184_notes:pc_energy

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184_notes:pc_energy [2018/06/25 18:44] – dmcpadden184_notes:pc_energy [2024/01/22 22:26] (current) – [Deriving Electric Potential Energy for Two Point Charges] tdeyoung
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 Sections 6.9 and 16.1 - 16.3 in Matter and Interactions (4th edition) Sections 6.9 and 16.1 - 16.3 in Matter and Interactions (4th edition)
  
-[[184_notes:pc_vefu|Next Page: Relating Electric Force, Electric Field, Electric Potential, and Electric Potential Energy]]+/*[[184_notes:pc_vefu|Next Page: Relating Electric Force, Electric Field, Electric Potential, and Electric Potential Energy]]
  
-[[184_notes:pc_force|Previous Page: Electric Force]]+[[184_notes:pc_force|Previous Page: Electric Force]]*/
  
 ===== Electric Potential Energy ===== ===== Electric Potential Energy =====
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 There are a few important features of this relationship: There are a few important features of this relationship:
-  * Energy is scalar (including electric energy), so to get from the vector force we must use the dot product with displacement to get energy. Dot products produce scalar quantities from two vector quantities. +  * Energy is scalar (including electric energy), so to get from the vector force we must use the [[184_notes:math_review#Vector_Multiplication|dot product]] with displacement to get energy. Dot products produce scalar quantities from two vector quantities. 
   * The electric force is **not constant**, it usually depends on $r$. This means we have to use the integral rather than just multiplying by the distance.   * The electric force is **not constant**, it usually depends on $r$. This means we have to use the integral rather than just multiplying by the distance.
   * If we are integrating from some initial location to some final location, we will get a **change** in electric potential energy between the two locations (not the energy at a single place). This change in energy is the important part - it does not matter how you get from the initial to the final location (squiggly or straight path), you will get the same change in energy. This means that electric potential energy is **path independent**.    * If we are integrating from some initial location to some final location, we will get a **change** in electric potential energy between the two locations (not the energy at a single place). This change in energy is the important part - it does not matter how you get from the initial to the final location (squiggly or straight path), you will get the same change in energy. This means that electric potential energy is **path independent**. 
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   * The units of electric potential energy are joules (J) just like all the other forms of energy.   * The units of electric potential energy are joules (J) just like all the other forms of energy.
  
-=== Deriving Electric Potential Energy for Two Point Charges ===+==== Deriving Electric Potential Energy for Two Point Charges ====
 [{{  184_notes:twocharges.png?300|Two charges are initially separated by $r_i$. After some time they are separated by $r_f$.}}] [{{  184_notes:twocharges.png?300|Two charges are initially separated by $r_i$. After some time they are separated by $r_f$.}}]
  
-Using the relationship between force and potential energy, we can derive the electric potential energy between two point charges from the electric force. Suppose we have two positive point charges $q_1$ and $q_2$, who are initially separated by a distance r. We will //__assume $q_1$ is fixed__// and let $q_2$ move to infinity. Starting with the general relationship:+Using the relationship between force and potential energy, we can derive the electric potential energy between two point charges from the electric force. Suppose we have two positive point charges $q_1$ and $q_2$, who are initially separated by a distance r. We will //__assume __//$q_1$//__ is fixed__// and let $q_2$ move to infinity. Starting with the general relationship:
  $$\Delta U_{elec} = U_f-U_i= -\int_i^f\vec{F}_{elec}\bullet d\vec{r}$$  $$\Delta U_{elec} = U_f-U_i= -\int_i^f\vec{F}_{elec}\bullet d\vec{r}$$
 we can plug in the electric force equation for the force from $q_1$ on $q_2$ (point charges), and we know that our initial location is $r_i=r$ and our final location is $r_f=\infty$. So we get: we can plug in the electric force equation for the force from $q_1$ on $q_2$ (point charges), and we know that our initial location is $r_i=r$ and our final location is $r_f=\infty$. So we get:
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 This energy then is the electric potential energy between two point charges $q_1$ and $q_2$ that are separated by a distance $r$. If $U$ is positive, $q_1$ and $q_2$ have the same sign and if $U$ is negative, $q_1$ and $q_2$ have opposite signs.  This energy then is the electric potential energy between two point charges $q_1$ and $q_2$ that are separated by a distance $r$. If $U$ is positive, $q_1$ and $q_2$ have the same sign and if $U$ is negative, $q_1$ and $q_2$ have opposite signs. 
  
-=== Getting from Energy to Force ===+==== Getting from Energy to Force ==== 
 We can also use the inverse of energy-force relationship to get the electric force from electric potential energy. If we know what the electric potential energy is in terms of $r$, you can calculate the electric force by taking the negative derivative of energy with respect to $r$, which will give you the electric force in the $\hat{r}$ direction. //__This assumes that your electric potential energy equation does not depend on an angle__//. (If your electric potential energy does depend on an angle, then you have to use the [[https://en.wikipedia.org/wiki/Gradient|gradient]].)  We can also use the inverse of energy-force relationship to get the electric force from electric potential energy. If we know what the electric potential energy is in terms of $r$, you can calculate the electric force by taking the negative derivative of energy with respect to $r$, which will give you the electric force in the $\hat{r}$ direction. //__This assumes that your electric potential energy equation does not depend on an angle__//. (If your electric potential energy does depend on an angle, then you have to use the [[https://en.wikipedia.org/wiki/Gradient|gradient]].) 
 $$\vec{F}=-\frac{dU}{dr}\hat{r}$$ $$\vec{F}=-\frac{dU}{dr}\hat{r}$$
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 $$\Delta U=q\Delta V$$ $$\Delta U=q\Delta V$$
  
-**Note that electric potential energy is NOT the same thing as electric potential**. Electric potential energy requires two charges or a charge interacting with potential, whereas electric potential is from a single charge. Electric potential energy has units of joules and electric potential has units of volts. That being said, electric potential is related to electric potential energy. Electric potential tells you about how much energy there could be, without needing to know charges are interacting.+**Note that electric potential energy is NOT the same thing as electric potential**. Electric potential energy requires two charges or a charge interacting with potential, whereas electric potential is from a single charge. **Electric potential energy has units of //joules// and electric potential has units of //volts//.** That being said, electric potential is related to electric potential energy. Electric potential tells you about how much energy there could be, without needing to know charges are interacting.
  
 ====Examples==== ====Examples====
-[[:184_notes:examples:Week3_particle_in_field|Particle Acceleration through an Electric Field]] +  * [[:184_notes:examples:Week3_particle_in_field|Particle Acceleration through an Electric Field]] 
- +    * Video Example: Particle Acceleration through an Electric Field 
-[[:184_notes:examples:Week3_spaceship_asteroid|Preventing an Asteroid Collision]]+  * [[:184_notes:examples:Week3_spaceship_asteroid|Preventing an Asteroid Collision]] 
 +    * Video Example: Preventing an Asteroid Collision 
 +{{youtube>_rghROIzNUk?large}} 
 +{{youtube>vf_b6k3iXeU?large}}
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