(a) A point charge `Q = 100 mu C` is placed at origin. Find electric field vector at point `( 6 m , 8 m)`.
(b) A point charge `Q = 169 mu C` is placed at `(1 m , 2m , 3m)`. Find the electric field vector at the point `(4m , 6m , 15m)`.

Explanation

`r = sqrt((6)^(2) + (8)^(2)) = 10 m`
`vec( r) = 6 hat(i) + 8 hat (j)`
Unit vector along `vec(r )` ,
`hat(r ) = (vec(r ))/(|vec(r )|) = ( 6 hat(i) + 8 hat(j))/(10)`
Magnitude of electric field at `P`:
` E = (1)/( 4 pi in_(0)) .(Q)/(r^(2)) = (( 9 xx 10^(9)) (100 xx 10^(-6)))/((10)^(2))`
`= 9 xx 10^(3) N//C`
`vec(E ) = E hat(r ) = 9 xx 10^(3) xx ((hat(6i) + 8 hat(j)))/(10)`
`= ( 54 hat(i) + 72 hat(j)) xx 10^(2) N//C`
(b) Position vector of change
`hat(r)_(1) = hat(i) + 2 hat(j) + 3 hat(k) m`
Position vector of point `P`
`hat(r )_(2) = 4 hat(i) + 6 hat(j) + 15 hat(k) m`
Position vector along which electric field is acting
`vec( r) = vec(r )_(2) - vec( r)_(1) = 3 hat (i) + 4 hat(j) + 12 hat(k) m`
Unit vector along field
`hat(r ) = (hat(r ))/(|vec( r)|) = ( 3 hat(i) + 4 hat(j) + 12 hat(k))/(13)`
Electric field at `P`:
`E = (1)/( 4 pi in_(0)) . (Q)/(r^(2)) = ( 9 xx 10^(9) xx 169 xx 10^(-6))/((13)^(2))`
`= 9 xx 10^(3) N//C`
`vec(E ) = E hat(r ) = 9 xx 10^(3) xx (( 3 hat(i) + 4 hat(j) + 12 hat(k)))/(13)`
` = ( 27 hat(i) + 36 hat(j) + 108 hat(k))/(13) xx 10^(3) N//C`