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 find 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 System
The electron is in a circular orbit around a proton, which means it is bound to the proton by electrostatic forces. The electron has a certain kinetic energy (E) while in this orbit.

### Step 2: Determine Total Energy
The total energy (T) of the electron in a bound state (such as in a circular orbit) is given by the relationship between kinetic energy and potential energy. For an electron in orbit, the total energy can be expressed as:
\[ T = K + U \]
where \( K \) is the kinetic energy and \( U \) is the potential energy.

In this case, the potential energy \( U \) is negative and is given by:
\[ U = -2K \]
Thus, the total energy can be expressed as:
\[ T = K - 2K = -K \]

Given that the kinetic energy \( K = E \), we can substitute this into the equation:
\[ T = -E \]

### Step 3: Energy Required to Escape
To escape to infinity, the electron must have a total energy of zero (since at infinity, the potential energy approaches zero). Therefore, the energy required to escape is equal to the negative of the total energy:
\[ \text{Energy required} = -T = -(-E) = E \]

### Conclusion
The minimum energy that must be supplied to the electron to escape to infinity is:
\[ \text{Energy required} = E \]

Thus, the correct answer is \( E \).

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