Finding the electric field

Question

The charge per unit length on a long, straight filament is -91.9 µC/m.

(a) Find the electric field 10.0 cm from the filament, where distances are measured perpendicular to the length of the filament. (Take radially inward toward the filament as the positive direction.) MN/C

(b) Find the electric field 50.0 cm from the filament, where distances are measured perpendicular to the length of the filament. MN/C

(c) Find the electric field 150 cm from the filament, where distances are measured perpendicular to the length of the filament.

$λ_q=\frac{dq}{dl}→dq=λ_qdl$

$\vec{E} =k_e\frac{q}{r^2}\hat{r}$

$||\vec{E}||=\frac{λ_qdl}{r^2}$ How can I solve this without knowing the length of the filament? Supposedly this is the suggested way of solving this problem--and I'd like to understand this method very much. Could someone help me?

Also, in addition to solving it by this expedient, is it possible to solve this problem by employing a Gaussian surface?

Answer 1

You have basically indicated the two natural ways to solve this problem:

1. Integrate the electric field due to small segments along the filament to find the total electric field at a specified point.

2. Use Gauss's law with an appropriate gaussian surface.

For method 1, you are basically almost there given your manipulations; you simply need to integrate $$ dE = k_e\frac{\lambda\,d\ell}{r^2} $$ But what are the limits of integration if we don't know the length of the filament? Well, the problem states that the wire is "long," which is just physics-speak for infinitely long. So you need to integrate from $-\infty$ to $\infty$. Just make sure to be careful that $r$ denotes the distance from the point where you want to find the field, to the point where $dq$ is located on the filament, so you need to use the Pythagorean theorem to related $r$ to the perpendicular distance to the filament, say $d$, and the length along the filament, say $\ell$.

For method 2, you simply pick a cylindrical Gaussian cylinder whose axis of symmetry is along the filament. I'll leave out the details for this method and let you try it. Let me know if you need more help, but this is something you can easily find by googling.

Hope that helps!

Physics Rocks.