ece106: add conductors

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).