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 known, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between angular momentum and kinetic energy**:
The angular momentum (J) of an electron in an orbit is given by the formula:
\[ J = \frac{nH}{2\pi} \]
where \( n \) is the principal quantum number and \( H \) is Planck's constant.

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:
\[ J = MVR \]
where \( V \) is the linear velocity of the electron and \( R \) is the radius of the orbit.

3. **Square the angular momentum equation**:
If we square the angular momentum equation, we have:
\[ J^2 = (MVR)^2 = M^2V^2R^2 \]

4. **Express kinetic energy**:
The kinetic energy (KE) of the electron is given by:
\[ KE = \frac{1}{2} MV^2 \]

5. **Relate kinetic energy to angular momentum**:
From the squared angular momentum equation, we can express \( MV^2 \) in terms of \( J \):
\[ MV^2 = \frac{J^2}{R^2} \]
Substituting this into the kinetic energy formula gives:
\[ KE = \frac{1}{2} M \left(\frac{J^2}{M^2R^2}\right) = \frac{J^2}{2MR^2} \]

6. **Final expression for kinetic energy**:
Therefore, we can express the kinetic energy in terms of angular momentum:
\[ KE = \frac{J^2}{2MR^2} \]