Solution : Study of a spherical inhomogeneous distribution - Corrected Exercises Gauss Theorem



1. Expression of the electrostatic field.

  All the planes containing the center O of the sphere and the point P are the load distribution of planes of symmetry. The electric field must belong simultaneously to all of these plans, it is therefore carried by their intersection which is the line OM.  we obtain.:
As this distribution has a rotational invariance around the point O, the electric field does not depend on the angular variables.

This result does not depend on the position of this point P.
We choose a Gaussian surface centered in O and radius r. The Gauss theorem is:


Then there are two regions of space:
● For r <R (one can put an equal sign here because the density distribution is what ensures the continuity of the normal component (here radial) of the electrostatic field).
For the domestic load:

Is obtained by applying the Gauss theorem:
The electric field is expressed in this region:
● For r> R, Inner charge on the surface Gauss is:

Is obtained by applying the Gauss theorem:

The electric field is expressed in this region:
2. Maximum field.
The field has an extreme value that we consider it maximum  when:



Where:

The coefficient k is expressed as:

In which r / R = 1/2 are obtained:




Comments

Post a Comment

Popular posts from this blog

CONDUIT AND SURFACE-CONTACT TRAMWAY SYSTEMS

Image
Roughly speaking, the arrangements for generating electricity, distributing it, and utilising it on the car, remain the same in conduit tramways and surface-contact tramways as on the overhead system. The differences between the three systems are, as already indicated, confined to the means of collecting the current for each car. Both the conduit and the surface-contact system were suggested as a means of escape from the main objection to the overhead system—the exposure of 'live' wires in the street. The cable tramway, with its concrete trough and slot, gave an obvious hint. There would be no difficulty, apparently, in carrying wires on insulators in the trough or conduit, and utilising the slot for a 'plough' which would slide along inside the conduit, keeping contact with the wires, and so conveying the current to the car. This was tried for the first time in Blackpool, where—in 1884—a length of conduit tramway was laid along the front street of the t
Read more

PETROL-ELECTRIC VEHICLES AND MAIN MARINE PROPULSION BY ELECTRICITY

Image
Between the petrol-driven vehicle and the electric automobile there is an interesting series of links provided by 'petrol-electric' systems. At one end of the chain, electricity plays an important part in supplying power to drive the car. At the other end, electrical apparatus is introduced merely as a form of transmission gear between the petrol engine and the driving axle. The reason for attempting the petrol-electric combination will be most readily understood by considering the latter arrangement first. The petrol engine is a high-speed engine, capable of working most satisfactorily when it runs at a uniform rate with a constant load. On the other hand, the speed of the driving axle of a car varies from a very much lower speed down to zero. It is therefore necessary, when driving a vehicle with a petrol engine, to arrange some forms of variable speed-reducing transmission gear between the engine and the driving axle. The problem is further complicated by the
Read more