Semi-infinite non-conducting rod

[kinof]

Homework Statement

A "semi-infinite" nonconducting rod has a uniform linear charge density λ. Show that the electric field E at point P a distance R above one end of the rod makes an angle of 45° with the rod and that this result is independent of the distance R.

Homework Equations

$$\vec{E}=\int \frac{k\lambda }{r^2}dx$$

The Attempt at a Solution

I have found that $$\vec{E_x}=\frac{-k\lambda}{\sqrt{R^2+x^2}}$$ and $$\vec{E_y}=\frac{k\lambda x}{R\sqrt{R^2+x^2}}$$

I have tried to show that, if Ex and Ey make a 45° angle, then they must be equal, but that leads nowhere. I have tried simply taking the arctan of Ey/Ex, but that leads to -x/R, which doesn't make sense. I have also tried evaluating the integrals at infinity and then using L'Hospital's rule, but it doesn't work for Ey.

 

[TSny]

After integrating, did you evaluate at the limits of integration? ##x## should not appear in the results.

 

[kinof]

Oh, woops. I got $$\frac{k\lambda}{R}$$ for both. Of course the ratio of these is 1, so the angle is 45 degrees. Thanks.

 

[TSny]

Looks good!

 

[Nickie]

I would approach this problem by first double-checking the equations and calculations used to find Ex and Ey. I would also make sure that the integrals are set up correctly and that the limits of integration are appropriate for a semi-infinite rod.

After confirming the calculations, I would then consider the geometry of the problem. Since the rod is non-conducting and has a uniform linear charge density, the electric field will be radial and perpendicular to the rod at all points. This means that the angle between the electric field and the rod must be 90°.

To show that the angle between the electric field and the rod is 45°, I would consider the components of the electric field in the x and y directions. Using trigonometry, I would show that the ratio of Ex and Ey is equal to 1, meaning that they are equal in magnitude. This would confirm that the angle between the electric field and the rod is indeed 45°.

Finally, to show that this result is independent of the distance R, I would vary R and recalculate the electric field at point P. If the angle between the electric field and the rod remains 45°, then we can conclude that this result is independent of the distance R.

Overall, approaching this problem with careful consideration of the equations, geometry, and experimental verification would provide a thorough and accurate response as a scientist.