The angular momentum of electron in a given orbit is J . Its kinetic energy will be :

Options

A. `1/2(J^(2))/(mr^(2))`
B. `(Jv)/(r)`
C. `(J^(2))/(2m)`
D. `(J^(2))/(2pi)`

Answer

To find the kinetic energy of an electron in a given orbit when its angular momentum is given as \( J \), we can follow these steps:

### Step 1: Understand the relationship between angular momentum and kinetic energy
The angular momentum \( J \) of an electron in a circular orbit can be expressed as:
\[ J = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 2: Relate angular momentum to linear momentum
The angular momentum can also be expressed in terms of the mass \( m \), velocity \( v \), and radius \( r \) of the orbit:
\[ J = m v r \]

### Step 3: Square the angular momentum equation
By squaring the angular momentum equation, we have:
\[ J^2 = (m v r)^2 = m^2 v^2 r^2 \]

### Step 4: Express kinetic energy in terms of velocity
The kinetic energy \( KE \) of the electron is given by:
\[ KE = \frac{1}{2} m v^2 \]

### Step 5: Isolate \( v^2 \) in terms of \( J \)
From the squared angular momentum equation, we can express \( v^2 \):
\[ v^2 = \frac{J^2}{m^2 r^2} \]

### Step 6: Substitute \( v^2 \) into the kinetic energy equation
Now, substitute \( v^2 \) into the kinetic energy equation:
\[ KE = \frac{1}{2} m \left(\frac{J^2}{m^2 r^2}\right) \]

### Step 7: Simplify the kinetic energy expression
This simplifies to:
\[ KE = \frac{1}{2} \frac{J^2}{m r^2} \]

### Final Result
Thus, the kinetic energy of the electron in terms of its angular momentum \( J \) is:
\[ KE = \frac{1}{2} \frac{J^2}{m r^2} \]