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1. Electric Charges and Fields
medium
The maximum value of electric field on the axis of a charged ring having charge $Q$ and radius $R$ is
A
$\frac{1}{{4\pi { \in _0}}}\frac{Q}{{{R^2}}}$
B
$\frac{1}{{4\pi { \in _0}}}\frac{{2Q}}{{3\sqrt 3 {R^2}}}$
C
$\frac{1}{{4\pi { \in _0}}}\frac{{2\sqrt 2 Q}}{{3{R^2}}}$
D
$\frac{1}{{4\pi { \in _0}}}\frac{Q}{{3{R^2}}}$
Solution
$\mathrm{E}=\mathrm{E}_{\max } \Rightarrow \mathrm{x}=\frac{\mathrm{R}}{\sqrt{2}}$ as $\frac{\mathrm{dE}}{\mathrm{dx}}=0$
${{\rm{E}}_{{\rm{axis }}}} = \frac{1}{{4\mu { \in _0}}}\frac{{{\rm{Qx}}}}{{{{\left( {{{\rm{x}}^2} + {{\rm{R}}^2}} \right)}^{3/2}}}}$
$\therefore {{\rm{E}}_{\max }} = \frac{2}{{3\sqrt 3 }}\left( {\frac{1}{{4\pi { \in _0}}}\frac{{\rm{Q}}}{{{{\rm{R}}^2}}}} \right)$
Standard 12
Physics
