This video is about the field and the potential inside a sphere.
We'll start by looking at the electric field and electric potential inside a hollow sphere,
and then move on to discuss gravitational field and gravitational potential inside a solid sphere.
Let's begin with electric field and electric potential.
When they're charged, conducting sphere, the field looks exactly the same as around a point charge.
The reason is that the electric field lines are perpendicular to the surface of the sphere,
therefore the field around the sphere is a radial field, just as around a point charge.
The first important point, when looking at the field and the potential inside the hollow sphere,
is that all the surplus charge is distributed on the outside of the sphere.
As a result, when we place a test charge inside a sphere, the forces acting on this test charge
completely cancel each other out. We can use the equations that we learned for electric fields,
along with some trigonometry to show this, but this mathematical reasoning is not part of the
IB physics course, so we won't go into its details here. Since the force acting on a test charge
inside a sphere is zero, the electric field strength inside a sphere is also zero.
Earlier in this topic we learned that electric field strength is equal to the negative of the
potential gradient. Since the electric field strength inside a sphere is zero, the potential
gradient is also zero. If we think about the potential gradient and the potential in terms of
calculus, we can conclude that when the potential gradient is zero, the potential must be constant.
Therefore the potential inside a sphere is constant and it is equal to the potential
on the surface of the sphere. These concepts often come up on IB physics exams in terms of graphs.
So based on what we discussed, we can draw the electric field strength versus distance graph
for the sphere, which looks like this. Lowercase r is the distance from the center of the sphere,
capital R is the radius of the sphere and E is the electric field strength.
As you can see, electric field strength inside a sphere is zero, while outside the sphere
we have a 1 over r squared graph just as we have seen for a point charge.
The potential versus distance graph for the sphere looks like this.
Inside a sphere the potential is constant, while outside the sphere we see a 1 over r
graph just as we did for a point charge. Next let's examine the gravitational field
and the gravitational potential inside a solid sphere. For gravitational field and
gravitational potential we consider a solid sphere because in this context we often work with planets
that can be considered as solid uniform spheres. Let's assume that we have a solid sphere for
example the earth with a radius of uppercase r. Now imagine a point inside a sphere at a distance
x from the center of the sphere. You can picture that we have dug a tunnel from the surface of the
earth towards its center and we have gone down this tunnel to a certain point that is at a distance
x from the center of the earth. So at this point basically we would be standing on the surface
of a smaller sphere or smaller earth that has a radius of x and above us we would have a spherical
shell. Let's think about the gravitational field strength at this point. There are two possible
sources for the gravitational field strength here. One is the spherical shell above us
and the other is the smaller sphere below us. The gravitational field strength due to the
spherical shell is zero. The reason is similar to what we explained about the electric field strength
inside a hollow sphere earlier in this video. The gravitational field strength at our chosen
point due to the smaller sphere of radius x is less than the gravitational field strength
at the surface of the solid sphere with radius r. Put more concisely as we get closer and closer
to the center of the original solid sphere of radius r or in our example towards the center of
the earth gravitational field strength decreases. This decrease is linear. The reason for this
linear decrease is that as we are decreasing the radius of the smaller inner sphere so as the value
of x gets smaller and smaller the mass of the smaller sphere decreases in a cubical manner
and this cubical decrease in mass combined with r squared in the denominator of the gravitational
field strength equation results in a linear decrease in the gravitational field strength.
No worries if you didn't fully get this explanation. The key thing to remember is the
linear decrease in gravitational field strength as we go from the surface towards the center
of a solid sphere. The gravitational field strength versus distance graph for a solid sphere
often comes up on ibophysics exams. This is how the graph looks like. Since gravitational field
strength is the negative of the potential gradient which in calculus terms means that it is the
negative derivative of the potential versus distance graph we must integrate the graph
that we see here in order to get the potential versus distance graph. Therefore the gravitational
potential versus distance graph looks like this. Once again no worries if you didn't fully get
this reasoning because it is not part of the ibophysics syllabus and the gravitational potential
versus distance graph for inside a solid sphere practically does not appear on ibophysics exams.
Let's summarize what we have learned in this video. We started by discussing the electric field
and the electric potential inside a hollow sphere. First we established that outside a charged
conducting sphere the field is the same as around a point charge and then we learned that the electric
field strength inside a sphere is zero. We added that the potential inside a sphere is constant
and that it is equal to the potential on the surface of the sphere. We concluded by examining
the electric field strength versus distance and the electric potential versus distance graphs.
Next we moved on to the gravitational field and gravitational potential inside a solid sphere
where similarly to an electric field around a sphere the gravitational field looks the same
as around a point mass. The key point that we discovered here was that gravitational field
strength decreases linearly as we approach the center of the sphere from the surface of the sphere.
Therefore the gravitational field strength versus distance graph looks like this.
We wrapped up by saying that using calculus and integration we can find the potential versus
distance graph which looks like this. A quick reminder that this final graph rarely comes up
on IB physics exams. This completes our discussion of field and potential inside a sphere.
In the next video we'll learn about orbiting and escaping a planet.