Overview
The total amount of charge is a conserved quantity. That is, given a region of space, unless a charge moves out or into that region, the total amount of measured charge within the region does not change. This is an exprimental fact, but can be derived from other assumptions.
Mathematical Description of Conservation of Charge
The total charge within a volume is given by
{% \displaystyle Q = \int_v \rho(\vec{r}) dV %}
The current that exits the volume is given by {% \int_v \vec{J} \cdot \vec{dA} %},
Given that charge is conserved, we have
{% \displaystyle \frac{dQ}{dt} = \int_v \vec{J} \cdot \vec{dA} %}
{% \displaystyle \int_v \frac{\partial{\rho}}{\partial{t}} dV = -\int_v \nabla \cdot \vec{J} dV %}
which also then leads to the continuity equation
{% \frac{\partial{\rho}}{\partial{t}} = - \nabla \cdot \vec{J} %}
Derivation from Maxwells Equations
Starting from the first Maxwell Equation
{% \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} %}
Taking the derivative with respect to time
{% \frac{d}{dt} \nabla \cdot \vec{E} = \nabla \cdot \frac{\partial \vec{E}}{\partial t} = \frac{1}{\epsilon_0} \frac{\partial \rho}{\partial t} %}
From the second of
Maxwells Equations we have
{% \frac{\partial \vec{E}}{\partial t} = c^2 [\nabla \times \vec{B} - \mu_0 \vec{j} ] %}
Now
{% \nabla \cdot \nabla \times \vec{B} = 0 %}
which leaves
{% \frac{1}{\epsilon_0} \frac{\partial \rho}{\partial t} = - \mu_0 c^2 \nabla \cdot \vec{j} %}
Charge Invariance
In addition to charge conservation, charge is a relativistic invariant. That is, the amount of measured charge is the same in each reference frame. See Purcell