Question

An infinitely long line of charge with uniform density, rho???????? lies in y-z plane parallel to the zaxis at y=1m. (a) Find the potential VAB at point A (4m, 2m, 4m) in Cartesian coordinates with respect to point B (0,0,0). (b) Find E filed at point B.

Answer 1

Answer with Explanation:

We are given that

Density=
`\rho l`

A(4m,2m,4m) and B(0,0,0)

y=1 m

a. Linear charge density=
`(\rho l)/(l)=\rho C/m`

Let a point P (0,1,4) on the line of charge and point Q (0,1,0)

Therefore,

Distance AP=
`√((4-0)^2+(2-1)^2+(4-4)^2)=√(17)`

Distance,BQ=
`√((0-0)^2+(1-0)^2+(0-0)^2)=1`

Electric field for infinitely long line

`E=(\rho)/(2\pi \epsilon_0 r)\cdot \hat{r}`

Therefore, potential

`V_(BA)=-\int_(a)^(b)E\cdot dl`

`V_(BA)=-\int_(√(17))^(1)(\rho)/(2\pi \epsilon_0 r)\hat{r}\cdot \hat{r} dr`

`V_(BA)=-\int_(√(17))^(1)(\rho)/(2\pi \epsilon_0 r)dr`

`V_(BA)=-(\rho)/(2\pi \epsilon_0)[\ln r]^(1)_(√(17))`

`V_(BA)=-(\rho)/(2\pi \epsilon_0)(ln 1-ln(√(17))=(\rho)/(2\pi \epsilon_0)(ln(√(17))`

`V_(BA)=V_B-V_A`

`V_(AB)=V_A-V_B=-V_(BA)=-(\rho)/(2\pi \epsilon_0)(ln(√(17))`

b.Electric field at point B

`E=(\rho)/(2\pi \epsilon_0 r)\cdot \hat{r}`

Unit vector r=
`-\hat{j}`

Therefore,

`E=(\rho)/(2\pi \epsilon_0 r)\cdot \hat{-j}`