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.

Solution

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

\frac{σ}{ε_{0}} \hat{n} .

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

E_{1} + E_{2} = \frac{σ}{ε_{0}}

∴

2 E_{1} = \frac{σ}{ε_{0}}

\Rightarrow

E_{1} = \frac{σ}{2 ε_{0}}

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

E_{1} = \frac{σ}{2 ε_{0}} \hat{n} (\hat{n} \to u n i t v e c t o r i n t h e o u t w a r d n o r m a l d i r e c t i o n)

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Electric Charges And Fields

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.

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

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For Daily Free Study Material Join wiredfaculty

Electric Charges And Fields

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.

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?b) Explain why two field lines never cross each other at any point?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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.

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?