Question

What current flows through a 2.54-cm-diameter rod of pure silicon that is 20 cmlong, when$\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{V}$is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)

Short answer

The current in the rod is obtained as: 1.10 mA .

Step-by-step solution

Define Resistance

In an electrical circuit, resistance is a measure of the resistance to current flow. The Greek letter omega$\left(\Omega \right)$is used to represent resistance in ohms.

Concepts and Principles

Ohm's Law: The current $I$through a circuit element (other than the battery) can be determined by dividing the potential difference $∆V$which is across the circuit element by its resistance$R$ :

$I=\frac{∆V}{R}$

The electrical resistance of the value $R$of a resistive circuit element depends on its geometric structure (its length symbolized as $L$and cross-sectional area symbolized as ) and on the resistivity $\rho$of the material of which it is made:

$R=\rho \frac{L}{A}$

The given data

• The diameter of the rod is:$D=2.54\mathrm{cm}\times \frac{1\mathrm{m}}{100\mathrm{cm}}=0.0254\mathrm{m}$.
• The radius of the rod is:$r=D/2=0.0127\mathrm{m}$.
• The length of the rod is:$L=20.0\mathrm{cm}\times \frac{1\mathrm{m}}{100\mathrm{cm}}=0.200\mathrm{m}$.
• The voltage applied to the rod is:$∆\mathrm{V}=1.00\times {10}^3\mathrm{V}$.
• The resistivity of silicon is: $\rho =2300\mathrm{\Omega }·\mathrm{m}$.

Evaluating the current in the rod

The current in the rod can be calculated as:

$I=\frac{∆V}{\rho {\displaystyle \frac{L}{\pi r^2}}}$

Entering the values, we obtain:

$$\begin{gathered} I=\frac{1.00\times {10}^3\mathrm{V}}{2300\mathrm{\Omega }·\mathrm{m}\times {\displaystyle \frac{0.200\mathrm{m}}{\mathrm{\pi }{\left(0.0127\mathrm{m}\right)}^2}}} \\ =1.10\times {10}^{-3}\mathrm{A} \\ =1.10\times {10}^{-3}\mathrm{A}\times \frac{1000.\mathrm{mA}}{1\mathrm{A}} \\ =1.10\mathrm{mA} \end{gathered}$$

Therefore,the current in the rod is: data-custom-editor="chemistry" $1.10\mathrm{mA}$.