Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
184_notes:pc_efield [2018/05/17 16:23] – [The Electric Field Equation for a Point Charge] curdemma | 184_notes:pc_efield [2021/05/26 13:39] (current) – schram45 | ||
---|---|---|---|
Line 2: | Line 2: | ||
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 ===== | ||
Line 19: | Line 19: | ||
==== 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 |
Line 30: | Line 30: | ||
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: |
- | * **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 charge to the location | + | * **Separation |
* **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, | ||
+ | |||
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, | + | 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, |
==== Electric Field Vectors ==== | ==== Electric Field Vectors ==== | ||
- | {{ 184_notes:efieldvectora.png? | + | [{{ :184_notes:efieldvectora_new.png?250|Electric Field at observation point A from Q}}] |
- | 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 $\vec{r_A}$, |
$$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$ | $$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$ | ||
$$r_A=d$$ | $$r_A=d$$ | ||
Line 64: | Line 65: | ||
$$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{d^2} \hat{y}$$ | $$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{d^2} \hat{y}$$ | ||
- | {{ 184_notes:efieldvectors.png?200}} | + | [{{ :184_notes:efieldvectors_new.png?250|Electric Field from a point charge}}] |
So we draw the electric field vector at Point A pointing straight up. If you follow the same steps for Points B-D, you find an important pattern from drawing this electric field vectors: **the electric field from a positive point charge points away from the charge**. If we were to look at points that were a distance of 2d away from the point charge, we would need to change the magnitude of the electric field by a factor of 4 (since it is $r^2$ in the denominator), | So we draw the electric field vector at Point A pointing straight up. If you follow the same steps for Points B-D, you find an important pattern from drawing this electric field vectors: **the electric field from a positive point charge points away from the charge**. If we were to look at points that were a distance of 2d away from the point charge, we would need to change the magnitude of the electric field by a factor of 4 (since it is $r^2$ in the denominator), | ||
==== Examples ==== | ==== Examples ==== | ||
- | [[184_notes: | + | * [[184_notes: |
+ | * Video Example: Electric Field from a Negative Point Charge | ||
+ | {{youtube> |