Question
An electron (mass m) with an initial velocity $\text{v}=\text{v}_0\hat{\text{i}}(\text{v}_0>0)$ is in an electric field $\text{E}=-\text{E}_0\hat{\text{i}}(\text{E}_0=\text{constant}>0)$. It’s de Broglie wavelength at time t is given by:
✓
Answer
- $\frac{\lambda_0}{\bigg(1+\frac{\text{eE}_0}{\text{m}}\frac{\text{t}}{\text{v}_0}\bigg)}$
The initial de-broglie wavelength of electron is given by $\lambda_0=\frac{1}{\text{mv}_0}$
Force on electron in electric field is given by F = -eE
$=-\text{E}[-\text{E}_0\text{i}]$
$=\text{e}\text{E}_0\text{i}$
Now, $\text{F}=\text{ma}$
$\Rightarrow\ \text{a}=\frac{\text{F}}{\text{m}}$
Substituting $\text{F}=\text{eE}_0\text{i}\text{ in a}=\frac{\text{F}}{\text{m}}$, we get
$\text{a}=\frac{\text{eE}_0\text{i}}{\text{m}}$
The velocity of electron after time t is given by
$\text{v}=\text{v}_0\hat{\text{i}}+\text{at}$
Substituting $\text{a}=\frac{\text{eE}_0\text{i}}{\text{m}}$, we get
$\text{v}=\text{v}_0\hat{\text{i}}+\Big(\frac{\text{eE}_0\text{i}}{\text{m}}\Big)\text{t}$
$=\bigg(\text{v}_0+\frac{\text{eE}_0}{\text{m}}\text{t}\bigg)\hat{\text{i}}$
$=\text{v}_0\bigg(1+\frac{\text{eE}_0}{\text{m}\text{v}_0}\text{t}\bigg)\hat{\text{i}}$
The de-Broglie wavelength associated with electron at time t is given by $\lambda=\frac{\text{h}}{\text{mv}}$
Substituting $\text{v}=\text{v}_0\bigg(1+\frac{\text{eE}_0}{\text{m}\text{v}_0}\text{t}\bigg)\hat{\text{i}}$ we get
$\lambda=\frac{\text{h}}{\text{m}\bigg[\text{v}_0\Big(\frac{\text{eE}_0}{\text{mv}_0}\text{t}\Big)\bigg]}$
$\Rightarrow \lambda=\frac{\lambda_0}{\Big(1+\frac{\text{eE}_0}{\text{mv}_0}\text{t}\Big)}\ \ \bigg[\because\ \lambda_0=\frac{\text{h}}{\text{mv}_0}\bigg]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Assertion : Anode of Coolidge tube gets heated up at time of emission of X-rays.
Reason : The anode of Coolidge tube is made of a material of high melting point.(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.(c) If assertion is true but reason is false.(d) If the assertion and reason both are false.
Reason : The anode of Coolidge tube is made of a material of high melting point.(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.(c) If assertion is true but reason is false.(d) If the assertion and reason both are false.
If in the circuit, power dissipation is $150 W,$ then $R$ is
→A galvanometer with a resistance of $12\Omega$ gives full scale deflection when a current of $3\ mA$ is passed. It is required to convert it into a voltmeter which can read up to $18\ V.$ the resistance to be connected is
→Wires $1$ and $2$ carrying currents $i_1$ and $i_2$ respectively are inclined at an angle $\theta$ to each other. What is the force on a small element dl of wire $2$ at a distance of $r$ from wire $1 ($as shown in figure$)$ due to the magnetic field of wire$1$
→If Fresnel’s biprism experiment as held in water inspite of air, then what will be the effect on fringe width (a) Decrease(b) Increase(c) No effect(d) None of these
The output of a step-down transformer is measured to be $24V$ when connected to a $12$ watt light bulb. The value of the peak current is:
→In the circuit shown in the figure, the current flowing in $2 \Omega$ resistance
→What is the electronic configuration of phosphorus?
A vertical wire carries a current in upward direction. An electron beam sent horizontally towards the wire will be deflected:
The magnetic field at the origin due to a current element $\text{i}\text{d}\ \vec{\text{l}}$ placed at a position $\overrightarrow{\text{r}}$ is:
→- $\frac{\mu_0\text{i}}{4\pi}\frac{\text{d}\overrightarrow{\text{l}}\times\overrightarrow{\text{r}}}{\text{r}^3}$
- $-\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$
- $\frac{\mu_0\text{i}}{4\pi}\frac{\overrightarrow{\text{r}}\times\text{d}\overrightarrow{\text{l}}}{\text{r}^3}$
- $-\frac{\mu_0\text{i}}{4\pi}\frac{\text{d}\overrightarrow{\text{l}}\times\overrightarrow{\text{r}}}{\text{r}^3}$