We will consider the possibility that a free electron acted on by an electric field could gain enough energy to ionize an air molecule in a collision. (a) Consider an electron that starts from rest in a region where there is an electric field (due to some charged objects nearby) whose magnitude is nearly constant. If the 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.

Step by step solution

01

Identification of given data

Electron travels a distance is$d$

Magnitude of the electric field is $E$

Charge of electron is$q$

02

Significance of electric potential energy

Electric potential energy is the total amount of energy that a unit charge would have if it were positioned anywhere in space.

$\Delta U=q·\Delta V$ ...(i)

Where, $\Delta V$is potential difference, $\Delta U$is the change in electric potential energy and $q$is the charge of electron

03

(b) Determining the change in potential energy of the system in this process

From equation (i)

$\Delta U=q·\Delta V$

Refer the subpart a, put the value of $\Delta V$in above equation

role="math" localid="1662206773548" $\begin{array}{rcl}\Delta U& =& q·\left(-Ed\right)\\ & =& -qEd\end{array}$

Hence, the change in potential energy of the system in this process is$\Delta U=-qEd$

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