Many enthusiasts may have already encountered the following statement in various books:
For an isotropic thin spherical shell, the gravitational force exerted by the shell on any point inside it is zero.
This is a remarkable phenomenon because it implies a uniform gravitational field. Although we assume uniform gravitational fields in many problems, it is often difficult to know how to construct a truly uniform one (and constructing a real uniform force field is very useful for analyzing certain problems, such as deriving proportional coefficients). Now, we have a uniform gravitational field right before us. Furthermore, by utilizing this property, we can calculate the gravitational force at a point inside a uniform solid sphere (which equals the gravitational force exerted by the sphere of matter interior to that point). While this can be proven using calculus, we will introduce an elementary method here, which we believe will make the wonders and beauty of physics even more apparent.
As shown in the figure, we want to prove that the sum of the gravitational forces at point A is zero. Through point A, we can construct two vertically opposite cones with very small solid angles. In a 2D projection, these are represented by "\triangle ABC and \triangle ADE" or "\triangle AGF and \triangle AHI". Taking the former as an example, when the vertical angle is very small, we can treat the segments DE and BC as point masses. We can also treat DE and BC as straight lines (note that they are not actually lines, but spherical caps; when treated as lines, they represent circles on a plane). We calculate their respective gravitational forces on point A: \begin{aligned} F_{ABC} &= \frac{GM_{BC}M_A}{AB^2} \\ F_{ADE} &= \frac{GM_{DE}M_A}{AD^2} \end{aligned}
Due to the uniform density: \frac{M_{BC}}{M_{DE}} = \frac{S_{BC}}{S_{DE}} = \frac{AB^2}{AD^2}
Substituting this, we find: F_{ABC} = F_{ADE}
In other words, the gravitational forces from these two parts cancel each other out. A spherical shell can be divided into an infinite number of such pairs, and their gravitational forces all cancel each other. Put another way, for every point on the shell exerting a force on A, there is another point whose force cancels it out. Consequently, the total gravitational force at point A is zero.
Since electrostatic fields possess properties similar to gravitational fields (being inversely proportional to the square of the distance), it is easy to draw an analogy: when a spherical shell is uniformly covered with the same type of charge, the net electrostatic force on any point charge inside the shell is zero. This leads us to a question: since charges can be positive or negative, if the spherical shell is divided into two equal halves, each uniformly distributed with an equal amount of opposite charge, what would the internal electric field look like?
Some friends might guess that it is also uniform, but that would be incorrect. Please look at the figure below:
The blue arc and the red arc represent equal amounts of opposite charges. Analyzing the forces at point A using the derivation method above: since the electrostatic forces from segments BC and DE (actually two spherical caps) would have canceled each other out if the charges were the same, the electrostatic forces from the red arc and the blue arc BD are now equal in magnitude and point in the same direction. Therefore, it can be seen that: the force at point A is equal to twice the resultant force exerted by the hemispherical shell CE (the red arc) on point A. If the force were uniform, it would mean the electrostatic force from the hemispherical shell CE on any point would be equal, which is clearly not the case.
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