184_notes:pc_energy

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184_notes:pc_energy [2021/01/29 20:38] – [Relating Energy Back to Potential] bartonmo184_notes:pc_energy [2024/01/22 22:26] (current) – [Deriving Electric Potential Energy for Two Point Charges] tdeyoung
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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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 ====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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