - V_B = 4.5 - 4.5 = 0$

So, the potential difference is $\boxed{0\,\text{volt}}$

6. Option check

• A: $4.5$ volts ❌
• B: $9$ volts ❌
• C: zero ✅
• D: $2$ volts ❌

Therefore, the correct option is C.