A very long, solid insulating cylinder has radius R; bored along its entire length is a cylindrical hole with radius a. The axis of the hole is a distance b from the axis of the cylinder, where a 6 b 6 R (Fig. P22.58). The solid material of the cylinder has a uniform volume charge density r. Find the magnitude and direction of the electric field E S inside the hole, and show that E S is uniform over the entire hole

The value of the electric field is

Explanation:

We know that the electric field inside a solid cylinder at a distance from the centre is given by

Let's consider the cross-section of the cylinder as shown in the figure. Let `O' be the centre of the long solid insulating cylinder having radius 'R'. Also consider that be the cetre of the hole of radius 'a' situated at a distance 'b' from 'O'. Given, the volume charge density of the material is 'r'. So, the volume charge density inside the hole will be '-r'. Let's consider 'P' be any arbitrary point inside the hole situated at a distance 's' from .

So, the electric field '' due to the long cylinder at point 'P' is given by

and the electric field ''due to the hole at point 'P' is given by

So the net electric field () inside the hole is given by

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