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In fiinding the electric field using Gau...

In fiinding the electric field using Gauses law the formula `|vec(E)|=(q_("enc"))/(in_(0)|A|)` is applicable. In the Formula. `in_(0)` is permittivity of free space, A is the area of Gaussian surface and `q_("enc")` is charge enclosed by the Gaussion surface. This equation can be used in which of the following situation?

A

Only when the Gaussian surface is an equipotential surface

B

only when `|vecE|` is constant on the surface

C

only when the the Gaussian surface is an equipotential surface `|vecE|` is constant of the surface

D

for any choice of Gaussian surface

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To determine the situations in which the formula \(|\vec{E}| = \frac{q_{\text{enc}}}{\epsilon_0 |A|}\) is applicable, we can analyze the conditions under which this equation holds true based on Gauss's Law. ### Step-by-Step Solution: 1. **Understanding Gauss's Law**: Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is expressed as: \[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\epsilon_0} \] where \(\Phi_E\) is the electric flux, \(\vec{E}\) is the electric field, \(d\vec{A}\) is the differential area vector, and \(q_{\text{enc}}\) is the charge enclosed by the surface. 2. **Rearranging the Equation**: From the equation of electric flux, we can derive the formula given in the question: \[ |\vec{E}| = \frac{q_{\text{enc}}}{\epsilon_0 |A|} \] Here, \(|A|\) is the total area of the Gaussian surface. 3. **Conditions for Applicability**: - **Uniform Electric Field**: The electric field \(\vec{E}\) must be uniform over the entire surface. This means that the magnitude and direction of \(\vec{E}\) do not change across the surface. - **Angle Between \(\vec{E}\) and \(d\vec{A}\)**: The angle \(\theta\) between the electric field vector \(\vec{E}\) and the area vector \(d\vec{A}\) should be zero. This implies that the electric field lines are perpendicular to the surface, leading to maximum flux. - **Equipotential Surface**: The surface should be equipotential, which means that the electric potential is constant across the surface. This ensures that there is no potential difference along the surface. 4. **Conclusion**: The formula \(|\vec{E}| = \frac{q_{\text{enc}}}{\epsilon_0 |A|}\) is applicable when: - The electric field is uniform across the Gaussian surface. - The electric field is perpendicular to the surface (angle \(\theta = 0\)). - The surface is equipotential. ### Final Answer: Thus, the formula can be used in situations where the electric field is uniform, perpendicular to the surface, and the surface is equipotential. ---
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