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GATE 2024 EC – Question 25

Electromagnetics · Maxwell's Equations · 1 mark · Multiple select

Let $\rho(x,y,z,t)$ and $u(x,y,z,t)$ represent density and velocity, respectively, at a point $(x,y,z)$ and time $t$. Assume $\dfrac{\partial\rho}{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat n$ be the outward unit normal of $S$.

Which of the following equations is/are equivalent to $\dfrac{\partial\rho}{\partial t}+\nabla\cdot(\rho u)=0$?

  1. $\int_V\dfrac{\partial\rho}{\partial t}dv=-\oint_S\rho u\cdot\hat n\,ds$
  2. $\int_V\dfrac{\partial\rho}{\partial t}dv=\oint_S\rho u\cdot\hat n\,ds$
  3. $\int_V\dfrac{\partial\rho}{\partial t}dv=-\int_V\nabla\cdot(\rho u)\,dv$
  4. $\int_V\dfrac{\partial\rho}{\partial t}dv=\int_V\nabla\cdot(\rho u)\,dv$

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Show answer and explanation

Correct answer: (A) $\int_V\dfrac{\partial\rho}{\partial t}dv=-\oint_S\rho u\cdot\hat n\,ds$; (C) $\int_V\dfrac{\partial\rho}{\partial t}dv=-\int_V\nabla\cdot(\rho u)\,dv$

Explanation

Integrating $\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho u)$ over $V$ gives $\int_V\frac{\partial\rho}{\partial t}dv=-\int_V\nabla\cdot(\rho u)dv$ (option C). By the divergence theorem the right side is $-\oint_S\rho u\cdot\hat n\,ds$ (option A). Options B and D have the wrong sign.