Question

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

• A. $360\epsilon _{0}$ Coulomb m$^{- 3}$
• B. $120\epsilon _{0}$ Coulomb m$^{- 3}$
• C. $240\epsilon _{0}$ Coulomb m$^{- 3}$
• D. $480\epsilon _{0}$Coulomb m$^{- 3}$

Answer 1

Answer: (D)

$\phi=EA$
$\phi=40x^{3}A$
$d\phi=\left(120 x^{2}\right)Adx$
$dq=\left(d \phi\right)\left(\epsilon \right)_{0}=\left(120 \left(\epsilon \right)_{0} x^{2}\right)\left(A d x\right)$
$\rho \left(x\right)=\frac{d q}{d v}=\frac{120 \left(\epsilon \right)_{0} x^{2}}{\left(A d x\right)}\left(A d x\right)$
$\rho \left(x\right)=120\left(\epsilon \right)_{0}x^{2}$
$\rho \left(x = 2\right)=480\left(\epsilon \right)_{0}$