1. Electric field due to an infinite non-conducting sheet

For an infinite sheet with surface charge density $-\sigma$, the magnitude of electric field is constant:

$E = \frac{\sigma}{2\varepsilon_0}$

Since the sheet is negatively charged, an electron (charge $-e$) experiences force away from the sheet, but we only need the magnitude.

2. Force and acceleration of the electron

Magnitude of force on the electron:

$F = eE = e\cdot \frac{\sigma}{2\varepsilon_0}$

Hence acceleration is constant:

$a = \frac{F}{m} = \frac{e\sigma}{2\varepsilon_0 m}$

So the electron moves with uniform acceleration from rest.

3. Momentum as a function of time

Released from rest, so

$v = at$

Therefore momentum is

$p = mv = mat$

Thus,

$p \propto t$

4. de-Broglie wavelength

The de-Broglie wavelength is

$\lambda = \frac{h}{p}$

Since $p \propto t$,

$\lambda \propto \frac{1}{t}$

Let

$\lambda = \frac{K}{t}$

for some constant $K$.

5. Rate of change of wavelength

Differentiate with respect to time:

$\frac{d\lambda}{dt} = -\frac{K}{t^2}$

Hence,

$\left|\frac{d\lambda}{dt}\right| \propto \frac{1}{t^2}$

So the rate of change of de-Broglie wavelength varies inversely as the 2nd power of time.

6. Final answer

$n = 2$