A point charge $+Q$ is placed on the positive $x$-axis at a distance $a$ from the centre $O$ of a neutral conducting sphere of radius $R$ (where $aR$). Using the method of images, the field outside the sphere is equivalent to that of the real charge $+Q$ and an image charge $-Q'$ located on the $x$-axis at a distance $b$ from $O$, where $Q' = Q\,\dfrac{R}{a}$ and $b = \dfrac{R^{2}}{a}$. Find the magnitude and direction of the electric field at point $P$ which lies on the same line, at a distance $2a$ from $O$ (i.e., beyond the real charge). Express your answer in terms of $Q$, $a$, $R$ and $\varepsilon_{0}$.

Question

A point charge $+Q$ is placed on the positive $x$-axis at a distance $a$ from the centre $O$ of a neutral conducting sphere of radius $R$ (where $a>R$). Using the method of images, the field outside the sphere is equivalent to that of the real charge $+Q$ and an image charge $-Q'$ located on the $x$-axis at a distance $b$ from $O$, where $Q' = Q\,\dfrac{R}{a}$ and $b = \dfrac{R^{2}}{a}$. Find the magnitude and direction of the electric field at point $P$ which lies on the same line, at a distance $2a$ from $O$ (i.e., beyond the real charge). Express your answer in terms of $Q$, $a$, $R$ and $\varepsilon_{0}$.

Accepted Answer

$E = \dfrac{1}{4\pi\varepsilon_{0}}\,Q\left[\dfrac{1}{a^{2}} - \dfrac{R a}{\bigl(2a^{2}-R^{2}\bigr)^{2}}\right]$ directed away from the sphere (positive x‑direction)

Neco Model Questions Vol1 2019 Physics Question 44

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