diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 35ee429..f908718 100644 --- a/docs/1b/ece106.md +++ b/docs/1b/ece106.md @@ -317,13 +317,13 @@ At a point $P$, the electrostatic potential $V_p$ or voltage is the work done pe $$ V_p=\lim_{q\to 0^+}\frac{W_i}{q} \\ -W_I=\int^p_\infty\vec F_I\bullet \vec{dl} +W_I=\int^p_\infty\vec F_I\bullet \vec{dl}=\Delta U=QV_p $$ Because the desired force acts opposite to the force from the electric field, as long as $\vec E$ is known at each point: $$ -V_p=-\int^p_\infty\vec E\bullet\vec{dl} +V_p=-\int^p_\infty\vec E\bullet\vec{dl} \\ V_p=-\int^p_\infty E\ dr $$ @@ -337,7 +337,7 @@ Therefore, the potential due to a point charge is equal to: $$V_p=-\int^p_\infty\frac{kQ}{r^2}dr=\frac{kQ}{r}$$ -**Positive** charges naturally move to **lower** potentials ($V$ decreases) while negative charges do the opposite. +**Positive** charges naturally move to **lower** potentials ($V$ decreases) while negative charges do the opposite. Potential energy always decreases. In order to calculate the voltage for charge distributions: @@ -348,3 +348,11 @@ $$V_p=-\int^p_\infty\vec E\bullet\vec{dl}$$ - If the charge is asymmetric: $$V_p=\int_\text{charge dist}\frac{kdQ}{r}$$ + +The electric field always points in the direction of **lower** potential, and is equal to the **negative gradient** of potential. + +$$\vec E=-\nabla V$$ + +If $\vec E$ is constant: + +$$\vec E=\frac{Q_{enc\ net}}{\epsilon_0\oint dS}$$