In hydrogen atom find magnetic field at center in ground. State if Bohr's radius is `r_(0) = 5xx 10^(-11)`m.

Options

A. 15.20 T
B. 10.90 T
C. 13.95 T
D. 20.00 T

Explanation

To find the magnetic field at the center of a hydrogen atom in the ground state, we can follow these steps:

### Step 1: Understand the Concept of Current Due to Electron Motion
An electron moving in a circular orbit around the nucleus constitutes a current. The magnetic field at the center of the orbit can be calculated using the formula for the magnetic field due to a circular loop of current.

### Step 2: Use the Formula for Magnetic Field
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( R \) is the radius of the orbit.

### Step 3: Calculate the Current
The current \( I \) due to the electron can be expressed as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron and \( T \) is the time period for one complete revolution.

### Step 4: Find the Time Period
The time period \( T \) can be calculated as:
\[ T = \frac{2\pi R}{v} \]
where \( v \) is the velocity of the electron and \( R \) is the radius of the orbit (Bohr radius \( r_0 \)).

### Step 5: Substitute for Current
Substituting \( T \) into the current formula gives:
\[ I = \frac{Qv}{2\pi R} \]

### Step 6: Substitute Current into Magnetic Field Formula
Now substituting \( I \) into the magnetic field formula:
\[ B = \frac{\mu_0 \cdot \frac{Qv}{2\pi R}}{2R} = \frac{\mu_0 Qv}{4\pi R^2} \]

### Step 7: Determine the Velocity of the Electron
Using Bohr's model, the velocity \( v \) of the electron in the ground state (n=1) can be calculated using:
\[ v = \frac{Z e^2}{2 h \epsilon_0 n} \]
For hydrogen, \( Z = 1 \) and \( n = 1 \).

### Step 8: Substitute Values
Substituting the known values:
- Charge of electron \( e = 1.6 \times 10^{-19} \) C
- Planck's constant \( h = 6.63 \times 10^{-34} \) J·s
- Permittivity of free space \( \epsilon_0 = 8.85 \times 10^{-12} \) C²/(N·m²)
- Bohr radius \( r_0 = 5 \times 10^{-11} \) m

Calculating \( v \):
\[ v = \frac{1 \cdot (1.6 \times 10^{-19})^2}{2 \cdot (6.63 \times 10^{-34}) \cdot (8.85 \times 10^{-12})} \approx 2.18 \times 10^6 \text{ m/s} \]

### Step 9: Calculate the Magnetic Field
Substituting \( v \) back into the magnetic field equation:
\[ B = \frac{(4\pi \times 10^{-7}) \cdot (1.6 \times 10^{-19}) \cdot (2.18 \times 10^6)}{4\pi (5 \times 10^{-11})^2} \]
Calculating this gives:
\[ B \approx 13.95 \text{ Tesla} \]

### Final Answer
The magnetic field at the center of the hydrogen atom in the ground state is approximately \( 13.95 \, \text{T} \).

---