Two long straight wires that lie along the same line have a separation at their tips of 0.25 nm.The potential energy of an electron in the gap is about 1eV higher than it is in the conduction band of the two wires.What is the probability that a conduction electron in one wire will be found in the other wire after arriving at the gap?
A)0.1%
B)1.0%
C)3.8%
D)7.7%
E)28%

Question

Quizplus · Accepted Answer

The probability of tunneling can be calculated using the formula:

$$P \propto e^{-2kd}$$

where $k$ is the wave vector of the electron and $d$ is the distance across which it must tunnel.

For an electron in the conduction band, we can estimate the wave vector as:

$$k \approx \sqrt{2mE}/\hbar$$

where $m$ is the mass of the electron, $E$ is the energy difference across the gap, and $\hbar$ is the reduced Planck constant.

Substituting the given values, we get:

$$k \approx 2.72 	imes 10^{10} 	ext{m}^{-1}$$
$$d = 0.25 	ext{ nm} = 2.5 	imes 10^{-10} 	ext{ m}$$
$$E = 1 	ext{ eV} = 1.6 	imes 10^{-19} 	ext{ J}$$

Plugging these values into the tunneling formula and simplifying, we get:

$$P \approx 7.7\%$$

Therefore, the correct answer is (D) 7.7%.

Two Long Straight Wires That Lie Along the Same Line