The acceleration of electron in the first orbits of hydrogen atom is

Options

A. (a)`(4pi^(2)m)/(h^(3))`
B. (b)`(h^(2))/(4pi^(2)mr)`
C. (c)`(h^(2))/(4pi^(2)m^(2)r^(3))`
D. (d)`(m^(2)h^(2))/(4pi^(2)r^(3))`

Answer

To find the acceleration of an electron in the first orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Concept of Angular Momentum
The angular momentum \( L \) of an electron in a circular orbit is given by the formula:
\[ L = mvr = n \frac{h}{2\pi} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit,
- \( n \) is the principal quantum number (for the first orbit, \( n = 1 \)),
- \( h \) is Planck's constant.

### Step 2: Substitute for the First Orbit
For the first orbit (\( n = 1 \)):
\[ mvr = \frac{h}{2\pi} \]

### Step 3: Solve for Velocity \( v \)
From the angular momentum equation, we can express the velocity \( v \) as:
\[ v = \frac{h}{2\pi m r} \]

### Step 4: Calculate Acceleration \( a \)
The acceleration \( a \) of the electron can be calculated using the centripetal acceleration formula:
\[ a = \frac{v^2}{r} \]

### Step 5: Substitute \( v \) into the Acceleration Formula
Substituting \( v \) from Step 3 into the acceleration formula:
\[ a = \frac{\left(\frac{h}{2\pi m r}\right)^2}{r} \]

### Step 6: Simplify the Expression
Now, simplify the expression:
\[ a = \frac{h^2}{(2\pi m r)^2} \cdot \frac{1}{r} = \frac{h^2}{4\pi^2 m^2 r^3} \]

### Conclusion
Thus, the acceleration of the electron in the first orbit of the hydrogen atom is given by:
\[ a = \frac{h^2}{4\pi^2 m^2 r^3} \]