A straight copper-wire of length 1 = 1000 m and corss-sectional area A = 1.0 `m m^(2)` carries a current i=4.5A. The sum of electric forces acting on all free electrons in the given wire is `M xx 10^(6)N`. Find M. Free electron density and resistivity of copper are `8.5xx10^(28)//m^(3)` and `1.7xx10^(-8)Omega`.m respectively.

Answer

To solve the problem step by step, we will follow the outlined steps based on the information provided in the question.

### Step 1: Calculate the number of free electrons per unit volume (n)

The formula for the number of free electrons per unit volume (n) is given by:

\[ n = \frac{N_A \cdot d}{M} \]

Where:
- \(N_A\) = Avogadro's number = \(6.022 \times 10^{23} \, \text{mol}^{-1}\)
- \(d\) = density of copper = \(8.5 \times 10^{28} \, \text{m}^{-3}\)
- \(M\) = atomic mass of copper = \(63.5 \, \text{g/mol} = 63.5 \times 10^{-3} \, \text{kg/mol}\)

Now, substituting the values:

\[ n = \frac{(6.022 \times 10^{23}) \cdot (8.5 \times 10^{28})}{63.5 \times 10^{-3}} \]

Calculating this gives:

\[ n \approx 8.49 \times 10^{28} \, \text{electrons/m}^3 \]

### Step 2: Calculate the total electric force acting on all free electrons in the wire

The formula for the total electric force (F) acting on all free electrons in the wire is given by:

\[ F = n \cdot e \cdot L \cdot \rho \cdot I \]

Where:
- \(e\) = charge of an electron = \(1.6 \times 10^{-19} \, \text{C}\)
- \(L\) = length of the wire = \(1000 \, \text{m}\)
- \(\rho\) = resistivity of copper = \(1.7 \times 10^{-8} \, \Omega \cdot \text{m}\)
- \(I\) = current = \(4.5 \, \text{A}\)

Substituting the values into the formula:

\[ F = (8.49 \times 10^{28}) \cdot (1.6 \times 10^{-19}) \cdot (1000) \cdot (1.7 \times 10^{-8}) \cdot (4.5) \]

Calculating this gives:

\[ F \approx 2.30 \times 10^{6} \, \text{N} \]

### Step 3: Relate the force to the given expression

According to the problem, the total electric force can also be expressed as:

\[ F = M \times 10^{6} \, \text{N} \]

From our calculation, we have:

\[ 2.30 \times 10^{6} = M \times 10^{6} \]

### Step 4: Solve for M

To find \(M\), we can divide both sides by \(10^{6}\):

\[ M = 2.30 \]

### Final Answer

Thus, the value of \(M\) is:

\[ M = 2.30 \]