- #1

Jan05

- 4

- 0

- Homework Statement
- Calculate the resistance in the ##x##-direction

- Relevant Equations
- ## R = \rho ~ \frac{l}{A} ##

I'm working on the following differential resistance problem and wondering wether I'm approaching the question the right way. The problem is to find an expression for the resistance in the ##x##-direction of a wire with electrical resistivity ## ~ \rho ~ ## in terms of ##~ z_0 ~##, ##y_0 ~ ##, ##x_0 ~ ## and ## ~ \theta##. I approached it the following way:

We can write ##x_i## in terms of the given variables as:

$$

x_i = \frac{z_0}{tan ~ \theta}

$$

Then add the contribution to the resistance of the two shapes together where we can write the contribution of the prism as an integral:

$$

R = \rho ~ \frac{l}{A} = \rho ~ ( \frac{x_0 - x_i}{y_0 z_0} + \int_{x_0 - x_i} ^{x_0} \frac{dx}{\frac{1}{2} x z_0})

$$

And finally:

\begin{equation}

R = \rho ~ ( \frac{x_0}{y_0 z_0} - \frac{z_0}{z_0 y_0 ~ tan ~ \theta} + \frac{2}{z_0} ~ ln ~ \frac{x_0}{x_0 - x_i}) = \rho ~ ( \frac{x_0}{y_0 z_0} - \frac{1}{y_0 ~ tan ~ \theta} + \frac{2}{z_0} ~ ln ~ \frac{x_0 ~ tan ~ \theta}{x_0 ~ tan ~ \theta - x_0})

\end{equation}