Question

We will consider the possibility that a free electron actedon by an electric field could gain enough energy to ionize anair molecule in a collision. (a) Consider an electron that startsfrom rest in a region where there is an electric field (due to somecharged objects nearby) whose magnitude is nearly constant. Ifthe electron travels a distance $\mathit{d}$and the magnitude of the electric field is $\mathit{E}$,what isthe potential difference through which the electron travels? (Pay attention to signs: Is the electron traveling with the electric field or opposite to the electric field?) (b) What is the change in potential energy of the system in this process? (c) What is the change in the kinetic energy of the electron in this process? (d) We found the mean free path of an electron in air to be about $\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ }\mathbf{\text{m}}$, and in the previous question you calculated the energy required to knock an electron out of an atom. What is the magnitude of the electric field that would be required in order for an electron to gain sufficient kinetic energy to ionize a nitrogen molecule? (e) The electric field required to cause a spark in air is observed to be about $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }\mathbf{\text{V/m}}$ at STP. What is the ratio of the magnitude of the field you calculated in the previous part to the observed value at STP? (f) What is it reasonable to conclude about this model of how air becomes ionized? (1) Since we used accurate numbers, this is a huge discrepancy, and the model is wrong. (2) Considering the approximations we made, this is pretty good agreement, and the model may be correct.

Short answer

(d) The energy required to knock an electron out of an atom is $2.3\times {10}^{-18} \text{J}$and the magnitude of the electric field that would be required in order for an electron to gain sufficient kinetic energy to ionize a nitrogen molecule is$2.87\times {10}^{7 }\text{V/m}$

Step-by-step solution

Identification of given data

Electron travels a distance is$d=5\times {10}^{-7} \text{m}$

Charge of electron $q=-1.6\times {10}^{-19} \text{C}$

Significance law of conservation of energy

It states that energy cannot be created and destroy, it can only convert from one form to other form

(d) Determining the energy required to knock an electron out of an atom and the magnitude of the electric field that would be required in order for an electron to gain sufficient kinetic energy to ionize a nitrogen molecule

Refer the subpart b,

$\Delta U=q·\Delta V$

Substitute $q=-1.6\times {10}^{-19} \text{C}$and $\Delta V=-14.4\text{V}$

$\begin{array}{rcl}\Delta U& =& \left(-1.6\times {10}^{-19} \text{C}\right)\times \left(-14.4\text{V}\right)\\ & =& 2.3\times {10}^{-18} \text{J}\end{array}$

Refer to subpart c,

$\begin{array}{rcl}\Delta K& =& -qEd\\ E& =& -\frac{\Delta K}{q·d}\end{array}$

Substitute$2.3\times {10}^{-18} \text{J}$for$\Delta K$and$q=-1.6\times {10}^{-19} \text{C}$,$d=5\times {10}^{-7} \text{m}$

role="math" localid="1662208509213" $\begin{array}{rcl}E& =& -\frac{2.3\times {10}^{-18} \text{J}}{\left(-1.6\times {10}^{-19} \text{C}\right)·\left(5\times {10}^{-7} \text{m}\right)}\\ & =& 2.87\times {10}^{7 }\text{V/m}\end{array}$

Hence, the energy required to knock an electron out of an atom is$2.3\times {10}^{-18} \text{J}$

And the magnitude of the electric field that would be required in order for an electron to gain sufficient kinetic energy to ionize a nitrogen molecule is$2.87\times {10}^{7 }\text{V/m}$