diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 47a52f9..7382946 100644 --- a/docs/1b/ece106.md +++ b/docs/1b/ece106.md @@ -359,4 +359,53 @@ $$\vec E=\frac{Q_{enc\ net}}{\epsilon_0\oint dS}$$ The **superposition** principle allows potential due to different charges to be calculated separately and summed together to achieve the same result. +## Conductors +An **ideal conductor** has electrons loosely bound to atoms such that an electric field causes them to freely move by $F=Q_e E$. However, this assumes that there are infinite electrons in the conductor, and that the electrons will move with **zero resistance** to the surface of the conductor but **not leave it**. + +A conductor placed in an external electric field will cause electrons to hop from atom to atom to reach the surface, charging one surface negatively and the other positively. The **induced electric field** from this imbalance opposes the external field force, slowing down electron movement until equilibrium is reached. + +$$\text{equilibrium}\iff \vec E_{ext}+\vec E_{ind}=\vec 0$$ + +At equilibrium, **every point in the conductor is equipotential**. Gauss's law implies that there is no volume charge inside a conductor. + +At its surface, $\vec E$ tangent to the surface must be zero. Normal to the surface: + +$$|\vec E_N|=\frac{|\rho_0|}{\epsilon_0}$$ + +- $\rho_0$ is negative if field lines **enter** the conductor. +- $\rho_0$ is positive if field lines exit the conductor. + +### Conductor cavities + +A cavity surface must have **zero surface charge**. This creates a Faraday cage — outside fields cannot affect the cavity, but fields from the cavity can affect the outside world. + +If there is a fixed/non-moving charge $Q$ in the cavity: + +- $\vec E=0$ inside the conductor, so the boundary surface charge must be $-Q$. +- Electrons are taken from the surface, so the surface charge outside the conductor must be $Q$, propagating the effect of the charge to the outside world. + +### Ground + +A **ground** is a reservoir or sink of charges that never changes, regardless of the quantity added or removed from it. At the connection point, $V=0$ is always guaranteed. + +Grounding a conductor means that it takes charges from the ground to balance an internal charge, neutralising it. + +A charge released into a conductor (e.g., battery into wire) will always go to the outside surface, regardless of the point of insertion. Two charged objects connected by a thin conductor will redistribute their charge such that: + +- their potentials are equal +- conservation of charge is followed. + +This implies that a larger object has more charge, but a smaller object has a denser charge and thus stronger electric field. + +$$Q_1=\frac {R_1} {R_2}Q_2$$ + +!!! example + For two spheres, as $\rho=\frac{Q_1}{4\pi R^2}$: + + $$\rho_1=\frac {R_2} {R_1}\rho_2$$ + +A non-uniform object, such as a cube, will have larger charge density / stronger electric field at sharper points in its shape. Symmetrical surfaces always have uniform charge density. + +!!! warning + An off-centre charge in a cavity will require a non-uniform induced charge to cancel out the internal field, but the external surface charge will be uniform (or non-uniform if the surface is odd).