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184_notes:pc_efield [2018/05/24 13:48] – [The Electric Field Equation for a Point Charge] dmcpadden184_notes:pc_efield [2021/05/26 13:39] (current) – schram45
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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:pc_potential|Next Page: Electric Potential]]+/*[[184_notes:pc_potential|Next Page: Electric Potential]]
  
-[[184_notes:charging_discharging|Previous Page: Charging and Discharging]]+[[184_notes:charge|Previous Page: Electric Interaction]]*/
  
 ===== 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:pc_force|related to the electric force]] between two charges. To begin to understand the electric field, we will start by making a //__point particle assumption about our objects - this means that we take our object and crush it down to a single small point that has the same mass and charge as what our original object does but does not have a shape__// ([[183_notes:point_particle|we often make this assumption in mechanics as well]]). Using a point particle assumption allows us to create a model first with very simple objects (which we call "point particles" or "point charges") before adding more complexity to our model. +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 an electric field**. As you will read soon, it can be [[184_notes:pc_force|related to the electric force]] between two charges. To begin to understand the electric field, we will start by making a //__point particle assumption about our objects - this means that we take our object and crush it down to a single small point that has the same mass and charge as what our original object does but does not have a shape__// ([[183_notes:point_particle|we often make this assumption in mechanics as well]]). Using a point particle assumption allows us to create a model first with very simple objects (which we call "point particles" or "point charges") before adding more complexity to our model. 
  
  
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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.) //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.
   * **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's constant: $ \frac{1}{4\pi\epsilon_0}=k=8.99 \cdot 10^9 \frac{N m^2}{C^2}$.   * **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's constant: $ \frac{1}{4\pi\epsilon_0}=k=8.99 \cdot 10^9 \frac{N m^2}{C^2}$.
  
 [{{  184_notes:efieldq.png?300|Distance vectors for finding E-field at Point P from charge q}}] [{{  184_notes:efieldq.png?300|Distance vectors for finding E-field at Point P from charge 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 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, you can plug the unit vector definition directly into the electric field equation to save a step in your calculations. $$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^3}\vec{r}$$   * **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, you can plug the unit vector definition directly into the electric field equation to save a step in your calculations. $$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^3}\vec{r}$$
 +
  
 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't matter which is used. The size of an electric field is difficult to judge without other information about the charge that is creating that field. For example, a large electric field could be due to either a large amount of charge or being a short distance from the charge. Likewise a small electric field could be due to a very small amount of charge or being far from the charge. +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't matter which is used. The size of an electric field is difficult to judge without other information about the charge that is creating that field. For example, a large electric field could be due to either a large amount of charge or being a short distance from the charge. Likewise, a small electric field could be due to a very small amount of charge or being far from the charge. 
  
 ==== Electric Field vs Distance Graph ==== ==== Electric Field vs Distance Graph ====
-[{{  184_notes:fieldgraph.png?200|Field vs. Distance Graph}}] +[{{  184_notes:fieldgraph.png?200|Electric Field vs. Distance Graph}}] 
-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, for 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't actually reach zero until you are infinitely far away. +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't actually reach zero until you are infinitely far away. 
  
 ==== Electric Field Vectors ==== ==== Electric Field Vectors ====
  
-[{{  184_notes:efieldvectora.png?250|Electric Field at A from Q}}] +[{{ :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 "electric field vectors" or just "field vectors." **The magnitude or length of these vectors represents the magnitude of the electric field, and the direction of the vector points in the same direction as the electric field** (shown in blue in the figures).+To understand the electric field around a point charge (or any other distribution of charge), we will often draw vectors around the charge called "electric field vectors" or just "field vectors." **The magnitude or length of these vectors represents the magnitude of the electric field, and the direction of the vector points in the same direction as the electric field** (shown with the dashed blue $\vec{E_A}$ arrow in the figure on the right).
  
-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 $\vec{r_A}$, which will help us find the magnitude $r_A$ and unit vector $\hat{r_A}$. The separation vector $\vec{r_A}$ will point from the charge to Point A (drawn as the red arrow in the figure). Using the traditional cartesian coordinate system (+x points to the right, +y points up, the origin at the positive point charge), then $\vec{r_A}$ points only in the +y-direction, so $\vec{r_A} = \langle 0, d, 0 \rangle$. The magnitude of $\vec{r_A}$ is then:+All we need now is to find the separation vector $\vec{r_A}$, which will help us find the magnitude $r_A$ and unit vector $\hat{r_A}$. The separation vector $\vec{r_A}$ will point from the charge to Point A (drawn as the solid red arrow in the figure). Using the traditional cartesian coordinate system (+x points to the right, +y points up, the origin at the positive point charge), then $\vec{r_A}$ points only in the +y-direction, so $\vec{r_A} = \langle 0, d, 0 \rangle$. The magnitude of $\vec{r_A}$ is then:
 $$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$ $$r_A=|\vec{r_A}|=\sqrt{r_{Ax}^2+r_{Ay}^2+r_{Az}^2}=\sqrt{0^2+d^2+0^2}$$
 $$r_A=d$$  $$r_A=d$$ 
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 $$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{d^2} \hat{y}$$ $$\vec{E_A} = \frac{1}{4 \pi\epsilon_0}\frac{Q}{d^2} \hat{y}$$
  
-[{{  184_notes:efieldvectors.png?200|Electric Field from a point charge}}]+[{{ :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), but the directions would stay the same. 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), but the directions would stay the same.
  
 ==== Examples ==== ==== Examples ====
-[[184_notes:examples:Week2_electric_field_negative_point|Electric Field from a Negative Point Charge]]+  * [[184_notes:examples:Week2_electric_field_negative_point|Electric Field from a Negative Point Charge]] 
 +    * Video Example: Electric Field from a Negative Point Charge 
 +{{youtube>a64SCwLdIe0?large}}
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