Calculating Electric Field Intensity with Two Spherical Shells

[john88]

Hi

Two spherical shells where the inner has a radius a and the outer a radius of b. The inner has a total charge of -Q whereas the outer shell has a total charge +Q. The question is to calculate the electric field intensity everywhere in space.
My question is now, how do I choose the limits for example $$a \leq R \leq b$$ I have two examples below

No point charge in the middle

$$E = 0, 0 \leq R \prec a$$ (not equal to a)

$$E = -\frac{Q}{4\pi\epsilon_{o}R^3} \bold{R}$$ $$a \leq R \leq b$$

$$E = 0, b \prec R \prec \infty$$

A point charge in the middle

$$E = \frac{Q}{4\pi\epsilon_{o}R^3} \bold{R}$$ $$0 \prec R \leq a$$ why set equal to a here and not when there aint no point charge in the middle?

$$E = 0$$ $$a \prec R \prec b$$

$$E = \frac{Q}{4\pi\epsilon_{o}R^3} \bold{R}$$ $$b \leq R \prec \infty$$

 

[Gear300]

A point within a spherical shell experiences 0N/C. For R between a and b, inner shell is effective and outer shell's net effect is 0N/C...this works for even insulators so long as charge is evenly distributed across the shell (question didn't specify whether it was a conductor or something else). If the thickness of the shells is negligible, then you could consider the effect of the inner shell as that due to a point charge.

 

[Moetasim]

As a scientist, it is important to consider all possible scenarios and understand the underlying principles behind them. In this case, we are dealing with electric fields generated by two spherical shells with different charges. The limits for calculating electric field intensity will depend on the position of the point in space relative to the two shells.

In the first scenario where there is no point charge in the middle, we can see that the electric field is zero for all points outside of the inner shell (0 ≤ R < a), as expected since there is no charge present there. The electric field is also zero for points beyond the outer shell (b < R < ∞) since the outer shell has no influence on those points.

For points between the two shells (a ≤ R ≤ b), we can see that the electric field is non-zero and is given by the formula -Q/4πε₀R³. This is because the inner shell with a negative charge will create an electric field that is directed towards the point, while the outer shell with a positive charge will create an electric field that is directed away from the point. The two fields will cancel each other out, resulting in a net electric field of zero.

In the second scenario where there is a point charge in the middle, the limits for calculating electric field intensity are different. For points within the inner shell (0 < R ≤ a), the electric field is given by Q/4πε₀R³, as the point charge will create an electric field that is directed away from the point. For points between the two shells (a < R ≤ b), the electric field is zero as the two fields from the shells will again cancel each other out. For points beyond the outer shell (b < R < ∞), the electric field is given by Q/4πε₀R³ as the point charge will be the only source of electric field.

In summary, the limits for calculating electric field intensity depend on the presence and position of charges in the system. It is important to carefully consider these factors in order to accurately calculate the electric field at any point in space.