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184_notes:pc_efield [2018/05/24 13:49] – [Electric Field vs Distance Graph] dmcpadden | 184_notes:pc_efield [2021/01/26 18:27] – [Electric Field Vectors] bartonmo | ||
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Sections 13.1 - 13.4 of Matter and Interactions (4th edition) | Sections 13.1 - 13.4 of Matter and Interactions (4th edition) | ||
- | [[184_notes: | + | /*[[184_notes: |
- | [[184_notes: | + | [[184_notes: |
===== Electric Field of a Point Charge ===== | ===== Electric Field of a Point Charge ===== | ||
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==== What is an electric field? ==== | ==== What is an electric field? ==== | ||
- | The **electric field** is a //vector// field (it has a magnitude and direction) that is generated by electric charges at every point in space. That is, **charges generate electric field**. As you will read soon, it can be [[184_notes: | + | The **electric field** is a //vector// field (it has a magnitude and direction) that is generated by electric charges at every point in space. That is, **charges generate |
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This equation has several pieces to it: | This equation has several pieces to it: | ||
- | * **Charge** - the variable q represents the charge that is creating the electric field. (The letter C is already used for the unit of coulombs and for capacitance so we use q or Q as the variable for charge.) q is a scalar number but can be positive or negative depending on the sign of the charge. q is also directly proportional to the electric field, so this tells us that the more charge you have, the bigger the electric field will be. | + | * **Charge** - the variable q represents the charge that is creating the electric field. (The letter C is already used for the unit of coulombs and for capacitance so we use q or Q as the variable for charge.) |
* **Constant** - the first part of this equation $\frac{1}{4\pi\epsilon_0}$ is a constant value, where $\epsilon_0 = 8.85 \cdot 10^{-12} \frac{C^2}{N m^2}$. This constant plays a similar role for electricity as the gravitational constant does for gravity. In other texts or previous classes, you may have seen this constant as k or Coulomb' | * **Constant** - the first part of this equation $\frac{1}{4\pi\epsilon_0}$ is a constant value, where $\epsilon_0 = 8.85 \cdot 10^{-12} \frac{C^2}{N m^2}$. This constant plays a similar role for electricity as the gravitational constant does for gravity. In other texts or previous classes, you may have seen this constant as k or Coulomb' | ||
[{{ 184_notes: | [{{ 184_notes: | ||
- | * **Separation Distance** - r is the scalar distance between the point charge and the location at which you want to know the electric field (Point P). In the equation, r is inversely related to electric field and squared, which means if you want the field at half the distance, the electric field is four times as large. You can find this distance using the **separation vector** $\vec{r}$ that points from the source charge to the observation location (Point P in this case); r is simply the magnitude of the separation vector $\vec{r}$. Mathematically you can calculate r from the positions of the charge and location of interest using: $$r=|\vec{r}|=|\vec{r}_{obs}-\vec{r}_{source}|=|\vec{r_{p}}-\vec{r_{q}}|$$ | + | * **Separation Distance** - r is the scalar distance between the point charge and the location at which you want to know the electric field (Point P). In the equation, r is squared and inversely related to electric field, which means if you want the field at half the distance, the electric field is four times as large. You can find this distance using the **separation vector** $\vec{r}$ that points from the source charge to the observation location (Point P in this case); r is simply the magnitude of the separation vector $\vec{r}$. Mathematically you can calculate r from the positions of the charge and location of interest using: $$r=|\vec{r}|=|\vec{r}_{obs}-\vec{r}_{source}|=|\vec{r_{p}}-\vec{r_{q}}|$$ |
* **Direction** - **Electric field is vector quantity**, so it has a direction associated with it, given by the $\hat{r}$ in the equation. $\hat{r}$ is the unit vector that points in the same direction as the separation vector $\vec{r}$ (from the charge to the location of interest) but has a magnitude of 1. You can calculate $\hat{r}$ by dividing $\vec{r}$ by its magnitude (definition of a unit vector): $\hat{r}=\frac{\vec{r}}{r}$. Alternatively, | * **Direction** - **Electric field is vector quantity**, so it has a direction associated with it, given by the $\hat{r}$ in the equation. $\hat{r}$ is the unit vector that points in the same direction as the separation vector $\vec{r}$ (from the charge to the location of interest) but has a magnitude of 1. You can calculate $\hat{r}$ by dividing $\vec{r}$ by its magnitude (definition of a unit vector): $\hat{r}=\frac{\vec{r}}{r}$. Alternatively, | ||
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Together, these pieces tell you how the electric field from a point charge changes in space. The main take away here is: **the electric field is created by a charge and can be calculated for every point in space around the charge.** | Together, these pieces tell you how the electric field from a point charge changes in space. The main take away here is: **the electric field is created by a charge and can be calculated for every point in space around the charge.** | ||
=== Units of Electric Field === | === Units of Electric Field === | ||
- | The units of electric field can either be given in $N/C$ (newtons per coulomb) or in $V/m$ (volts per meter). These units are equivalent, so it doesn' | + | The units of electric field can either be given in** $N/C$ (newtons per coulomb) or in $V/m$ (volts per meter)**. These units are equivalent, so it doesn' |
==== Electric Field vs Distance Graph ==== | ==== Electric Field vs Distance Graph ==== | ||
[{{ 184_notes: | [{{ 184_notes: | ||
- | Because electric field is a vector quantity, it can be difficult to graph. Generally we get around this by choosing a path and graphing the magnitude of the electric field vs distance //along that path// (the other option being to make three graphs, one for each vector component of the field). For any path moving away from a charge, we graph the magnitude of $\vec{E}$ vs the position away from the charge ( r ). In this case, $|\vec{E}|=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$ because the separation between the point charge and where we are calculating the field is given by r along our path. This gives us a graph that is very similar to a $y=\frac{1}{x^2}$ graph - This means that as you get closer to the charge, the magnitude of the electric field gets incrementally bigger. Eventually, the electric field gets infinitely large if you get extremely close to the charge. Likewise, the electric field rapidly gets smaller as you move away from the charge, but doesn' | + | Because electric field is a vector quantity, it can be difficult to graph. Generally, we get around this by choosing a path and graphing the magnitude of the electric field vs distance //along that path// (the other option being to make three graphs, one for each vector component of the field). For any path moving away from a charge, we graph the magnitude of $\vec{E}$ vs the position away from the charge ( r ). In this case, $|\vec{E}|=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$ because the separation between the point charge and where we are calculating the field is given by r along our path. This gives us a graph that is very similar to a $y=\frac{1}{x^2}$ graph, which shows that as you get closer to the charge, the magnitude of the electric field gets incrementally bigger. Eventually, the electric field gets infinitely large if you get extremely close to the charge. Likewise, the electric field rapidly gets smaller as you move away from the charge, but doesn' |
==== Electric Field Vectors ==== | ==== Electric Field Vectors ==== | ||
- | [{{ 184_notes:efieldvectora.png? | + | [{{ :184_notes:efieldvectora_new.png? |
- | To understand the electric field around a point charge (or any other distribution of charge), we will often draw vectors around the charge called " | + | To understand the electric field around a point charge (or any other distribution of charge), we will often draw vectors around the charge called " |
- | For a positive point charge Q, consider Points A-D, each a distance d (shown in red) from the charge. To draw the electric field vectors around this charge, we need to find the magnitude **//and//** direction of the electric field at each point. Starting with the electric field equation, we can find the electric field for Point A. We already know that the charge is Q, so we have: | + | For a positive point charge Q, consider Points A-D, each a distance d from the charge. To draw the electric field vectors around this charge, we need to find the magnitude **//and//** direction of the electric field at each point. Starting with the electric field equation, we can find the electric field for Point A. We already know that the charge is Q, so we have: |
$$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{r_A^2} \hat{r_A}$$ | $$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{r_A^2} \hat{r_A}$$ | ||
- | All we need now is to find the separation vector $\vec{r_A}$, | + | All we need now is to find the separation vector $\vec{r_A}$, |
$$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$ | $$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$ | ||
$$r_A=d$$ | $$r_A=d$$ |