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### Two charged spherical conductors of radius $R_1$ and $R_2$ ....

Two charged spherical conductors of radius $R_1$ and $R_2$ are connected by wire. Then the ratio of surface charge densities of the spheres $\frac {\sigma_1}{\sigma_2}$ is:

(1) $\frac {R_1}{R_2}$
(2) $\frac {R_2}{R_1}$
(3) $\sqrt \frac {R_1}{R_2}$
(4) $\sqrt \frac {R_2}{R_1}$

Solution

$V_1=V_2$

$\therefore k\frac {Q_1}{R_1}=k\frac {Q_2}{R_2}$

$\Rightarrow \frac {\sigma_1 . A_1}{R_1}=\frac {\sigma_2 . A_2}{R_2}$

$\Rightarrow \frac {\sigma_1 . 4\pi {R_1}^2}{R_1}=\frac {\sigma_2 . 4\pi {R_2}^2}{R_2}$

$\Rightarrow \sigma_1 . R_1 = \sigma_2 . R_2$

$\Rightarrow \frac {\sigma_1}{\sigma_2} = \frac {R_2}{R_1}$

### $f(x)=x^6+2x^4+x^3+2x+3 $$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$$n=?$

Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R$. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$ is _ _ _ _ . Solution Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable. Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $\Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$