đź’Ľ Case: consider an arbitrary surface charge distribution where we define an infinitesimal surface $\text d \vec a$, an infinitesimal volume $\delta \tau '$ and where $\vec j$ is the flux coming out of the surface
$$ \frac{\text d Q}{\text d t} = \int _V \frac{\partial \rho}{\partial t} \,\text d\tau '=-\oint _S \vec j \cdot \text d \vec a=-\int _V \vec \nabla \cdot \vec j \, \text d \tau' $$
where we used the divergence theorem
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/7a457349-e601-4cba-ad0c-ec7bdf2f655b/Continuity_equation_for_charge.gif" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/7a457349-e601-4cba-ad0c-ec7bdf2f655b/Continuity_equation_for_charge.gif" width="40px" />
Continuity equation for charge: true for any arbitrary choice of volume
$$ \boxed{\frac{\partial \rho}{\partial t} + \vec \nabla \cdot \vec j =0} $$
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$$ 0=\vec \nabla\cdot (\vec \nabla \times \vec B)=\mu_0 ( \vec \nabla\cdot \vec j + \frac{\partial \rho}{\partial t})= \mu_0 \left ( \vec \nabla \cdot \vec j + \epsilon_0 \frac{\partial}{\partial t} (\vec \nabla \cdot \vec E)\right ) $$
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/caea8eac-0c3a-430a-bdb5-522b33d21a95/Ampres_law_with_Maxwells_correction.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/caea8eac-0c3a-430a-bdb5-522b33d21a95/Ampres_law_with_Maxwells_correction.png" width="40px" />
Ampère's law with Maxwell's correction:
$$ \boxed{\vec \nabla \times \vec B = \mu_0\vec j + \epsilon_0 \mu_0 \frac{\partial \vec E}{\partial t}} $$
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đź’Ľ Case: consider a region of space with a magnetic field with some loop of wire going through it
If we changed the system is in the follow way:
We get a reduction in the flux of the magnetic field through the loop wire
We get an induced EMF which is the rate of change of magnetic flux through the loop
$$ \varepsilon=\oint \vec E\cdot \text d \vec l =\int_S \vec \nabla \times \vec E\cdot \text d \vec a=-\frac{\text d \Phi_m}{\text dt}=-\int _S\frac{\partial B}{\partial t}\cdot \text d \vec a $$
where we used Stoke’s theorem and $\Phi_m$ is the magnetic flux
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/0745c2c1-40a9-414b-adf9-745da7959036/Faradays_Law.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/0745c2c1-40a9-414b-adf9-745da7959036/Faradays_Law.png" width="40px" />
Faraday’s Law:
$$ \boxed{\vec \nabla \times \vec E = - \frac{\partial \vec B}{\partial t}} $$
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🗒️ Note: direction of $\varepsilon, I$ is such that they oppose the change in flux $\Phi_m$ through the loop
