The net electric field at point \(P\) is the vector sum of the fields due to the three charges:
Step 1: Electric Field Due to Individual Charges
Charge \(q\) at \(x = -a\):
\[E_1 = \frac{q}{4\pi\epsilon_0 (x+a)^2}.\]
Charge \(-2q\) at \(x = 0\):
\[E_2 = \frac{-2q}{4\pi\epsilon_0 x^2}.\]
Charge \(q\) at \(x = a\):
\[E_3 = \frac{q}{4\pi\epsilon_0 (x-a)^2}.\]
Step 2: Approximation for \(x \gg a\)
For \(x \gg a\), expand the denominators using binomial approximation:
\((x+a)^2 \approx x^2 \left(1 + \frac{2a}{x}\right)\),
\[E_1 \approx \frac{q}{4\pi\epsilon_0 x^2} \left(1 - \frac{2a}{x}\right).\]
\((x-a)^2 \approx x^2 \left(1 - \frac{2a}{x}\right)\),
\[E_3 \approx \frac{q}{4\pi\epsilon_0 x^2} \left(1 + \frac{2a}{x}\right).\]
\(E_2 = \frac{-2q}{4\pi\epsilon_0 x^2}\) (no approximation needed).
Step 3: Net Electric Field
Adding all contributions:
\[E = E_1 + E_2 + E_3.\]
Substitute the approximations:
\[E \approx \frac{q}{4\pi\epsilon_0 x^2} \left(1 - \frac{2a}{x}\right) + \frac{-2q}{4\pi\epsilon_0 x^2} + \frac{q}{4\pi\epsilon_0 x^2} \left(1 + \frac{2a}{x}\right).\]
Simplify:
\[E \approx \frac{q}{4\pi\epsilon_0 x^2} \left[1 - \frac{2a}{x} + 1 + \frac{2a}{x} - 2\right].\]
\[E \approx \frac{q}{4\pi\epsilon_0 x^2} \left(-\frac{4a}{x}\right).\]
\[E \approx \frac{-4qa}{4\pi\epsilon_0 x^3}.\]
Step 4: Substitution for \(qa^2 = Q\)
Given \(qa^2 = Q\), substitute \(q = \frac{Q}{a^2}\):
\[E \approx \frac{-4 \left(\frac{Q}{a^2}\right) a}{4\pi\epsilon_0 x^3}.\]
\[E \approx \frac{-4Q}{4\pi\epsilon_0 x^3}.\]
Step 5: Compare with the Given Form
The given form is:
\[E = \frac{-\alpha Q}{4\pi\epsilon_0 x^3}.\]
By comparison:
\[\alpha = 2, \quad \beta = 3.\]
Step 6: Relationship Between \(\alpha\) and \(\beta\)
\[\alpha = \frac{2}{3}\beta.\]
Final Answer
\[\boxed{\alpha = \frac{2}{3}\beta.}\]
Match List - I with List - II:
List - I:
(A) Electric field inside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
(B) Electric field at distance \( r > 0 \) from a uniformly charged infinite plane sheet with surface charge density \( \sigma \).
(C) Electric field outside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
(D) Electric field between two oppositely charged infinite plane parallel sheets with uniform surface charge density \( \sigma \).
List - II:
(I) \( \frac{\sigma}{\epsilon_0} \)
(II) \( \frac{\sigma}{2\epsilon_0} \)
(III) 0
(IV) \( \frac{\sigma}{\epsilon_0 r^2} \) Choose the correct answer from the options given below: