Two copper wires with their lengths in the ratio 1 : 2 and resistances in the ratio 1 : 2 are connected (i) in series (ii) in parallel with a battery. What will be the ratio of drift velocities of free electrons in two wires in (i) and (ii) ?

Explanation

To find the ratio of drift velocities of free electrons in two copper wires connected in series and parallel, we can follow these steps:

### Step 1: Understand the Given Ratios
We have two copper wires:
- Lengths in the ratio \( L_1 : L_2 = 1 : 2 \)
- Resistances in the ratio \( R_1 : R_2 = 1 : 2 \)

### Step 2: Analyze the Series Connection
In a series connection, the current \( I \) through both wires is the same. The current can be expressed in terms of drift velocity using the formula:
\[ I = n e A v_d \]
where:
- \( n \) = number of charge carriers per unit volume (constant for copper)
- \( e \) = charge of an electron (constant)
- \( A \) = cross-sectional area of the wire
- \( v_d \) = drift velocity

For wire 1:
\[ I = n e A_1 v_{d1} \]
For wire 2:
\[ I = n e A_2 v_{d2} \]

Since the current is the same:
\[ n e A_1 v_{d1} = n e A_2 v_{d2} \]
This simplifies to:
\[ A_1 v_{d1} = A_2 v_{d2} \]

### Step 3: Relate Areas to Resistances
Using the resistance formula:
\[ R = \rho \frac{L}{A} \]
For wire 1:
\[ R_1 = \rho \frac{L_1}{A_1} \]
For wire 2:
\[ R_2 = \rho \frac{L_2}{A_2} \]

Given \( R_1 : R_2 = 1 : 2 \), we can write:
\[ \frac{R_1}{R_2} = \frac{L_1 / A_1}{L_2 / A_2} = \frac{1}{2} \]
This leads to:
\[ \frac{A_2}{A_1} = \frac{L_2}{L_1} = \frac{2}{1} \]

### Step 4: Substitute Area Ratios
Substituting \( A_2 = 2 A_1 \) into the current equation:
\[ A_1 v_{d1} = (2 A_1) v_{d2} \]
This simplifies to:
\[ v_{d1} = 2 v_{d2} \]

### Step 5: Find the Ratio of Drift Velocities in Series
Thus, the ratio of drift velocities in series is:
\[ \frac{v_{d1}}{v_{d2}} = 2 \]

### Step 6: Analyze the Parallel Connection
In a parallel connection, the potential difference \( V \) across both wires is the same. Using Ohm's law:
\[ V = I_1 R_1 = I_2 R_2 \]

For wire 1:
\[ V = n e A_1 v_{d1} R_1 \]
For wire 2:
\[ V = n e A_2 v_{d2} R_2 \]

Since \( V \) is the same:
\[ n e A_1 v_{d1} R_1 = n e A_2 v_{d2} R_2 \]
This simplifies to:
\[ A_1 v_{d1} R_1 = A_2 v_{d2} R_2 \]

### Step 7: Substitute Resistance Ratios
Using \( R_1 : R_2 = 1 : 2 \):
\[ A_1 v_{d1} (1) = A_2 v_{d2} (2) \]
Substituting \( A_2 = 2 A_1 \):
\[ A_1 v_{d1} = (2 A_1) v_{d2} \]
This leads to:
\[ v_{d1} = 2 v_{d2} \]

### Step 8: Find the Ratio of Drift Velocities in Parallel
Thus, the ratio of drift velocities in parallel is:
\[ \frac{v_{d1}}{v_{d2}} = 2 \]

### Final Result
The ratio of drift velocities in both series and parallel connections is:
- In series: \( \frac{v_{d1}}{v_{d2}} = 1 \)
- In parallel: \( \frac{v_{d1}}{v_{d2}} = 2 \)