Question

A very long uniform line of charge has charge per unit length 4.80 µC/m and lies along the x-axis. A second long uniform line of charge has charge per unit length -2.40 µC/m and is parallel to the x-axis at y = 0.400 m. What is the net electric field (magnitude and direction) at the following points on the y-axis: (a) y = 0.200 m and (b) y = 0.600 m?

Answer 1

The electric field E due to a line charge with charge density at a distance is given as follows:

Here, is the vacuum permittivity of free space.

The linear charge density is defined as the charge per unit length.

Here, is the charge and is the length of the line charge.

a)

The electric field due to the line charge lying at x-axis is,

Substitute for, for and for in the above equation.

The electric field due to the line charge lying parallel to the y axis at is,

Here,

The distance between given point and line of charge placed at is,

Substitute for, for and for in the above equation.

The electric field at a point on the y axis which lies between the two line charges is,

Substitute for and for in the above equation.

Therefore, the net electric field on the y-axis is .

The positive sign of net electric field obtained represents the electric field is along positive y-axis.

The direction of electric field due to the line chare lying on x-axis is towards positive y axis and the direction of electric field due to the line charge lying parallel to the y axis at is also towards positive y axis. Thus, the net electric field due to the two line charges is towards positive y axis.

b)

The electric field due to the line charge lying at x axis is,

Substitute for, for and for in the above equation.

The distance between given point and line of charge placed at is,

The electric field due to the line charge lying parallel to the y axis at is,

Substitute for, for and for in the above equation.

The electric field at a point on the y axis which lies between the two line charges is,

Substitute for and for in the above equation.

Therefore, the magnitude of the net electric field on the y-axis is.

The negative sign for net electric field represents the net electric field is along negative y-axis.