[YMP/EM-04001] Grounded Conducting Sphere and a Point Charge

For page specific messages
For page author info


Two point charges \(Q\) and \(Q^\prime\) are located at positions given by \(\vec{a}, \vec{b},  (a\ne b) \). Find the conditions on \(Q, Q^\prime, a, b \)  so that the potential on the surface of a sphere of radius \(R\) with center at the origin may be zero.

Note the result of this problem is used in solution for electric potential of a grounded conducting sphere in presence of a point charge by the method of images.

The potential due to the two charges at point \(\vec{r}\) is given by \[ \Phi(\vec{r}) = \frac{1}{4\pi\epsilon_0}\frac{Q}{|\vec{r}-\vec{a}|} + \frac{1}{4\pi\epsilon_0}\frac{Q^\prime}{|\vec{r}-\vec{b}|}\] The condition that the potential at an arbitrary point P, position vector \(= R \hat{n}\), on the sphere be zero is

 \frac{1}{4\pi\epsilon_0}\frac{Q}{|\vec{r}-\vec{a}|} +
\frac{1}{4\pi\epsilon_0}\frac{Q^\prime}{|\vec{r}-\vec{b}|}&=&0 \\
  Q^\prime {|R\vec{n}-\vec{a}|} +

By symmetry the centre of the sphere and the positions of the two point charges must be in a straight line. Thus vectors \(\vec{a}\) and \(\vec(b)\) must be parallel and we write \[ \vec{a} = a\hat{m}, \qquad \vec{b} = b \hat{m}.\] Squaring the relation \eqref{EQ01} we demand that

  Q^{\prime 2}\{ R^2+ b^2 -2Rb\, \hat{m}\cdot\hat{n}\}  =  Q^2\{ R^2+ a^2 -2Ra\,

The above relation must hold for all \(\hat{m} \cdot \hat{n}\). This gives two relations

  Q^{\prime 2} b  &=&  Q^2 a \label{EQ02}\\
    Q^{\prime 2}\{ R^2+ b^2 \}  &=& Q^2\{ R^2+ a^2 \}\label{EQ03}

Substituting \(Q^{\prime2}=Q^2(a/b)\), from \eqRef{EQ02}, in \eqref{EQ03}, gives

  &&  Q^2 \frac{a}{b}(R^2+b^2)= Q^2 (R^2+a^2)\\
\Rightarrow &&   aR^2+ab^2 =bR^2 + ba^2\\
\Rightarrow && (a-b)R^2 - ab(a-b) =0 \\
\Rightarrow && R^2 = ab \\

It is obvious that the sign of \(Q\) and \(Q^\prime\) must be opposite. Hence

  Q^{\prime} &=& - Q \sqrt{a/b}

Thus we have the answer \begin{equation} \boxed{ Q^\prime = - \frac{Q a^2}{R}, \qquad   b =  \frac{R^2}{a}.} \end{equation}

Exclude node summary :