Question

When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) T equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of T = 1 occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies E that will result in T = 1. You assume a barrier height of $E = 12$ eV and a width of $1.7 \times 10^{-10}$ m. Part A Calculate the first lowest value of E for which T = 1. Express your answer in electron volts. $E_1 =$ $eV$ Submit Request Answer Part B Calculate the second lowest value of E for which T = 1. Express your answer in electron volts. $E_2 =$ Submit Request Answer Part C Calculate the third lowest value of E for which T = 1. Express your answer in electron volts. $E_3 =$ $eV$ Submit Request Answer

          When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas
as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) T equal
to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of T = 1 occurs
when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the
barrier equal the width of the barrier. You are planning an experiment to measure this effect. To assist you
in designing the necessary apparatus, you estimate the electron energies E that will result in T = 1. You
assume a barrier height of $E = 12$ eV and a width of $1.7 \times 10^{-10}$ m.
Part A
Calculate the first lowest value of E for which T = 1.
Express your answer in electron volts.
$E_1 =$
$eV$
Submit
Request Answer
Part B
Calculate the second lowest value of E for which T = 1.
Express your answer in electron volts.
$E_2 =$
Submit
Request Answer
Part C
Calculate the third lowest value of E for which T = 1.
Express your answer in electron volts.
$E_3 =$
$eV$
Submit
Request Answer

Show more…

When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas
as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) T equal
to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of T = 1 occurs
when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the
barrier equal the width of the barrier. You are planning an experiment to measure this effect. To assist you
in designing the necessary apparatus, you estimate the electron energies E that will result in T = 1. You
assume a barrier height of E = 12 eV and a width of 1.7 × 10^-10 m.
Part A
Calculate the first lowest value of E for which T = 1.
Express your answer in electron volts.
E1 =
eV
Submit
Request Answer
Part B
Calculate the second lowest value of E for which T = 1.
Express your answer in electron volts.
E2 =
Submit
Request Answer
Part C
Calculate the third lowest value of E for which T = 1.
Express your answer in electron volts.
E3 =
eV
Submit
Request Answer

Added by Rachel C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Daniel Sneed

07/02/2019

Step 1

The de Broglie wavelength is given by the formula λ = h / p, where h is the Planck constant (6.626 x 10^-34 m^2 kg / s) and p is the momentum of the electron. Since the electron is low-energy, we can use the non-relativistic formula for momentum: p = sqrt(2mE), Show more…

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When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren' ◻ t there and thus have transmission coefficients (tunneling probabilities) T equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of T=1 occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width L of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies E that will result in T=1. You assume a barrier height of E=12eV and a width of 1.7 imes 10^(-10)m. Part A Calculate the first lowest value of E for which T=1. Express your answer in electron volts. E_(1)= eV Request Answer Part B Calculate the second lowest value of E for which T=1. Express your answer in electron volts. oot(◻)()ASigma phi ? E_(2)= eV Request Answer Part C Calculate the third lowest value of E for which T=1. Express your answer in electron volts. oot(◻)(◻) ASigma phi eV Part A When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities)T' equal to unity.The gas ions can be modeled approximately as a rectangular barrier.The value of T= 1 occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width I of the barrier You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies that will result in =1. You Calculate the first lowest value of E for which T = 1. Express your answer in electron volts. E= Submit Request Answer Part B Calculate the second lowest value of E for which T=1 Express your answer in electron volts. E Submit Request Answer Part C Calculate the third lowest value of E for which T = 1. Express your answer in electron volts. E3 Submit Request Answer

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Transcript

00:02 All right.

00:03 So problem number 63 states that when low energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there, and thus have transmission coefficients, tunneling probabilities t equal to unity.

00:19 The gas ions can be modeled approximately as a rectangular barrier.

00:22 The value of t equals 1 occurs when an integral or half integral number of debroli wavelengths of the electron, as it passes over the barrier equal the width l of the barrier.

00:34 You are planning an experiment to measure this effect.

00:37 To assist you in designing the necessary apparatus, you estimate the electron energies e that will result in t equals 1.

00:44 You assume a barrier height of 10 electron volts and a width of 1 .8 times 10 to the minus 10 meters.

00:50 Calculate the three lowest values of e for which t equals 1.

00:53 All right.

00:54 So basically what this problem is saying is that if you're, debroli wavelength lambda is equal to some half -integer numbers of l, then you will get, and as long as you are above, your energy is higher than the barrier height, you will get 100 % transmission.

01:20 Now, so this tells you that the debroli wavelength has to be in 2l over n, i'm sorry, has to be the debroli wavelength.

01:37 Let me rewrite this here a little bit better.

01:39 So the debroli wavelength lambda has to be equal to 2l over n, where n is some integer number.

01:46 Now, the debroli wavelength is given by h over p.

01:51 P is equal to the square root of 2m .e.

01:55 So that's how you can relate your wavelength to your energy.

01:59 So now we can make the statement that 2l over n is going to be equal to h over the square root of 2m .e.

02:15 This has to be the condition.

02:17 Let me rewrite that again.

02:22 2m .e.

02:25 So this is the condition that must be met in order for there to be a transmission coefficient of of unity or 100 % transmission.

02:36 So now what we know is that the energy barrier here is 10 electron volts.

02:42 So u is equal to 10 ev.

02:47 Now, this is going to be the lowest energy that the electron can be in in order to have 100 % transmission.

02:55 So what we're going to do now is we're going to solve for what energy state that actually corresponds to.

03:02 So we're going to solve for n.

03:04 So n is equal to 2l times the square root of 2m .e over h...

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