An electric charge 2 x 10^-8 C is placed at the point (1, 2, 4). At the point (3, 2, 1), the electric

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field will be along the y-axis

potential will be 49.9 V

field will have no y-component

field will increase by a factor K if the space between the points is filled with a dielectric of dielectric constant K

answer is C, D.

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Detailed Solution

${\vec{r}}_{P} = \hat{i} + 2 \hat{j} + 4 \hat{k}; {\vec{r}}_{q} = 3 \hat{i} + 2 \hat{j} + \hat{k}$

$|{\vec{r}}_{q} - {\vec{r}}_{P}| = \sqrt{[(2)^{2} + (- 3)^{2}]} = \sqrt{13}$

As $q = 2 \times 10^{- 8} C$

$V = \frac{1}{4 {πε}_{0}} \frac{q}{|{\vec{r}}_{q} - {\vec{r}}_{p}|}$

$= \frac{9 \times 10^{9} \times 2 \times 10^{- 8}}{\sqrt{13}} = 49 .9 V$

$\vec{E} = \frac{1}{4 {πε}_{0} {|{\vec{r}}_{q} - {\vec{r}}_{P}|}^{2}} \frac{({\vec{r}}_{q} - {\vec{r}}_{P})}{|{\vec{r}}_{q} - {\vec{r}}_{P}|}$

$= \frac{1}{4 {πε}_{0}} \frac{q}{{|{\vec{r}}_{q} - {\vec{r}}_{p}|}^{3}} (2 \hat{i} - 3 \hat{k})$

(By filling a dielectric of dielectric constant K, electric field gets decreased as $E = E_{0} / K and K > 1$ .)

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