Electric Fields

Question

Find the electric field vector at the origin due to an infinitely long, straight line bearing a uniform linear charge density of 360 fC/m. The line charge is tilted at π/4 radians such that it runs diagonally from the lower left to the upper right with respect to the usual x-y coordinate axes (+x to the right, +y upward). Its point of closest approach to the origin is below and to the right of (0,0) at a distance of 2 μm. Use 9 pC2/J·m for the permittivity of free space.

 

 

Answer 1

You can find the magnitude of the field at a distance r from a straight line, and then break it down into 2 components, one parallel to the x-axis, and one parallel to the y-axis.

 

The simplest way to find the field is to use Gauss' law.  

 

Φ = q/ε

 

∫E·dA = λL/ε, λ = linear charge density

 

For the Gaussian surface, choose a cylinder of radius r and length L with axis along the line of charge.  We will let L→∞.  The field is uniform in magnitude over this surface, and by symmetry, perpendicular to this surface.

 

E(r) A(r) = λL/ε

 

E(r) (2πrL) = λL/ε

 

Let L→∞.  Note L cancels out.

 

E(r) = λ / (2πεr)

 

We have the magnitude.  We still need its direction.  Since λ is positive, the field points away from the line and perpendicular to the line.  From the orientation of the line, 

 

E_x(r) = -E(r)sinθ = -[λ / (2πεr)] sinθ

 

E_y(r) = E(r)cosθ = [λ / (2πεr)] cosθ

 

Note that in the geometry of this problem, sinθ is associated with the x direction, and cosθ is associated with the y direction.

 

r = 2 μm

θ = π/4

 

The other numbers are given to you.  Plug and chug to get the final numbers.

 

 

 

 

Answer 2

Here ya go:

https://en.wikipedia.org/wiki/Coulomb%27s_law

 

 

Coilomb's Laws:

 

e0 = 9 pC2.J

 

r1 = (0,0) and r2 = (2 um, 0)

or vice versa.

 

Q2 = 360

 

If you need a meat and potato function,

 it's the identity line y = x,