diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index eeff21f..3c5adc3 100644 --- a/docs/1b/ece106.md +++ b/docs/1b/ece106.md @@ -310,3 +310,22 @@ Generally: 1. Determine $\vec E$ outside the slab. 2. Set one outside surface and one inside surface as a pillbox and apply rules. + +## Electrostatic potential + +At a point $P$, the electrostatic potential $V_p$ or voltage is the work done per unit positive test charge from infinity to bring it to point $P$ by an external agent. + +$$ +V_p=\lim_{q\to 0^+}\frac{W_i}{q} \\ +W_I=\int^p_\infty\vec F_I\bullet \vec{dl} +$$ + +Because the desired force acts opposite to the force from the electric field, as long as $\vec E$ is known at each point: + +$$V_p=-\int^p_\infty\vec E\bullet\vec{dl}$$ + +The work done only depends on initial and final positions — it is conservative, thus implying Kirchoff's voltage law. + +Where $\vec dl$ is the path of the test charge and $\vec dr$ is the direct path from infinity through the point to the charge, because $dr=|dl|\cos\theta$: + +$$\vec E\bullet\vec{dl}=Edr$$