1. Magnetic field due to a dipole

For a magnetic dipole of moment $m$:

• On its axial line, $B_{\text{axial}}=\left(\frac{\mu_0}{4\pi}\right)\frac{2m}{r^3}$
• On its equatorial line, $B_{\text{equatorial}}=\left(\frac{\mu_0}{4\pi}\right)\frac{m}{r^3}$

Here, the midpoint $P$ is at distance $r=\frac d2$ from each dipole.

2. Field at $P$ due to dipole $X$

From the figure description, dipole $X$ has its axis horizontal, and dipole $Y$ has its axis vertical, with axes perpendicular.

At midpoint $P$, for dipole $X$, the point lies on its axial line. If dipole moment of $X$ is $M$, then $B_X=\left(\frac{\mu_0}{4\pi}\right)\frac{2M}{(d/2)^3}$ Direction: along the horizontal axis.

3. Field at $P$ due to dipole $Y$

Dipole $Y$ has moment twice that of $X$, so $m_Y=2M$ At $P$, the point lies on the equatorial line of dipole $Y$. Hence $B_Y=\left(\frac{\mu_0}{4\pi}\right)\frac{2M}{(d/2)^3}$ Direction: opposite to the dipole moment direction, i.e. along the horizontal axis opposite to $B_X$.

So the two fields are equal in magnitude and opposite in direction: $B_X=B_Y$ therefore, $B_{\text{net}}=0$

4. Force on the moving charge

Magnetic force on a charge is $\vec F=q\,\vec v\times \vec B$ Since the net magnetic field at $P$ is zero, $\vec F=q\,\vec v\times \vec 0=0$ Hence magnitude of force is $F=0$

5. Option check

• A: Non-zero ❌
• B: Non-zero ❌
• C: Non-zero ❌
• D: $0$ ✅

Therefore, the correct answer is D.