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

`E = 7.2*10^(10)N/C`

Step-by-step explanation:

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

`dE = k(dQ)/(x^2)`

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

`dE = k(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.}`