A potential difference of 30V is applied between the ends of a conductor of length 100 m and resistance `0.5Omega` . The conductor has a uniform cross-section. Find the total linear momentum of free electrons.

Options

A. `10^(-9)kg.m//s`
B. `6.8xx10^(-8)kg.m//s`
C. `3.4xx10^(-9)kg.m//s`
D. `3.4xx10^(-8)kg.m//s`

Answer

To find the total linear momentum of free electrons in a conductor with a given potential difference, resistance, and length, we can follow these steps:

### Step 1: Calculate the Current (I)
Using Ohm's Law, we can calculate the current flowing through the conductor. The formula is:

\[ I = \frac{V}{R} \]

Where:
- \( V = 30 \, \text{V} \) (potential difference)
- \( R = 0.5 \, \Omega \) (resistance)

Substituting the values:

\[ I = \frac{30 \, \text{V}}{0.5 \, \Omega} = 60 \, \text{A} \]

### Step 2: Use the Formula for Linear Momentum (P)
The total linear momentum of free electrons can be expressed as:

\[ P = n \cdot m_e \cdot A \cdot L \cdot v_d \]

Where:
- \( n \) = number density of free electrons
- \( m_e \) = mass of a single electron \( \approx 9.1 \times 10^{-31} \, \text{kg} \)
- \( A \) = cross-sectional area of the conductor
- \( L \) = length of the conductor \( = 100 \, \text{m} \)
- \( v_d \) = drift velocity of electrons

We also know from the current formula that:

\[ I = n \cdot A \cdot e \cdot v_d \]

Where \( e \) is the charge of an electron \( \approx 1.6 \times 10^{-19} \, \text{C} \).

### Step 3: Rearranging the Formula
From the current equation, we can express \( n \cdot A \cdot v_d \):

\[ n \cdot A \cdot v_d = \frac{I}{e} \]

### Step 4: Substitute into the Momentum Equation
Now we can substitute this expression into the momentum formula:

\[ P = \left(\frac{I}{e}\right) \cdot m_e \cdot L \]

### Step 5: Substitute the Known Values
Now substituting the known values into the equation:

- \( I = 60 \, \text{A} \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)
- \( L = 100 \, \text{m} \)

So,

\[ P = \left(\frac{60}{1.6 \times 10^{-19}}\right) \cdot (9.1 \times 10^{-31}) \cdot 100 \]

### Step 6: Calculate the Momentum
Calculating the expression step by step:

1. Calculate \( \frac{60}{1.6 \times 10^{-19}} \):

\[ \frac{60}{1.6} = 37.5 \times 10^{19} \]

2. Multiply by \( 9.1 \times 10^{-31} \):

\[ 37.5 \times 10^{19} \cdot 9.1 \times 10^{-31} = 3.4125 \times 10^{-11} \]

3. Finally, multiply by \( 100 \):

\[ 3.4125 \times 10^{-11} \cdot 100 = 3.4125 \times 10^{-9} \]

Thus, the total linear momentum \( P \) is approximately:

\[ P \approx 3.4 \times 10^{-8} \, \text{kg m/s} \]

### Final Answer
The total linear momentum of free electrons in the conductor is:

\[ P \approx 3.4 \times 10^{-8} \, \text{kg m/s} \]

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