An electron moves in a circular orbit at a distance from a proton with kinetic energy E to escape to infinity, the minimum energy which must be supplied to the electron is

Options

A. E
B. 2E
C. 0.5E
D. `Esqrt(2)`

Explanation

To solve the problem of determining the minimum energy that must be supplied to an electron moving in a circular orbit around a proton to escape to infinity, we can follow these steps:

### Step 1: Understand the Energy Components
The total energy (E_total) of an electron in a circular orbit is the sum of its kinetic energy (K.E.) and potential energy (P.E.).

### Step 2: Write the Expressions for Kinetic and Potential Energy
1. The kinetic energy (K.E.) of the electron is given as \( E \).
2. The potential energy (P.E.) of the electron in the electric field of the proton is given by the formula:
\[ P.E. = -\frac{ke^2}{r} \]
where \( k \) is Coulomb's constant, \( e \) is the charge of the electron, and \( r \) is the distance from the proton.

### Step 3: Calculate the Total Energy
The total energy (E_total) of the electron can be expressed as:
\[ E_{total} = K.E. + P.E. = E - \frac{ke^2}{r} \]

### Step 4: Determine the Minimum Energy Required to Escape
To escape to infinity, the total energy must be zero or greater. Therefore, we set up the equation:
\[ E_{total} + E_{min} = 0 \]
where \( E_{min} \) is the minimum energy supplied to the electron to escape.

Rearranging gives:
\[ E_{min} = -E_{total} \]

Substituting the expression for \( E_{total} \):
\[ E_{min} = -\left(E - \frac{ke^2}{r}\right) \]
\[ E_{min} = \frac{ke^2}{r} - E \]

### Step 5: Conclusion
Thus, the minimum energy that must be supplied to the electron to escape to infinity is:
\[ E_{min} = \frac{ke^2}{r} - E \]