Solved: A very long, solid insulating cylinder has radius

Chapter 22, Problem 62P

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QUESTION:

A very long, solid insulating cylinder with radius  has a cylindrical hole with radius  bored along its entire length. The axis of the hole is a distance  from the axis of the cylinder, where   (Fig. P22.62). The solid material of the cylinder has a uniform volume charge density  Find the magnitude and direction of the electric field \(\bar{E}\) inside the hole, and show that \(\bar{E}\) is uniform over the entire hole. (Hint: See Problem 22.61.)

Equation transcription:

Text transcription:

bar{E}

Questions & Answers

QUESTION:

A very long, solid insulating cylinder with radius  has a cylindrical hole with radius  bored along its entire length. The axis of the hole is a distance  from the axis of the cylinder, where   (Fig. P22.62). The solid material of the cylinder has a uniform volume charge density  Find the magnitude and direction of the electric field \(\bar{E}\) inside the hole, and show that \(\bar{E}\) is uniform over the entire hole. (Hint: See Problem 22.61.)

Equation transcription:

Text transcription:

bar{E}

ANSWER:

Solution 62P

Step 1

Consider that according to Gauss law, the electric field at point  inside the cylinder in x-y plane and in the radial direction.

 …… (1)

Here,  is the electric field vector,  is the position vector in the x-y plane,  is the charge density, and  is the permittivity of free space.

Substitute,  for .

If the cylinder axis goes through point  in the x-y plane, then electric field off-axis is given as.

Here,  is the axis- off position vector and  is the electric field vector at off-axis.

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