A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε₀) $\overset{⏞}{n},$ where $\overset{⏞}{n}$ is the unit vector in the outward normal direction, and σ is the surface charge density near the hole.

Let us take a charged conductor with the hole filled up, as shown by shaded portion in the figure.

We find with the application of Gaussian theorem that field inside is zero and just outside is

straight sigma over straight epsilon subscript 0 straight n with hat on top.

This field can be viewed as the superposition of the field E₂ due to the filled up hole plus the field E₁ due to the rest of the charged conductor.
The two fields (E₁ and E₂) must be equal and opposite as the field vanishes inside the conductor. Thus, E₁ – E₂ = 0
Now, the field outside the conductor is given by

space space space space space space space space space space space space space space space space space space space straight E subscript 1 plus straight E subscript 2 space equals space straight sigma over straight epsilon subscript 0

therefore

2 straight E subscript 1 space equals space straight sigma over straight epsilon subscript 0

straight E subscript 1 space equals fraction numerator straight sigma over denominator 2 straight epsilon subscript 0 end fraction

Therefore, field in the hole (due to the rest of the conductor) is given as:

straight E subscript 1 space equals space fraction numerator straight sigma over denominator 2 straight epsilon subscript 0 end fraction straight n with hat on top space left parenthesis straight n with hat on top space rightwards arrow space space space unit space vector space in space the space outward space normal space direction right parenthesis

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A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε0) n⏞, where n⏞ is the unit vector in the outward normal direction, and σ is the surface charge density near the hole.

Electric Charges and Fields

Physics Part I

A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε₀) $\overset{⏞}{n},$ where $\overset{⏞}{n}$ is the unit vector in the outward normal direction, and σ is the surface charge density near the hole.