Answer
$\lambda = -3.72\times 10^{-23}~C/m$
Work Step by Step
Since the electron is suspended, the electric force exerted on the electron is equal in magnitude to the electron's weight. We can find the magnitude of the linear charge density:
$E~\vert q \vert = mg$
$(\frac{\lambda}{2\pi~\epsilon_0~r})~(\vert q \vert) = mg$
$\lambda = \frac{2\pi~\epsilon_0~r~mg}{\vert q \vert}$
$\lambda = \frac{(2\pi)~(8.854\times 10^{-12}~F/m)~(0.012~m)~(9.109\times 10^{-31}~kg)(9.80~m/s^2)}{1.6\times 10^{-19}~C}$
$\lambda = 3.72\times 10^{-23}~C/m$
Since the electric force exerted on the electron is directed upward, the linear charge density must have a negative value. Therefore, $\lambda = -3.72\times 10^{-23}~C/m$
