A current of 5 A of passes through a copper conductor (resistivity =`1.7xx10^(-8)Omegam)` of radius of cross-section 5 mm.Find the mobility of the charges if their drift velocity is `1.1xx10^(-3)m//s`

Options

A. `1.8 m^2//Vs`
B. `1.5 m^2//Vs`
C. `1.3 m^2//Vs`
D. `1.0 m^2//Vs`

Answer

To solve the problem, we need to find the mobility of the charges in a copper conductor given the current, resistivity, radius of the conductor, and drift velocity. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Current (I) = 5 A
- Resistivity (ρ) = \(1.7 \times 10^{-8} \, \Omega \cdot m\)
- Radius (r) = 5 mm = \(5 \times 10^{-3} \, m\)
- Drift velocity (v_d) = \(1.1 \times 10^{-3} \, m/s\)

### Step 2: Calculate the cross-sectional area (A) of the conductor
The cross-sectional area \(A\) of a circular conductor is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (5 \times 10^{-3})^2 = \pi (25 \times 10^{-6}) = 25\pi \times 10^{-6} \, m^2 \]

### Step 3: Calculate the current density (J)
Current density \(J\) is defined as the current per unit area:
\[ J = \frac{I}{A} \]
Substituting the values:
\[ J = \frac{5}{25\pi \times 10^{-6}} = \frac{5}{25\pi} \times 10^{6} = \frac{1}{5\pi} \times 10^{6} \, A/m^2 \]

### Step 4: Calculate the conductivity (σ)
The conductivity \(σ\) is the reciprocal of resistivity:
\[ \sigma = \frac{1}{\rho} = \frac{1}{1.7 \times 10^{-8}} \, S/m \]

### Step 5: Calculate the electric field (E)
The electric field \(E\) can be calculated using the formula:
\[ E = \rho J \]
Substituting the values:
\[ E = (1.7 \times 10^{-8}) \left(\frac{1}{5\pi} \times 10^{6}\right) \]
Calculating this gives:
\[ E = \frac{1.7 \times 10^{-2}}{5\pi} \, V/m \]

### Step 6: Calculate the mobility (μ)
The mobility \(μ\) is defined as:
\[ \mu = \frac{v_d}{E} \]
Substituting the drift velocity and the electric field:
\[ \mu = \frac{1.1 \times 10^{-3}}{E} \]
Substituting the expression for \(E\):
\[ \mu = \frac{1.1 \times 10^{-3}}{\frac{1.7 \times 10^{-2}}{5\pi}} = \frac{1.1 \times 10^{-3} \times 5\pi}{1.7 \times 10^{-2}} \]

### Step 7: Calculate the final value of mobility
Calculating the above expression gives:
\[ \mu \approx 1.01 \, m^2/V \cdot s \]

### Final Answer
The mobility of the charges is approximately \(1.01 \, m^2/V \cdot s\).

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