Electric Field

Question

Two point charges $+q$ and $-q$ are placed at positions $(d, 0)$ and $(-d, 0)$ respectively. At which location on the y-axis is the net electric field directed purely along the negative x-direction?

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The electric field due to each charge at a point $(0, y)$ on the y-axis must be resolved into components. The $+q$ charge at $(d,0)$ produces a field of magnitude $E_1 = kq/(y^2+d^2)$ directed from $(d,0)$ towards $(0,y)$. Its x-component is $-E_1 \sin\theta$ (negative x) and y-component is $+E_1 \cos\theta$. The $-q$ charge at $(-d,0)$ produces a field of magnitude $E_2 = kq/(y^2+d^2)$ directed from $(0,y)$ towards $(-d,0)$. Its x-component is also $-E_2 \sin\theta$ (negative x) and y-component is $-E_2 \cos\theta$. Since $E_1 = E_2$, the y-components cancel and the x-components add. Therefore, the net electric field at every point on the y-axis is directed purely in the negative x-direction. Option 'Only at the origin' is incorrect because the field at the origin is also along $-x$ but by no means exclusively. Option 'Only at \infty' is incorrect because the field vanishes at \infty. Option 'At no point' is incorrect because the symmetry argument clearly shows x-direction field everywhere on the y-axis. This JEE Advanced style problem tests vector component analysis. Plausibility check: the dipole field at midpoint of the axis is anti-parallel to the dipole moment, consistent with the $-x$ direction here.

Electric Field

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