A circular coil of 30 turns and radius 15 cm carries a current of 5 A. placed in a uniform magnetic field of 0.2 T normal to the plane of coil. , the coil has an aren of cross section `2 xx 10^(-6)m^(2)` and electron density of `10^(27) m^(-3)` . Calculate the average force on each electron in the coil due to magnetic field.

Answer

To solve the problem of calculating the average force on each electron in a circular coil due to a magnetic field, we can follow these steps:

### Step 1: Calculate the Length of the Coil
The length of the coil (L) can be calculated using the formula for the circumference of a circle:
\[ L = n \times 2\pi r \]
Where:
- \( n = 30 \) (number of turns)
- \( r = 0.15 \, \text{m} \) (radius in meters)

Substituting the values:
\[ L = 30 \times 2\pi \times 0.15 \]
\[ L = 30 \times 2 \times 3.14 \times 0.15 \approx 28.27 \, \text{m} \]

### Step 2: Calculate the Total Force on the Coil
The force (F) on the coil in a magnetic field can be calculated using the formula:
\[ F = I \times L \times B \times \sin(\theta) \]
Where:
- \( I = 5 \, \text{A} \) (current)
- \( B = 0.2 \, \text{T} \) (magnetic field)
- \( \theta = 90^\circ \) (angle between current direction and magnetic field direction, hence \(\sin(90^\circ) = 1\))

Substituting the values:
\[ F = 5 \times 28.27 \times 0.2 \times 1 \]
\[ F \approx 28.27 \, \text{N} \]

### Step 3: Calculate the Volume of the Coil
The volume (V) of the coil can be calculated using the formula:
\[ V = A \times L \]
Where:
- \( A = 2 \times 10^{-6} \, \text{m}^2 \) (cross-sectional area)

Substituting the values:
\[ V = 2 \times 10^{-6} \times 28.27 \]
\[ V \approx 5.654 \times 10^{-5} \, \text{m}^3 \]

### Step 4: Calculate the Total Number of Electrons in the Coil
The total number of electrons (N) can be calculated using the formula:
\[ N = \text{electron density} \times V \]
Where:
- Electron density \( \rho_e = 10^{27} \, \text{m}^{-3} \)

Substituting the values:
\[ N = 10^{27} \times 5.654 \times 10^{-5} \]
\[ N \approx 5.654 \times 10^{22} \]

### Step 5: Calculate the Average Force on Each Electron
The average force (f) on each electron can be calculated using:
\[ f = \frac{F}{N} \]

Substituting the values:
\[ f = \frac{28.27}{5.654 \times 10^{22}} \]
\[ f \approx 5 \times 10^{-22} \, \text{N} \]

### Final Result
The average force on each electron in the coil due to the magnetic field is approximately:
\[ \boxed{5 \times 10^{-22} \, \text{N}} \]