If the kinetic energy of a moving particle is E , then the de-Broglie wavelength is

Options

A. `lamda=hsqrt(2mE)`
B. `lamda=sqrt((2mE)/h)`
C. `lamda=h/(sqrt(2mE))`
D. `lamda=(hE)/(sqrt(2mE))`

Answer

To find the de-Broglie wavelength in terms of the kinetic energy \( E \) of a moving particle, we can follow these steps:

### Step 1: Understand the de-Broglie wavelength formula
The de-Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

**Hint:** Recall that momentum \( p \) can be expressed in terms of mass and velocity.

### Step 2: Relate kinetic energy to momentum
The kinetic energy \( E \) of a particle is given by:
\[ E = \frac{1}{2} mv^2 \]
We can also express momentum \( p \) as:
\[ p = mv \]

**Hint:** You can manipulate the kinetic energy formula to express \( v \) in terms of \( E \) and \( m \).

### Step 3: Express velocity in terms of kinetic energy
From the kinetic energy equation, we can rearrange it to find \( v \):
\[ E = \frac{1}{2} mv^2 \implies v^2 = \frac{2E}{m} \implies v = \sqrt{\frac{2E}{m}} \]

**Hint:** Substitute this expression for \( v \) back into the momentum equation.

### Step 4: Substitute \( v \) into the momentum formula
Now substituting \( v \) into the momentum formula:
\[ p = mv = m\sqrt{\frac{2E}{m}} = \sqrt{2Em} \]

**Hint:** This gives you a direct expression for momentum in terms of kinetic energy.

### Step 5: Substitute momentum into the de-Broglie wavelength formula
Now, substitute \( p \) back into the de-Broglie wavelength formula:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2Em}} \]

**Hint:** This is the final expression for the de-Broglie wavelength in terms of kinetic energy.

### Final Result
Thus, the de-Broglie wavelength \( \lambda \) in terms of the kinetic energy \( E \) is:
\[ \lambda = \frac{h}{\sqrt{2Em}} \]