A very long charged solid cylinder of radius 'a' contains a uniform charge density $\rho$. Dielectric constant of the material of the cylinder is $\mathrm{k}$. What will be the magnitude of electric field at a radial distance ' $x^{\prime}(x < a)$ from the axis of the cylinder?

• A

  $\rho \frac{x}{\varepsilon_{0}}$
• Correct answer
  B

  $\rho \frac{x}{2 k \varepsilon_{0}}$
• C

  $\rho \frac{\mathrm{x}^{2}}{2 \mathrm{a} \varepsilon_{0}}$
• D

  $\rho \frac{\mathrm{x}^{2}}{2 \mathrm{k}}$

Solution

Hint: Using Gauss's Law $\mathrm{E}(2 \pi \times \ell)=\frac{\rho\left(\pi \mathrm{X}^{2} \ell\right)}{\mathrm{k} \varepsilon_{0}} \quad \therefore \mathrm{E}=\frac{\rho \mathrm{x}}{2 \mathrm{k} \varepsilon_{0}}$

Hence, the correct answer is option: 'B'.