For a short dipole placed at origin O, the dipole moment P is along the X-axis, as shown in the figure. If the electric potential and electric field at A are V and E respectively, then the correct combination of the electric potential and electric field, respectively, at point B on the Y-axis is given by:
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For a short dipole:
- The potential along the perpendicular bisector is always zero.
- The electric field along the perpendicular bisector follows \( E \propto \frac{1}{r^3} \).
Step 1: Compute the electric potential at point \( B \).
The electric potential due to a dipole at any point is given by:
\[
V = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{r}}{r^2}.
\]
Since point \( B \) is on the perpendicular bisector of the dipole, \( \mathbf{p} \cdot \hat{r} = 0 \), implying:
\[
V_B = 0.
\]
Step 2: Compute the electric field at point \( B \).
The magnitude of the electric field along the perpendicular bisector of a dipole is:
\[
E = \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2 + d^2)^{3/2}}.
\]
For small dipole approximation \( d \ll r \), we use:
\[
E_B = \frac{E_0}{16}.
\]
Thus, the answer is \( \boxed{0, \frac{E_0}{16}} \).
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