Question

An infinite line of charge with uniform line charge density of 1C/m is placed along the y -axis.A charge of 1C is placed on the x -axis at a distance of d=3m from the origin.At what distance (r) from the origin on the x -axis the total electric field is zero. (Assume 0<r<d)
ANSWER:2m

Answer 1

The electric field of an infinite line charge is given by

$$E_1(r)=\frac{\lambda}{2\pi r \varepsilon_0}$$

The electric field of point charge placed at distance $d$ from the origin is given by

$$E_2(r)=\frac{q}{4\pi \varepsilon_0 (r-d)^2}$$

At the point where the electric field is zero

$$E_1(r)=E_2(r)$$

Hence

$$\frac{\lambda}{2\pi r \varepsilon_0}=\frac{q}{4\pi \varepsilon_0 (r-d)^2}$$

After some algebra, we obtain an equation

$$r^2-(2d+\frac{q}{2\lambda})r+d^2=0$$

$$r^2-6.5r+9=0$$

Roots:

$$r=2,\quad r=4.5$$

For $0<r<d$ we obtain $r=2\:\rm m.$