Question Number 222191 by wewji12 last updated on 20/Jun/25
![((d )/dt) ∫_( V^( 3) ) ρ_q (r,t)dV=−∮_( ∂V) J_q (r,t)∙da+∫_( V^( 3) ) S_q (r,t)dV ∫_( V^( 3) ) ((∂ρ_q (r,t))/∂t) dV=−∫_V^( 3) ▽^→ ∙J_q (r,t)dV+∫_( V^( 3) ) S_q (r,t)dV ∫_( V^( 3) ) [ (∂ρ/∂t)+▽^→ ∙J^ (r,t)−S(r,t)]dV=0 ∴((∂ρ_q (r,t))/∂t)+▽^→ ∙J_q (r,t)=S_q (r,t)](https://www.tinkutara.com/question/Q222191.png)
$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\:\int_{\:{V}^{\:\mathrm{3}} } \rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V}=−\oint_{\:\partial{V}} \:\boldsymbol{\mathrm{J}}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\centerdot\mathrm{d}\boldsymbol{\mathrm{a}}+\int_{\:{V}^{\:\mathrm{3}} } \:{S}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V} \\ $$$$\int_{\:{V}^{\:\mathrm{3}} } \:\frac{\partial\rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)}{\partial{t}}\:\mathrm{dV}=−\int_{{V}^{\:\mathrm{3}} } \overset{\rightarrow} {\bigtriangledown}\centerdot\boldsymbol{\mathrm{J}}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V}+\int_{\:{V}^{\:\mathrm{3}} } {S}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V} \\ $$$$\int_{\:\mathcal{V}^{\:\mathrm{3}} } \left[\:\frac{\partial\rho}{\partial{t}}+\overset{\rightarrow} {\bigtriangledown}\centerdot\boldsymbol{\mathrm{J}}^{\:} \left(\boldsymbol{\mathrm{r}},{t}\right)−{S}\left(\boldsymbol{\mathrm{r}},{t}\right)\right]\mathrm{d}{V}=\mathrm{0} \\ $$$$\therefore\frac{\partial\rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)}{\partial{t}}+\overset{\rightarrow} {\bigtriangledown}\centerdot\boldsymbol{\mathrm{J}}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)={S}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right) \\ $$