Question
Two spheres A and B with radii $a$ and $b$ respectively are at the same potential. What will be the ratio of surface charge densities at $A$ and $B$ ?
✓
Answer
It is given that the potentials of the spheres are equal $i$.e.
$V _{ A }= V _{ B }$
$\Rightarrow \quad \frac{1}{4 \pi \in_0} \times \frac{ Q _1}{a}=\frac{1}{4 \pi \in_0} \times \frac{ Q _2}{b}$
$\Rightarrow \quad \frac{ Q _1}{ Q _2}=\frac{a}{b}$
$\because$ Surface charge density $\sigma=\frac{ Q }{4 \pi r^2}$
$\Rightarrow \quad \frac{\sigma_1}{\sigma_2}=\frac{ Q _1}{ Q _2} \times \frac{b^2}{a^2}$
$=\frac{a}{b} \times \frac{b^2}{a^2}$
$\frac{\sigma_1}{\sigma_2}=\frac{b}{a}$
$V _{ A }= V _{ B }$
$\Rightarrow \quad \frac{1}{4 \pi \in_0} \times \frac{ Q _1}{a}=\frac{1}{4 \pi \in_0} \times \frac{ Q _2}{b}$
$\Rightarrow \quad \frac{ Q _1}{ Q _2}=\frac{a}{b}$
$\because$ Surface charge density $\sigma=\frac{ Q }{4 \pi r^2}$
$\Rightarrow \quad \frac{\sigma_1}{\sigma_2}=\frac{ Q _1}{ Q _2} \times \frac{b^2}{a^2}$
$=\frac{a}{b} \times \frac{b^2}{a^2}$
$\frac{\sigma_1}{\sigma_2}=\frac{b}{a}$
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