If the electric flux entering and leaving an enclosed surface respectively is ${\text{ϕ}}_1$ and ${\mathrm{\varphi }}_2$ , the electric charge inside the surface will be

To determine the electric charge inside an enclosed surface when the electric flux entering and leaving the surface are given as ϕ₁ and ϕ₂ respectively, we can use Gauss's Law.

1. Understand the Concept of Electric Flux

Electric flux (Φ) through a surface is the measure of the electric field passing through that surface. It is mathematically defined as the product of the electric field (E) and the area (A) through which the electric field lines pass. The formula for electric flux is:

Φ = E ⋅ A

Where:

• E is the electric field.
• A is the area through which the electric field passes.

2. Apply Gauss's Law

According to Gauss's Law, the net electric flux (Φ_net) through a closed surface is proportional to the net charge (Q) enclosed within the surface. Mathematically, Gauss's Law is expressed as:

Φ_net = Q / ε₀

Where:

• Φ_net is the net electric flux through the closed surface.
• Q is the charge enclosed within the surface.
• ε₀ is the permittivity of free space, a constant value.

3. Calculate the Net Electric Flux

The net electric flux (Φ_net) through the surface is calculated as the difference between the flux leaving the surface (ϕ₂) and the flux entering the surface (ϕ₁).

Φ_net = ϕ₂ - ϕ₁

4. Relate the Net Electric Flux to Charge

Using Gauss's Law, we can now relate the net electric flux to the charge enclosed within the surface. Substituting the expression for net electric flux into Gauss's Law gives:

ϕ₂ - ϕ₁ = Q / ε₀

5. Solve for the Enclosed Charge (Q)

To find the electric charge (Q) enclosed within the surface, we rearrange the equation as follows:

Q = (ϕ₂ - ϕ₁) ⋅ ε₀

Final Answer:

Thus, the electric charge inside the enclosed surface is:

Q = (ϕ₂ - ϕ₁) ⋅ ε₀