If 2000 electric field lines enter a given volume of closed space and 4000 field lines diverge from it, then the total charge within the volume is

(Given $ε_0 = 8.85 × 10^{-12} C^2m^{-2}N$)

$1.77 × 10^{-8} C$

The correct answer is Option (2) → $1.77 × 10^{-8} C$

Given:

Number of electric field lines entering: $N_{\text{in}} = 2000$

Number of electric field lines leaving: $N_{\text{out}} = 4000$

Permittivity: $\varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2$

Net number of field lines leaving: $N_{\text{net}} = N_{\text{out}} - N_{\text{in}} = 4000 - 2000 = 2000$

Using Gauss's law: $\Phi_E = \frac{Q}{\varepsilon_0}$

Assuming 1 field line corresponds to flux unit $\phi_0$, total flux: $\Phi_E = N_{\text{net}} \cdot \phi_0$

Then charge:

$Q = \varepsilon_0 \Phi_E = \varepsilon_0 \cdot N_{\text{net}} \cdot \phi_0$

For standard proportionality, taking each field line as 1 unit of flux,

$Q = \varepsilon_0 \cdot 2000 = 8.85 \times 10^{-12} \cdot 2000 = 1.77 \times 10^{-8} \, \text{C}$

Answer: $Q = 1.77 \times 10^{-8} \, \text{C}$