Question

. A non-uniform positive line charge of length 2 m is put along the x-axis as shown in the figure, where x​0​=1.5 m. The linear charge density is given by λ(x)=4x​2 C/m​3​. Find the magnitude of the total electric field, E, created by the line charge at the origin using integration. (Take k=9x10​9 ​N m​2​ /C​2​)

Answer 1

Answer:

$E = 7.2*10^{10}N/C$

Explanation:

The differential electric field $dE$ due to differential charge $dQ$ at distance $x$ from the origin is

$dE = k\dfrac{dQ}{x^2}$

but since $dQ = \lambda dx = 4x^2dx$ we have

$dE = k\dfrac{4x^2dx}{x^2}$

$dE = 4k\: dx$

integrating this from $x_0$ to $x_0+L$ we get

$E = \int^{x_0+L}_{x_0} {4k} \, dx$

$E = 4k[(x_0+L)-x_0]$

$E =4kL$

putting in $k = 9*10^9Nm^2/C^2$ and $L =2m$ we get

$\boxed{E = 7.2*10^{10}N/C.}$