Electrostatics 1 Question 15
16. Let $E _1(r), E _2(r)$ and $E _3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. If $E _1\left(r _0\right)=E _2\left(r _0\right)$ $=E _3\left(r _0\right)$ at a given distance $r _0$, then
(2014 Adv.)
(a) $Q=40 \pi r _0^{2}$
(b) $r _0=\frac{\lambda}{2 \pi \sigma}$
(c) $E _1\left(r _0 / 2\right)=2 E _2\left(r _0 / 2\right)$
(d) $E _2\left(r _0 / 2\right)=4 E _3\left(r _0 / 2\right)$
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Answer:
Correct Answer: 16. (a)
Solution:
- $\frac{Q}{4 \pi \varepsilon _0 r _0^{2}}=\frac{\lambda}{2 \pi \varepsilon _0 r _0}=\frac{\sigma}{2 \varepsilon _0} \Rightarrow Q=2 \pi \sigma r _0^{2}$
(a) is incorrect, $r _0=\frac{\lambda}{\pi \sigma}$
(b) is incorrect, $E _1 \frac{r _0}{2}=4 E _1\left(r _0\right)$
As $\quad E _1 \propto \frac{1}{r^{2}}$
$$ E _2 \frac{r _0}{2}=2 E _2\left(r _0\right) \text { as } E _2 \propto \frac{1}{r} $$
(c) is correct
$$ E _3 \frac{r _0}{2}=E _3\left(r _0\right)=E _2\left(r _0\right) $$
as $E _3 \propto r^{0} \Rightarrow(d)$ option is incorrect
