GATE 2026 PH – Question 48
A positive point charge is fixed at the origin. At some distance from it on the x axis, a point dipole is kept pointing in the +y direction. The force on the dipole is
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Correct answer: (B) in the +y direction
Explanation
The force on a small dipole $\mathbf p$ in an electric field $\mathbf E$ is
$$\mathbf F=(\mathbf p\cdot\nabla)\mathbf E .$$
For a point charge $Q$ at the origin, $\mathbf E=\dfrac{kQ\,\mathbf r}{r^3}$. The dipole is on the $x$ axis at $(x,0,0)$, pointing along $+y$, so $\mathbf p=p\hat y$ and
$$\mathbf F=p\,\frac{\partial\mathbf E}{\partial y}.$$
**Components:** the $y$ component of the field is $E_y=\dfrac{kQ\,y}{(x^2+y^2)^{3/2}}$, so at $y=0$
$$\frac{\partial E_y}{\partial y}=\frac{kQ}{x^3}>0 .$$
(The $x$ component $E_x=kQx/(x^2+y^2)^{3/2}$ has $\partial E_x/\partial y=0$ at $y=0$.)
$$F_y=p\,\frac{kQ}{x^3}>0 .$$
The force is in the **$+y$ direction** (option B). Physically, the field of the charge fans out: the transverse component $E_y$ grows with $y$, so the positive end of the $+y$ dipole (at larger $y$) feels a larger force than the negative end, and the net force is along $+y$.