Find the electric field vector at P (a,a,a) due to three infinitely long lines of charges along the x-, y- and z-axis, respectively. The charge density i.e. charge per unit length of each wire is `(lambda)`,

Options

A. `(lambda)/(3 pi epsilon_(0)a)(hat(i)+hat(j)+hat(k))`
B. `(lambda)/(2 pi epsilon_(0)a)(hat(i)+hat(j)+hat(k))`
C. `(lambda)/(2sqrt(2) pi epsilon_(0)a)(hat(i)+hat(j)+hat(k))`
D. `(2 lambda)/( pi epsilon_(0)a)(hat(i)+hat(j)+hat(k))`

Answer

B

Explanation

Let us consider the electric field due to wire (3) only

`vecE_(3)=E hat(u)`
`=(lambda)/(2 pi epsilon_(0)(a^(2)+a^(2))^(1//2)) (hat (i) cos 45^(@)+hat (j)45^(@))`
`(lambda)/(2sqrt(2)pi epsilon_(0)a)(1)/(sqrt(2))(hat(i)+hat(j))`
`(lambda)/(4 pi epsilon_(0)a)(hat(i)+hat(j))`
Similarly, electric field due to wires (1) and (2)
`vec(E_1)=(lambda)/(4 pi epsilon_(0)a)(hat(j)+hat(k))` and `vec(E_2)=(lambda)/(4 pi epsilon_(0)a)(hat(i)+hat(k))`
`vecE_("net")=vecE_(1)+vecE_(2)+vecE_(3)=(lambda)/(4 pi epsilon_(0)a)(hat(i)+hat(j)+hat(k))`.