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This video is about the field and the potential inside a sphere.

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We'll start by looking at the electric field and electric potential inside a hollow sphere,

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and then move on to discuss gravitational field and gravitational potential inside a solid sphere.

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Let's begin with electric field and electric potential.

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When they're charged, conducting sphere, the field looks exactly the same as around a point charge.

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The reason is that the electric field lines are perpendicular to the surface of the sphere,

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therefore the field around the sphere is a radial field, just as around a point charge.

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The first important point, when looking at the field and the potential inside the hollow sphere,

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is that all the surplus charge is distributed on the outside of the sphere.

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As a result, when we place a test charge inside a sphere, the forces acting on this test charge

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completely cancel each other out. We can use the equations that we learned for electric fields,

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along with some trigonometry to show this, but this mathematical reasoning is not part of the

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IB physics course, so we won't go into its details here. Since the force acting on a test charge

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inside a sphere is zero, the electric field strength inside a sphere is also zero.

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Earlier in this topic we learned that electric field strength is equal to the negative of the

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potential gradient. Since the electric field strength inside a sphere is zero, the potential

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gradient is also zero. If we think about the potential gradient and the potential in terms of

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calculus, we can conclude that when the potential gradient is zero, the potential must be constant.

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Therefore the potential inside a sphere is constant and it is equal to the potential

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on the surface of the sphere. These concepts often come up on IB physics exams in terms of graphs.

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So based on what we discussed, we can draw the electric field strength versus distance graph

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for the sphere, which looks like this. Lowercase r is the distance from the center of the sphere,

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capital R is the radius of the sphere and E is the electric field strength.

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As you can see, electric field strength inside a sphere is zero, while outside the sphere

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we have a 1 over r squared graph just as we have seen for a point charge.

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The potential versus distance graph for the sphere looks like this.

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Inside a sphere the potential is constant, while outside the sphere we see a 1 over r

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graph just as we did for a point charge. Next let's examine the gravitational field

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and the gravitational potential inside a solid sphere. For gravitational field and

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gravitational potential we consider a solid sphere because in this context we often work with planets

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that can be considered as solid uniform spheres. Let's assume that we have a solid sphere for

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example the earth with a radius of uppercase r. Now imagine a point inside a sphere at a distance

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x from the center of the sphere. You can picture that we have dug a tunnel from the surface of the

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earth towards its center and we have gone down this tunnel to a certain point that is at a distance

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x from the center of the earth. So at this point basically we would be standing on the surface

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of a smaller sphere or smaller earth that has a radius of x and above us we would have a spherical

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shell. Let's think about the gravitational field strength at this point. There are two possible

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sources for the gravitational field strength here. One is the spherical shell above us

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and the other is the smaller sphere below us. The gravitational field strength due to the

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spherical shell is zero. The reason is similar to what we explained about the electric field strength

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inside a hollow sphere earlier in this video. The gravitational field strength at our chosen

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point due to the smaller sphere of radius x is less than the gravitational field strength

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at the surface of the solid sphere with radius r. Put more concisely as we get closer and closer

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to the center of the original solid sphere of radius r or in our example towards the center of

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the earth gravitational field strength decreases. This decrease is linear. The reason for this

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linear decrease is that as we are decreasing the radius of the smaller inner sphere so as the value

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of x gets smaller and smaller the mass of the smaller sphere decreases in a cubical manner

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and this cubical decrease in mass combined with r squared in the denominator of the gravitational

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field strength equation results in a linear decrease in the gravitational field strength.

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No worries if you didn't fully get this explanation. The key thing to remember is the

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linear decrease in gravitational field strength as we go from the surface towards the center

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of a solid sphere. The gravitational field strength versus distance graph for a solid sphere

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often comes up on ibophysics exams. This is how the graph looks like. Since gravitational field

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strength is the negative of the potential gradient which in calculus terms means that it is the

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negative derivative of the potential versus distance graph we must integrate the graph

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that we see here in order to get the potential versus distance graph. Therefore the gravitational

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potential versus distance graph looks like this. Once again no worries if you didn't fully get

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this reasoning because it is not part of the ibophysics syllabus and the gravitational potential

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versus distance graph for inside a solid sphere practically does not appear on ibophysics exams.

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Let's summarize what we have learned in this video. We started by discussing the electric field

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and the electric potential inside a hollow sphere. First we established that outside a charged

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conducting sphere the field is the same as around a point charge and then we learned that the electric

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field strength inside a sphere is zero. We added that the potential inside a sphere is constant

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and that it is equal to the potential on the surface of the sphere. We concluded by examining

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the electric field strength versus distance and the electric potential versus distance graphs.

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Next we moved on to the gravitational field and gravitational potential inside a solid sphere

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where similarly to an electric field around a sphere the gravitational field looks the same

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as around a point mass. The key point that we discovered here was that gravitational field

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strength decreases linearly as we approach the center of the sphere from the surface of the sphere.

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Therefore the gravitational field strength versus distance graph looks like this.

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We wrapped up by saying that using calculus and integration we can find the potential versus

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distance graph which looks like this. A quick reminder that this final graph rarely comes up

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on IB physics exams. This completes our discussion of field and potential inside a sphere.

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In the next video we'll learn about orbiting and escaping a planet.