The total energy of an electron in the excited state corresponding to n =3 state is E. What is its potential energy with proper sign?

Options

A. -2E
B. 2E
C. -E
D. E

Answer

To find the potential energy of an electron in the excited state corresponding to n = 3, we can follow these steps:

### Step 1: Understand the relationship between total energy and potential energy
In the hydrogen atom model, the total energy \( E \) of an electron in a given energy level \( n \) is related to its potential energy \( U \) by the equation:
\[ E = \frac{U}{2} \]
This means that the potential energy is twice the total energy, but with the opposite sign.

### Step 2: Express potential energy in terms of total energy
From the relationship established in Step 1, we can express the potential energy as:
\[ U = 2E \]
Since the total energy \( E \) is negative for bound states (like in the case of an electron in a hydrogen atom), we write:
\[ U = -2|E| \]
where \( |E| \) is the magnitude of the total energy.

### Step 3: Determine the sign of potential energy
Since the total energy \( E \) is negative, the potential energy \( U \) will also be negative. Thus, we can conclude:
\[ U = -2E \]

### Final Answer
If the total energy of the electron in the excited state corresponding to \( n = 3 \) is \( E \), then the potential energy \( U \) is:
\[ U = -2E \]

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