MyOpenMath/Solutions/Electromotive force/Helicopter

The following problem is inspired by CHECK YOUR UNDERSTANDING 13.4 in University Physics Volume II Chapter 13:



If the moving blade develops a motional emf caused by the magnetic field, why doesn't a current flow? At one level, the answer is simple: Electrostatic charges build up to create an internal electric field to oppose the motional emf. But how do wires manage this? Consider an infinitely long thin wire and calculate how any electric field inside of that wire can be generated by a surface charge density. To keep things simple, consider only the electric field at the center of the wire. We begin with the well known formula for the electric field on the axis of a charged loop: Integrate to find the electric field at z=0: $$E_z(0)=\int dE_z$$, where:


 * $$dE_z=\frac{-1}{4\pi\varepsilon_0} \frac {z\lambda dz}{\left(z^2+R^2\right)^{3/2}},$$

where $$dq=\lambda dz$$ is a charge distributed on the surface of the wire. We shall now show that an arbitrary electric field at the center of the wire can be created by a suitable choice of the line density, expressed as a function of z.

This expression for $$dE_z$$ diverges as $$R\rightarrow 0$$ when $$z=0$$. This suggests that a thin wire, perhaps the dominant contribution to the electric field at $$z=0$$ is due to charges located in the range $$-R\gtrsim z\lesssim R$$. To verify this suggestion, we use integration by parts to introduce the Dirac delta function:


 * $$\int u \, dv \ =\ uv - \int v du.$$


 * $$u=\lambda (z)$$


 * $$dv=\frac {-z dz}{\left(z^2+R^2\right)^{3/2}}\Rightarrow v = \frac{1}{\sqrt{z^2+R^2}}$$


 * $$4\pi\varepsilon_0E_z(0) =

\left.\frac{\lambda (z)}{\sqrt{z^2+R^2}}\right|_{-a}^b +\int_{-a}^b \frac{d\lambda}{dz}\frac{dz}{\sqrt{z^2+R^2}}$$

We can discard the first term by integrating past the wire, allowing $$\lambda (z)$$ to be a Heavyside function


 * Students are encouraged to finish this problem right here on Wikiversity. See also this question posed on OpenStax.}}

Footnotes and references

 * Andrews, Mark. (1997). Equilibrium charge density on a conducting needle. American Journal of Physics. 65. 846. 10.1119/1.18671.
 * Jackson, John David. "Charge density on a thin straight wire: The first visit." American Journal of Physics 70.4 (2002): 409-410.
 * https://physics.stackexchange.com/questions/135569/what-is-the-square-root-of-the-dirac-delta-function/548644#548644