Question

(2) Find the work needed to move a 6 nC charge from point P(2-23) for 2 μm along the unit vectors (i)퀴帰川k Given the electric field in the systemis E-5 (8xyzi + 4x":ノー4xzyk ) Vim. I+ラ/tラK (ii)줘悖川k vectors (i) and and (3) Let the electric potential field be V-100 r cos φ volt. Find E and at P(r-3 m, ф 60°, z-2 m)

Answer 1

2)

Charge, $\small Q=6nC$

Electric Field, $\small \vec{E}=(1/z^{2})(8xyz\hat{i}+4x^{2}z\hat{j}-4x^{2}y\hat{k})$

Point 1, $\small P= 2\hat{i}-2\hat{j}+3\hat{k}$

Work Done, $\small W=q\vec{E}.\vec{d};$

(i)

$\small -6/7\hat{i}+3/7\hat{j}+2/7\hat{k}$

$\small \vec{d} = 2*10^{-6}(-6/7\hat{i}+3/7\hat{j}+2/7\hat{k})$

$\small W = 6*10^{-9}*2*10^{-6}*(1/z^{2})(8xyz(-6/7)+4x^{2}z(3/7)-4x^{2}y(2/7))$

=$\small 1.49*10^{-13}Joules$

(ii)

$\small 3/7\hat{i}-6/7\hat{j}+2/7\hat{k}$

$\small \vec{d} = 2*10^{-6}(3/7\hat{i}-6/7\hat{j}+2/7\hat{k})$

$\small W = 6*10^{-9}*2*10^{-6}*(1/z^{2})(8xyz(3/7)+4x^{2}z(-6/7)-4x^{2}y(2/7))$

$\small =-9.75*10^{-14}Joule$

(3)

Potential, $\small V= 100 r cos(\phi ) volt$

Electric Field, $\small \vec{E} = -(\partial V/\partial r.\hat{r} +(1/r)\partial V/\partial \phi .\hat{\phi }+\partial V/\partial z.\hat{z})$

$\small \partial V/\partial r=100*cos(\phi )$

(1/r).OV/до =-100 * sin(o);

$\small \partial V/\partial \z =0;$

$\small \vec{E} = 100.sin(\phi )\hat{\phi } -100.cos(\phi )\hat{r}$

$\small r= 3m ; \phi =60^{\circ};z=2m;$

$\small \vec{E}= -50\hat{r}+86.6025\hat{\phi }$

$\small \rho _{v}=\varepsilon _{o}(\partial E_{r}.\partial r+(1/r))\partial E_{\phi }.\partial \phi +\partial E_{z}.\partial z)$

$\small \rho _{v} = 8.854*10^{-12}(-50+86.6025)=3.24*10^{-10}C/m^{3}$