Question

Assume that an electric field $\vec{E}=40x^3\hat{i}N{C}^{-1}$ exists in space. The charge density at $x=2m$ is

• A. $360{ϵ}_0$ Coulomb m${}^{-3}$
• B. $120{ϵ}_0$ Coulomb m${}^{-3}$
• C. $240{ϵ}_0$ Coulomb m${}^{-3}$
• D. $480{ϵ}_0$Coulomb m${}^{-3}$

Answer 1

Answer: (D)

$\varphi =EA$
$\varphi =40x^{3}A$
$d\varphi =\left(120x^2\right)Adx$
$dq=\left(d\varphi \right){\left(ϵ\right)}_0=\left(120{\left(ϵ\right)}_0{x}^2\right)\left(Adx\right)$
$\rho \left(x\right)=\frac{dq}{dv}=\frac{120{\left(ϵ\right)}_0{x}^2}{\left(Adx\right)}\left(Adx\right)$
$\rho \left(x\right)=120{\left(ϵ\right)}_0{x}^2$
$\rho \left(x=2\right)=480{\left(ϵ\right)}_0$