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8.7 - Applications of Gauss's Law

8.7.1 - Gauss's Law in Continuous Charge Distributions​

Essentially Gauss's Law can also be used to find electric fields inside high symmetry charge distributions. What this means is that the distribution of charges within an object is somewhat "even" in terms of placement through the medium.

For example, in a problem regarding electric flux through a spherically symmetric charge distribution, ∮E⋅dA\oint \mathbf{E} \cdot d\mathbf{A} is able to be represented in a more simpler form in which E\mathbf{E} is able to be taken out of the integral, making it E∮dA\mathbf{E} \oint d\mathbf{A} which evens out into a form which resembles the simple expression EA\mathbf{EA} such that A\mathbf{A} is represented by the surface area of the sphere. Through this it reaches a form in which we are easily able to evaluate and find the electric flux through a continuous charge distribution. Using similar approaches to this, we will cover how to find the electric field of complex charged objects using Gauss's Law

The main idea is that for any continuously charged high symmetry object, the equation ΦE=EA⊥\Phi_E = EA_{\perp} is true, but Gauss's Law still applies. Using this, and the surface area of the object, we can solve for the electric field.

8.7.2 - Gauss's Law Applied to Conductors in Electrostatic Equilibrium​

Earlier we covered conductors with respect to the important role they play in the transfer of electrons in different electronic systems. In basic terms, they help facilitate electron transfer through the presence of free electrons which are present in their respective medium. Electrostatic equilibrium is NOT when electrons through a conductor stop moving, but instead when the charges inside the conductor "cancel" such that no electric field is produced. Hence, electrons stop moving in a way which produces current. Current, I\mathbf{I}, is the rate of change in charge with respect to time. It will play a more important role when we cover circuits. For now, just think of it as dQdt\frac{dQ}{dt}.

One interesting application of Gauss's law is that we can use it to find the electric field of a conductor in electrostatic equilibrium. There is a separation between the way we calculate the electric field inside and outside of a conductor in electrostatic equilibrium because of a discontinuous jump between the two. Below is the setup for them in a spherical conductor in electrostatic equilibrium.

In: The electric field for a point inside the radius of this sphere will be 0

Out: ∮E⋅dA=Qencε0\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

E⋅4πr2=Qε0\mathbf{E} \cdot 4 \pi r^2 = \frac{Q}{\varepsilon_0}

E(r)=Q4πε0r2 r^,r≥R\mathbf{E}(r) = \frac{Q}{4 \pi \varepsilon_0 r^2} \,\hat{\mathbf{r}}, \quad r \ge R

Esurface=Q4πε0R2 r^=σε0 r^,σ=Q4πR2\mathbf{E}_{\text{surface}} = \frac{Q}{4 \pi \varepsilon_0 R^2} \,\hat{\mathbf{r}} = \frac{\sigma}{\varepsilon_0} \,\hat{\mathbf{r}}, \quad \sigma = \frac{Q}{4 \pi R^2}

8.7.3 - The Formal Derivation of Gauss's Law​

ΔΦE=E⋅ΔA=E ΔAcos⁡θ\Delta \Phi_E = \mathbf{E} \cdot \Delta \mathbf{A} = E \, \Delta A \cos\theta

E=keqr2E = k_e \frac{q}{r^2}

ΔΦE=keqr2 ΔAcos⁡θ\Delta \Phi_E = k_e \frac{q}{r^2} \, \Delta A \cos\theta

ΔΩ=ΔAcos⁡θr2\Delta \Omega = \frac{\Delta A \cos\theta}{r^2}

ΔΦE=keq ΔΩ\Delta \Phi_E = k_e q \, \Delta \Omega

ΦE=∑ΔΦE=keq∑ΔΩ\Phi_E = \sum \Delta \Phi_E = k_e q \sum \Delta \Omega

∑ΔΩ=4π\sum \Delta \Omega = 4\pi

ΦE=4πkeq=qε0\Phi_E = 4 \pi k_e q = \frac{q}{\varepsilon_0}

∮E⋅dA=qencε0\oint \mathbf{E} \cdot d\mathbf{A} = \frac{q_{\text{enc}}}{\varepsilon_0}