The total energy (E) of the electron is an orbit of radius r in hydrogen atom is

Options

A. `e^2/(8πepsilon_0r)`
B. `-e^2/(4πepsilon_0r)`
C. `-e^2/(8πepsilon_0r)`
D. zero

Answer

To find the total energy (E) of the electron in an orbit of radius \( r \) in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Forces Acting on the Electron
The electron in a hydrogen atom experiences a Coulombic force of attraction towards the nucleus (proton). This force can be expressed using Coulomb's law:
\[ F = \frac{k \cdot e^2}{r^2} \]
where:
- \( k \) is Coulomb's constant,
- \( e \) is the charge of the electron (and proton),
- \( r \) is the radius of the orbit.

### Step 2: Relate the Coulombic Force to Centripetal Force
This Coulombic force provides the necessary centripetal force for the electron to maintain its circular motion:
\[ F = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the speed of the electron.

### Step 3: Set the Forces Equal
Since the Coulombic force provides the centripetal force, we can set the two expressions equal:
\[ \frac{k \cdot e^2}{r^2} = \frac{mv^2}{r} \]
Multiplying both sides by \( r \) gives:
\[ \frac{k \cdot e^2}{r} = mv^2 \]

### Step 4: Calculate the Kinetic Energy (KE)
The kinetic energy of the electron can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Substituting \( mv^2 \) from the previous equation:
\[ KE = \frac{1}{2} \cdot \frac{k \cdot e^2}{r} \]
Thus, we have:
\[ KE = \frac{k \cdot e^2}{2r} \]

### Step 5: Calculate the Potential Energy (PE)
The potential energy of the electron in the electric field of the proton is given by:
\[ PE = -\frac{k \cdot e^2}{r} \]
The negative sign indicates that the force is attractive.

### Step 6: Calculate Total Energy (E)
The total energy \( E \) of the electron is the sum of its kinetic and potential energies:
\[ E = KE + PE \]
Substituting the expressions for kinetic and potential energy:
\[ E = \frac{k \cdot e^2}{2r} - \frac{k \cdot e^2}{r} \]
This simplifies to:
\[ E = \frac{k \cdot e^2}{2r} - \frac{2k \cdot e^2}{2r} = -\frac{k \cdot e^2}{2r} \]

### Step 7: Substitute the Value of \( k \)
The value of \( k \) is given by:
\[ k = \frac{1}{4 \pi \epsilon_0} \]
Substituting this into the expression for total energy:
\[ E = -\frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{2r} = -\frac{e^2}{8 \pi \epsilon_0 r} \]

### Final Result
Thus, the total energy \( E \) of the electron in an orbit of radius \( r \) in a hydrogen atom is:
\[ E = -\frac{e^2}{8 \pi \epsilon_0 r} \]

---