Verified answer

EXPLANATION.

Two point charges + q and + 4q.

Are at a distance = l.

Third charge is also placed that the entire system is in equilibrium.

As we know that,

Third charge is placed at the middle of both charge along the small charges.

We can write equation as,

⇒ F₁ = F₂.

⇒ kqQ/x² = k4qQ/(l - x)².

⇒ 1/x² = 4/(l - x)².

⇒ (l - x)²/(x)² = 4.

⇒ [(l - x)/x] = √4.

⇒ [(l - x)/x] = ± 2.

⇒ (l - x)/x = 2.

⇒ l - x = 2x.

⇒ l = 2x + x.

⇒ l = 3x.

⇒ x = l/3.

F(net) om q is zero,

We can write equation as,

⇒ F₁ + F₂ = 0.

⇒ k(q Q)/x² + k(4q²)/(l)² = 0.

⇒ Q/x² + q/l² = 0.

Put the values in the equation, we get.

⇒ Q/(l/3)² + 4q/l² = 0.

⇒ 9Q/l² + 4q/l² = 0.

⇒ 9Q + 4q = 0.

⇒ 9Q = - 4q.

⇒ Q = - 4q/9.

Value of n = 9.

MORE INFORMATION.

Electric field due to :

(1) infinite line of charge : 2kλ/r.

(2) semi infinite line of charge √2kλ/r.

(3) Uniformly charged ring : E(center) = 0  &  E(axis) = [kQx/(x² + R²)^3/2].

(4) Continuous charge distribution E = k∫dE,  dE = electric field due to an elementary charge.

Note : E ≠ ∫dE because E is a vector quantity.

[tex]\huge\mathfrak\red{ANSWER}[/tex]

SOLUTION IN DETAILS:-

Charges =q and 4q distance =d

third charge =Q

distance =x

charges =q

[tex] \frac{kq}{ {x }^{2} } = \frac{k(4q)}{(l - {x}^{2}) } [/tex]

[tex]x = \frac{l}{3} [/tex]

[tex]\small\mathfrak\pink{important \: thing}[/tex]

This is the position where the third charge should be placed. Now, we have to determine the nature and magnitude of it. For this just try to satisfy the condition of equilibrium on any one of the remaining two charges. Let's take +q charges. The force on it due to +4q charge is directed left wards so force on it due to the third charge should be directed right wards. This implies that the force between +q charge and third charge should be of attractive nature. Thus the nature of the 3rd charge is (−ve).

Taking the third charge to be −Q (say) and then on applying the condition of equilibrium on +q charge

KQ = K(4q)

x² L²

KQ = K(4q)

(L/3)² L²

PUTTING EQU

9KQ = 4Kq

L² L²

➡️ 9Q = 4q

HENCE PROVED ...