Question

Two beads, one carrying charge +q and other carrying charge +4q, are separated by distance d that is much greater than the radius of each bead

Is ththere any location along the line between them where the electric field magnitude is zero?

Answer 1

The electric field produced by the first bead at some distance $x$ from it is given as follows:

$$E_1 = k\dfrac{q}{x^2}$$

where $k$ is the constant and $x$ is the distance from the first bead. The second bead produces the following field at the same point $x$:

$$E_2 = k\dfrac{4q}{(d-x)^2}$$

These fields are directed in opposite direction (in the region between beads), thus, the resulting field is zero if

$$E_1 = E_2$$

Solving for $x$, obtain:

$$k\dfrac{q}{x^2} = k\dfrac{4q}{(d-x)^2}\\ \dfrac{1}{x^2} = \dfrac{4}{(d-x)^2}\\ (d - x)^2 = 4x^2\\ d^2 - 2dx + x^2 = 4x^2\\ 3x^2 + 2dx - d^2 = 0\\ x = \dfrac{d + \sqrt{4d^2 +12d^2}}{6} = \dfrac{d + 4d}{6} = \dfrac{5}{6}d\\$$

Answer. At distance $\dfrac{5}{6}d$ from the first bead.