184_notes:dist_charges

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Over the last three pages of notes, we have talked about how we use superposition to find the electric field or electric potential from a line of charge, how you set up the dQ and the $\vec{r}$, and how to use those steps in a specific example. For this class, we will expect you to be able to set up these kinds of integrals for a line a charge (1D), but we will not go into the mathematics for 2D or 3D distributions of charge. Even though we won't go into the integral set up or analytical derivation of these fields, it is useful to have an idea of what the electric field would look like around some of these shapes. These notes will show what the electric field looks like for two common shapes of charge (spheres and cylinders) and talk about what would change if the shape was made of an insulator or a conductor.

In the first week of notes, we talked about what it means for an object to be an insulator or conductor. When we make a point charge assumption, these properties don't matter because all of the charge is assumed to be at a single location (whether or not the object is a conductor or an insulator). However, when talking about shapes and distributions of charge, whether the object is an insulator or conductor can significantly change what the electric field looks like. We'll start by considering the electric field for a conducting sphere (remember that a conductor is a material where the excess charges are free to move) and then consider an insulating sphere (where the excess charges are not free to move).

Conducting Sphere of Charge

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For the sake of illustration, let's say that we placed a collection of electrons in the center of a conducting, metal ball so that the ball has an overall net negative charge. What would you expect to happen to the charges in the ball? The electrons would feel a strong repulsion force from one another because they all have a negative charge (“like” charges repel). Since the ball is made of metal, the electrons can easily move through the material, so we would expect the electrons to move as far away from one another as possible.

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This means that the electrons will move to the surface of the metal ball and will evenly distribute around the surface, so that they have the maximum space possible between the electrons. Now that we know where the electrons are located, we can say what the electric field would look like around the metal ball.

Outside the metal ball, we would hypothesize that the electric field should point in towards the metal ball since the electric field points in toward a negative point charge. If you actually do the math (either with an integral over the volume of the sphere or with a computational code), you will see exactly this. The electric field will point radially towards the metal ball and get stronger the closer you are to the ball. In fact, if you are looking for the electric field outside the metal ball, it will look exactly the same as if there were a point charge (with the same net negative charge) at the center of the ball. Thus, outside a conducting sphere we can still use the equation: $$\vec{E}_{outside}=\frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\hat{r}$$

However, what would happen to the electric field inside the sphere? We know that all of the charges should be located on the surface of the metal ball. When inside the sphere, there will always be a contribution to the electric field from the electrons on one side of the sphere that opposes the electric field contribution from the electrons on the other side of sphere. This means that on the inside, the electric field from the electrons on the surface perfectly cancels out, leaving a net field of zero. $$\vec{E}_{inside}=0$$ This is actually the primary idea behind shielding sensitive electronics (also referred to as a Faraday Cage). If you surround something with a sheet of charged metal, then you can guarantee that there will be no electric field on the inside.

Insulating Sphere of Charge

If instead we have an insulating, plastic sphere (rather than a metal, conducting one), we would see a very different charge distribution. In an insulator, excess charges cannot move freely and are stuck where they were placed. Thus, if we place a collection of electrons inside the ball, they will stay distributed through the volume of the sphere rather moving to the surface. For the purposes of our class, we will assume that any charge on an insulator will be evenly distributed - in future courses you may talk about what would happen if there was an uneven distribution of charge.

In constrast to conductors, insulators are materials where the excess charges are NOT free to move - they stay in place and cannot move through the material. We will talk about how the electric field changes for a sphere and cylinder made of insulating material rather than a conducting material.

Insulating Sphere of Charge

Insulating Cylinder of Charge

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