Question

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

Is there any location along the line between them where the electromagnetic field magnitude is zero. If so, what is the distance from the particle carrying +q to the location?

Write answer in term of d(unless there is no answer enter none)

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. $\dfrac{5}{6}d$.