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184_notes:examples:week3_particle_in_field [2018/01/24 23:18] – [Solution] tallpaul | 184_notes:examples:week3_particle_in_field [2018/05/24 14:59] – [Example: Particle Acceleration through an Electric Field] curdemma | ||
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- | =====Particle Acceleration through an Electric Field===== | + | [[184_notes: |
+ | |||
+ | =====Example: | ||
Suppose you have a particle with a mass $m$, charge $Q$, initially at rest in an electric field $\vec{E}=E_0\hat{x}$. The electric field extends for a distance $L$ in the $+\hat{x}$-direction before dropping off abruptly to 0. So, the magnitude of the electric field is exactly $E_0$, and then exactly $0$. There is no in-between. What happens to the particle if $Q>0$? What if $Q<0$? What if $Q=0$? In one of these cases, the particle exits the electric field. What is its velocity when it reaches the region of zero electric field? | Suppose you have a particle with a mass $m$, charge $Q$, initially at rest in an electric field $\vec{E}=E_0\hat{x}$. The electric field extends for a distance $L$ in the $+\hat{x}$-direction before dropping off abruptly to 0. So, the magnitude of the electric field is exactly $E_0$, and then exactly $0$. There is no in-between. What happens to the particle if $Q>0$? What if $Q<0$? What if $Q=0$? In one of these cases, the particle exits the electric field. What is its velocity when it reaches the region of zero electric field? | ||
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* It has charge $Q$, which can be positive or negative or zero. | * It has charge $Q$, which can be positive or negative or zero. | ||
* The particle is a distance $L$ from the boundary of the electric field. | * The particle is a distance $L$ from the boundary of the electric field. | ||
- | * We can write the change in electric potential energy (from an initial location " | + | * We can write the change in electric potential energy (from an initial location "$i$" to a final location "$f$") for a point charge two ways here: |
\begin{align*} | \begin{align*} | ||
\Delta U &= -\int_i^f\vec{F}\bullet d\vec{r} &&&&&& | \Delta U &= -\int_i^f\vec{F}\bullet d\vec{r} &&&&&& | ||
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We have listed two expressions (Equations (1) and (2)) for the electric potential energy of the particle. You can check for yourself that both lead us to the following equation (using Equations (3) and (4)): $$\Delta U=-q\int_i^f \vec{E}\bullet d\vec{r}$$ | We have listed two expressions (Equations (1) and (2)) for the electric potential energy of the particle. You can check for yourself that both lead us to the following equation (using Equations (3) and (4)): $$\Delta U=-q\int_i^f \vec{E}\bullet d\vec{r}$$ | ||
- | We choose the path of integration to start at the current location of the particle, and end at place where the particle exits the electric field, as outlined in the Plan. The particle will follow a straight line in the $+x$-direction (when $Q>0$, as explained earlier). So we can set $\text{d}\vec{r}=\hat{x}dx$. We also need to define the path endpoints, so we'll say the particle starts at $x_i=x_0$, which means the integration will end at $x_f=x_0+L$. You'll see that it won't matter in the end what $x_0$ is, since it drops out. In fact, it is common to set $x_0=0$, for simplicity of calculation. In this example, we'll leave it as $x_0$. | + | We choose the path of integration to start at the current location of the particle, and end at place where the particle exits the electric field, as outlined in the Plan. The particle will follow a straight line in the $+x$-direction (when $Q>0$, as explained earlier). So we can set $\text{d}\vec{r}=\text{d}x \cdot \hat{x}$. We also need to define the path endpoints, so we'll say the particle starts at $x_i=x_0$, which means the integration will end at $x_f=x_0+L$. You'll see that it won't matter in the end what $x_0$ is, since it drops out. In fact, it is common to set $x_0=0$, for simplicity of calculation. In this example, we'll leave it as $x_0$. |
\begin{align*} | \begin{align*} | ||
- | \Delta U &= -q\int_i^f \vec{E}\bullet d\vec{r} \\ | + | \Delta U &= -q\int_i^f \vec{E}\bullet |
- | & | + | & |
- | & | + | & |
& | & | ||
& | & | ||
& | & | ||
\end{align*} | \end{align*} | ||
- | The physical significance of this result is that the particle " | + | The physical significance of this result is that the particle " |
Remember that $\vec{v}_i=0$, | Remember that $\vec{v}_i=0$, | ||
\begin{align*} | \begin{align*} | ||
- | & 0=-QE_0L+\frac{1}{2}M(v_f^2-0) \\ | + | & 0=-QE_0L+\frac{1}{2}m(v_f^2-0) \\ |
- | \Rightarrow \text{ } & QE_0L=\frac{1}{2}Mv_f^2 \\ | + | \Rightarrow \text{ } & QE_0L=\frac{1}{2}mv_f^2 \\ |
- | \Rightarrow \text{ } & \frac{2QE_0L}{M}=v_f^2 | + | \Rightarrow \text{ } & \frac{2QE_0L}{m}=v_f^2 |
\end{align*} | \end{align*} | ||
We reasoned earlier that the particle will be traveling in the $+x$-direction, | We reasoned earlier that the particle will be traveling in the $+x$-direction, | ||
- | $$\vec{v}_f=\sqrt{\frac{2QE_0L}{M}}\hat{x}$$ | + | $$\vec{v}_f=\sqrt{\frac{2QE_0L}{m}}\hat{x}$$ |