Applications Of Gauss's Law

Problem: An infinite line charge produces an electric field of $9 \times 10^4 \text{ N/C}$ at a distance of $2 \text{ cm}$. Calculate the linear charge density $\lambda$.

Solution: We know the formula for the electric field of an infinite line charge is $E = \frac{\lambda}{2 \pi \epsilon_0 r}$. To find the linear charge density, we rearrange the formula to isolate $\lambda$: $\lambda =$ . Now, we substitute the given values: $E = 9 \times 10^4 \text{ N/C}$ and $r = 0.02 \text{ m}$. Using the electrostatic constant $\frac{1}{4 \pi \epsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$, we can write $2 \pi \epsilon_0 = \frac{1}{18 \times 10^9}$. Plugging these into our rearranged equation gives $\lambda = \frac{9 \times 10^4 \times 0.02}{18 \times 10^9}$. Calculating this yields the final linear charge density: $\lambda =$ $\text{ C/m}$.