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===== Distributions of Charges ===== | ===== Distributions of Charges ===== | ||
- | Over the last three pages of notes, we have talked about [[184_notes: | + | Over the last set of notes, we have talked about [[184_notes: |
==== Sphere of Charge ==== | ==== Sphere of Charge ==== | ||
In the [[184_notes: | In the [[184_notes: | ||
- | === Conducting Sphere of Charge === | + | ==== Conducting Sphere of Charge ==== |
- | FIXME Add Figure | + | |
- | For the sake of illustration, | + | [{{ 184_notes: |
- | + | For the sake of illustration, | |
- | FIXME Add Figure | + | |
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. | 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 | + | Now, in practice, it need not be the electrons we placed in the metal the move to the surface of the metal. In fact, it isn't those electrons that move large distances. The entire electron distribution in the metal shifts outward and those that are already near the edge of the material are the ones that produce the effect. The net result is the same whether we model the metal as having the excess electrons move to the edge or we recognize it's a shift of all the electrons in the material including the newly introduced ones. |
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+ | [{{ 184_notes: | ||
+ | Outside the metal ball, we would hypothesize that the electric field should point in towards the metal ball since the electric field points 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: | ||
$$\vec{E}_{outside}=\frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\hat{r}$$ | $$\vec{E}_{outside}=\frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\hat{r}$$ | ||
where $Q$ is the total charge on the ball and $\vec{r}$ points from the center of the sphere to the observation location. | where $Q$ is the total charge on the ball and $\vec{r}$ points from the center of the sphere to the observation location. | ||
- | 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. | + | 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. |
$$\vec{E}_{inside}=0$$ | $$\vec{E}_{inside}=0$$ | ||
- | This is actually the primary idea behind shielding sensitive electronics (also referred to as a [[https:// | + | This is actually the primary idea behind shielding sensitive electronics (also referred to as a [[https:// |
- | === Insulating Sphere of Charge === | + | ==== Insulating Sphere of Charge ==== |
+ | [{{ 184_notes: | ||
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__// | 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__// | ||
- | Now that we know where the charges are located, we can think about what the electric field should look like around the plastic charged ball. Similar to the metal ball, we would expect the electric field to point in towards the plastic ball since the electric field points in toward a negative point charge. If you actually do the math (again either with an integral over the volume of the sphere or with a computational code), you will exactly the same electric field outside the plastic ball as you would if the ball (with the same amount of charge) were metal: | + | [{{ 184_notes: |
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+ | Now that we know where the charges are located, we can think about what the electric field should look like around the plastic charged ball. Similar to the metal ball, at any point outside of the plastic ball we would expect the electric field to point in towards the plastic ball since the electric field points in toward a negative point charge. If you actually do the math (again either with an integral over the volume of the sphere or with a computational code), you will exactly the same electric field outside the plastic ball as you would if the ball (with the same amount of charge) were metal: | ||
$$\vec{E}_{outside}=\frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\hat{r}$$ | $$\vec{E}_{outside}=\frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\hat{r}$$ | ||
where $Q$ is the total charge on the ball and $\vec{r}$ points from the center of the sphere to the observation location. | where $Q$ is the total charge on the ball and $\vec{r}$ points from the center of the sphere to the observation location. | ||
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==== Cylinders of Charge ==== | ==== Cylinders of Charge ==== | ||
+ | [{{ 184_notes: | ||
Another common example that we will frequently talk about is the idea of a charged cylinder. This will become particularly relevant when we start talking about wires in circuits next week. For now, we'll talk about the shape of the electric field for a conducting and an insulating cylinder of charge. | Another common example that we will frequently talk about is the idea of a charged cylinder. This will become particularly relevant when we start talking about wires in circuits next week. For now, we'll talk about the shape of the electric field for a conducting and an insulating cylinder of charge. | ||
- | === Conducting Cylinder of Charge === | + | ==== Conducting Cylinder of Charge ==== |
- | Much like what happened with the metal sphere, if we place an excess charge - let's say it's positive this time - on a metal cylinder (like a wire), those charges will spread out as far as they possibly can from one another (" | + | |
+ | Much like what happened with the metal sphere, if we place an excess charge - let's say it's positive this time - on a metal cylinder (like a wire), those charges will spread out as far as they possibly can from one another (" | ||
+ | [{{ 184_notes: | ||
If we think about the electric field inside the cylinder, we would see a similar effect as we did with the metal ball. There will always be a contribution to the electric field from the charges on one side of the cylinder that opposes the electric field contribution from the charges on the other side of cylinder. This means that: | If we think about the electric field inside the cylinder, we would see a similar effect as we did with the metal ball. There will always be a contribution to the electric field from the charges on one side of the cylinder that opposes the electric field contribution from the charges on the other side of cylinder. This means that: | ||
$$E_{inside} = 0$$ | $$E_{inside} = 0$$ | ||
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Outside the cylinder, we would expect the electric field to generally point away from the positively charged cylinder since the electric field points in away from a positive charge. If you actually do the math (either with an integral over the volume of the cylinder or with a computational code), you will see something like this. Particularly, | Outside the cylinder, we would expect the electric field to generally point away from the positively charged cylinder since the electric field points in away from a positive charge. If you actually do the math (either with an integral over the volume of the cylinder or with a computational code), you will see something like this. Particularly, | ||
- | === Insulating Cylinder of Charge === | + | |
- | For an insulating cylinder (like a plastic pipe) if we add excess charges, those will stay in place (as charges are not free to move in an insulator). We will again make the assumption in these cases of __//a uniform charge | + | ==== Insulating Cylinder of Charge ==== |
+ | [{{ 184_notes: | ||
+ | For an insulating cylinder (like a plastic pipe) if we add excess charges, those will stay in place (as charges are not free to move in an insulator). We will again // | ||
Just like with the sphere, if we are outside the cylinder, the electric field will look the same as if the cylinder were a conductor (as long as it has the same amount of charge). Namely that it points radially away from the positively charged cylinder, expect near the ends of the cylinder. | Just like with the sphere, if we are outside the cylinder, the electric field will look the same as if the cylinder were a conductor (as long as it has the same amount of charge). Namely that it points radially away from the positively charged cylinder, expect near the ends of the cylinder. | ||
Inside the insulating cylinder, the electric field would be non-zero. The contributions to the electric field from the charges in the middle of the cylinder do not completely cancel out, leaving an electric field that points away from the center of the cylinder for a positive charge (it would point towards the center for a negative charge). The electric field inside will be strongest at the edge of the cylinder and will be smallest (or exactly zero) in the center. | Inside the insulating cylinder, the electric field would be non-zero. The contributions to the electric field from the charges in the middle of the cylinder do not completely cancel out, leaving an electric field that points away from the center of the cylinder for a positive charge (it would point towards the center for a negative charge). The electric field inside will be strongest at the edge of the cylinder and will be smallest (or exactly zero) in the center. | ||
+ | [{{184_notes: |