>[!summary] Conductors are a type of materials that transfer something through them. > **Key equations:** > Electric field of a conductor with empty cavities (Inside and outside the cavity): $\vec{E} = 0$ > Electric field of a conductor with a charged cavity: $\vec{E} = \frac{Q}{4 \pi r^2 \epsilon_0}$ Inside cavity $\vec{E} = 0$ Outside cavity >[!info]+ Read Time **⏱ 4 mins** # Definition Conductors are a type of material that transfers something through them (conduct). Often, when using the term conductors, we assume the transfer of electricity ([[Current|current]]), but it can also be the transfer of heat. Some materials that act as conductors are copper, iron & gold. # Conductors With Empty Cavities >[!bug] Note Assumption Assume that [[Charge|charges]] inside a conductor will move freely until they reach a configuration where it makes the [[Electric Fields|electric field]] inside the conductor zero. Suppose we have a conductor that has a cavity in it. Inside the cavity, there is no charge. To find the [[Electric Fields|electric field]] the easiest way if by using [[Gauss's Law|Gauss's law]]. Assuming a spherical Gaussian surface to match with the conductor, there can be two cases. First case of a Gaussian surface inside the cavity and one outside the cavity but inside the conductor. ![[con_1.png|400]] [^1] >[!note] Explanation Example of a hollow cavity (no charge) in the middle of a sphere ## Case 1: Gaussian Surface Inside the Cavity >[!warning] Assumptions Assume a Gaussian surface of sphere that expands to the edge of inside the cavity, not touching the rim that the cavity and the rest of the conductor meet. $ \begin{array}{c} \oint \vec{E}\cdot \vec{dA} = \frac{Q_{enc}}{\epsilon _0} \\ \text{Because our cavity is empty, $Q_{enc} = 0$} \\ \\ \vec{E} \cdot (4\pi r^2) = \frac{Q_{enc}}{\epsilon_0}\\ \vec{E} \cdot (4\pi r^2) = 0 \\ \vec{E} = 0 \end{array} $ ## Case 2: Gaussian Surface Outside the Cavity >[!warning] Assumptions Assume a Gaussian area a sphere that expands to the edge of the conductor not reaching outside the conductor. Since our Gaussian area is a sphere, the cavity is enclosed in it. > Remember at the start of this section we assume that a conductor has enough [[Charge|charges]] to configure charges to create [[! Equilibrium|equilibrium]]. $ \begin{array}{c} \oint \vec{E} \cdot \vec{dA} = \frac{Q_enc}{\epsilon_0} \\ \text{Our gassian surface enclosed the entire sphere, no free charges so:} \\ \\ Q_{enc} = 0 \\ \vec{E} = 0 \end{array} $ # Conductors With Charged Cavities >[!bug] Note Assumption We are assuming that [[Charge|charges]] inside a conductor will move freely until they reach a configuration where it makes the [[Electric Fields|electric field]] inside the conductor zero ([[! Equilibrium|equilibrium]]) Suppose we have a conductor will a cavity again, but now the cavity has a charge inside. This [[Charge|charges]] inside the conductor will still want to arrange themselves to configure for [[! Equilibrium|equilibrium]]. ![[con_2.png|400]] [^1] >[!note] Explanation Example of conductor with a charge enclosed in a cavity If we wanted to find the electric field inside this conductor, there are two cases to use Gauss's law. A case where we choose a spherical Gaussian area inside the cavity and outside the cavity. ## Case 1: Gaussian Surface Inside the Cavity >[!warning] Assumptions Assume a spherical gaussian surface inside the cavity (till the edge not touching the conductor part) > Assume our charge inside the cavity is $Q$ $ \begin{array}{c} \oint \vec{E} \cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_0} \\ \\ Q_{enc} = +Q \\ \oint \vec{E} \cdot \vec{dA} = \frac{Q}{\epsilon_0} \\ \vec{E} \cdot (4\pi r^2) = \frac{Q}{\epsilon_0} \\ \vec{E} = \frac{Q}{4 \pi r^2 \epsilon_0} \end{array} $ Notice the difference in having [[Conductors#Case 1 Gaussian Surface Inside the Cavity|no charge]] and charge inside the cavity. ## Case 2: Gaussian Surface Outside the Cavity >[!warning] Assumptions Assume a spherical Gaussian surface outside the cavity but still inside the conductor. Assume the charge inside the cavity is still Q > Remember at the start of this section we assume that a conductor has enough [[Charge|charges]] to configure charges to create [[! Equilibrium|equilibrium]]. $ \begin{array}{c} \oint \vec{E} \cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_0} \end{array} $ >[!bug] Note Our $Q_{enc}$ are the charges inside the conductor (including the cavity). The cavity has a charge $Q$ and in order for the conductor to maintain equilibrium the conductor will push $-Q$ towards the edge of the cavity. > Therefore, $Q_{enc} = Q + (-Q) = 0$ $ \begin{array}{c} \oint \vec{E} \cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_0} \\ \vec{E} \cdot (4\pi r^2) = \frac{0}{\epsilon_0} \\ \vec{E} = 0 \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/PafSqL1riS4?si=rvvglT-VwcCFqe3O" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <iframe width="560" height="315" src="https://www.youtube.com/embed/_b9Pldu1vV0?si=mvickm_AOZAlsOzu" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidan-MathMatter) to support the project and help others discover it! --- [^1]: Taken from https://tikz.net/electric_field_sphere/?utm_source=chatgpt.com by Izaak Neutelings (February, 2020)