Find out the surface charge density at the intersection of point x = 3 m plane and x-axis, in the region of uniform line charge of 8 nC/m lying along the z-axis in free space.

Solution

Physics sir · 4 steps

Here 'surface charge density' means flux density D, so use Gauss's law without the ε₀.

1. What is asked. The unit C m⁻² tells you this is the electric flux density D = ε₀ E. It is the charge per unit area that the field lines carry through a surface at that point.
2. Gauss's law. Take a cylinder of radius r and length L around the z-axis. All the flux leaves through the curved surface, so D × 2π r L = λ L.

   D = λ/(2π r)

3. Put in numbers. At x = 3 m on the x-axis, the distance from the line is r = 3 m. So D = (8 nC m⁻¹)/(2π × 3 m) = 8/18.85 nC m⁻².
4. So. D ≈ 0.424 nC m⁻², which is option (a).

The idea to keep. For a line charge, D = λ/(2π r) and E = λ/(2πε₀ r). Look at the unit asked for: C m⁻² means D, and N C⁻¹ means E.

Answer from the NTA final answer key. Solution by PrepWiser.