An electric field in free space is E=(2x+5y+6z) i+(5x+4y+10z) j+(6x+10y+2z) k\ V/m. The charge density is C/m3. ( 0 is the permittivity of free space.)

GATE 2026 IN – Question 37

Electricity and Magnetism · Electrostatics and capacitance · 2 marks · Multiple choice

An electric field in free space is $\vec E=(2x+5y+6z)\hat i+(5x+4y+10z)\hat j+(6x+10y+2z)\hat k\ \mathrm{V/m}$. The charge density is ________ $\mathrm{C/m^3}$. ($\varepsilon_0$ is the permittivity of free space.)

$8\varepsilon_0$
$10\varepsilon_0$
$6\varepsilon_0$
$4\varepsilon_0$

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Correct answer: (A) $8\varepsilon_0$

Explanation

Gauss’s law in differential form is $\rho=\varepsilon_0\nabla\cdot\vec E$. The required derivatives are $\partial E_x/\partial x=2$, $\partial E_y/\partial y=4$, and $\partial E_z/\partial z=2$.

Thus $\nabla\cdot\vec E=8$ and $\rho=8\varepsilon_0\ \mathrm{C/m^3}$. The correct option is **A**.