A wire of length l and area A is connected to an ideal battery . The drift speed of electrons is v. If the wire is twisted to half its length, what will be the new drift speed ?

Options

A. v
B. 2v
C. `(v )/(2)`
D. `(v)/(4)`

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the relationship between current, drift velocity, and wire properties
The current \( I \) flowing through a wire can be expressed as:
\[ I = n \cdot A \cdot e \cdot v_d \]
where:
- \( n \) = number of charge carriers per unit volume,
- \( A \) = cross-sectional area of the wire,
- \( e \) = charge of an electron,
- \( v_d \) = drift velocity of the electrons.

### Step 2: Relate current to resistance and voltage
From Ohm's law, we know that:
\[ I = \frac{V}{R} \]
where \( V \) is the voltage of the battery and \( R \) is the resistance of the wire. The resistance \( R \) of the wire can be expressed as:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) = resistivity of the material,
- \( L \) = length of the wire.

### Step 3: Substitute resistance into the current equation
Substituting the expression for resistance into the current equation gives us:
\[ I = \frac{V}{\frac{\rho L}{A}} = \frac{V \cdot A}{\rho L} \]

### Step 4: Set the two expressions for current equal to each other
Now we can set the two expressions for current equal to each other:
\[ n \cdot A \cdot e \cdot v_d = \frac{V \cdot A}{\rho L} \]
Here, we can cancel \( A \) from both sides:
\[ n \cdot e \cdot v_d = \frac{V}{\rho L} \]

### Step 5: Express the relationship between drift velocity and length
From the above equation, we can express \( v_d \):
\[ v_d = \frac{V}{\rho n e L} \]
This shows that drift velocity \( v_d \) is inversely proportional to the length \( L \) of the wire.

### Step 6: Analyze the change in length of the wire
If the wire is twisted to half its length, the new length \( L_2 \) becomes:
\[ L_2 = \frac{L}{2} \]

### Step 7: Relate the new drift velocity to the original drift velocity
Using the relationship derived earlier, we can write:
\[ v_{d2} = \frac{V}{\rho n e L_2} \]
Substituting \( L_2 = \frac{L}{2} \):
\[ v_{d2} = \frac{V}{\rho n e \left(\frac{L}{2}\right)} = \frac{2V}{\rho n e L} \]

### Step 8: Compare the new drift velocity with the original drift velocity
Since the original drift velocity \( v_d = \frac{V}{\rho n e L} \), we can relate the new drift velocity to the original:
\[ v_{d2} = 2 \cdot v_d \]

### Final Result
Thus, the new drift speed \( v_{d2} \) when the wire is twisted to half its length is:
\[ v_{d2} = 2v \]