Given, Line charge density of rod,
λ=q∕l . . . (i)
where,
q is total charge of rod and
L is the total length of rod.
Now,
dq=λdx where,
dq is elemental charge at distance
x from origin and
dx is elemental length.
∵‌‌dEy=‌‌cos‌θ where,
dE is elemental electric field.
∴‌‌dE‌=k‌⋅‌ ⇒‌‌dE=kλ⋅‌. . . (ii)
Let,
‌‌x2+y2=r2 On differentiating both sides w.r.t
x, we get
2x+0‌‌=2r‌ ⇒‌‌xdx‌‌=rdr Substituting the value Eq. (ii), we get
dE=kλ‌ On integrating both sides, we get
dE‌‌=kλ‌=kλ‌r−2dr=kλ(‌)0r=−‌ ‌‌=−‌‌=‌‌ ‌‌=‌‌=‌‌=‌‌