A straight conductor of uniform cross section carries a time varying current, which varies at the rate dI//dt = I. If s is the specific charge that is carried by each charge carries of the conductor and l is the length of the condcutor, then the totalt force experienced by all the charge carries per unit length of the conductor due to their drift velocities only is

Options

A. `F = I s`
B. `F = (I)/(2sqrtls)`
C. `F = (I)/(s)`
D. `F = (2I l)/(s)`

Explanation

To solve the problem, we need to determine the total force experienced by all the charge carriers per unit length of a straight conductor carrying a time-varying current. Here’s the step-by-step solution:

### Step 1: Define the parameters
Let:
- \( I \) = current flowing through the conductor
- \( \frac{dI}{dt} = I \) = rate of change of current
- \( s \) = specific charge carried by each charge carrier
- \( l \) = length of the conductor
- \( n \) = number of charge carriers per unit volume
- \( A \) = cross-sectional area of the conductor
- \( e \) = charge of an individual charge carrier (electron)

### Step 2: Calculate the number of charge carriers in the conductor
The total volume of the conductor is given by \( A \times l \). Therefore, the total number of charge carriers in the conductor is:
\[ N = n \cdot A \cdot l \]

### Step 3: Calculate the drift velocity
The current \( I \) can be expressed in terms of the drift velocity \( v_d \):
\[ I = n \cdot A \cdot e \cdot v_d \]
From this, we can solve for the drift velocity \( v_d \):
\[ v_d = \frac{I}{n \cdot A \cdot e} \]

### Step 4: Calculate the momentum of charge carriers
The momentum \( P \) of all the charge carriers can be expressed as:
\[ P = N \cdot m \cdot v_d = (n \cdot A \cdot l) \cdot m \cdot v_d \]
Substituting the expression for \( v_d \):
\[ P = (n \cdot A \cdot l) \cdot m \cdot \left(\frac{I}{n \cdot A \cdot e}\right) \]
This simplifies to:
\[ P = \frac{l \cdot m \cdot I}{e} \]

### Step 5: Calculate the force using Newton's second law
According to Newton's second law, the force \( F \) is the rate of change of momentum:
\[ F = \frac{dP}{dt} \]
Substituting the expression for \( P \):
\[ F = \frac{d}{dt}\left(\frac{l \cdot m \cdot I}{e}\right) = \frac{l \cdot m}{e} \cdot \frac{dI}{dt} \]

### Step 6: Relate \( \frac{dI}{dt} \) to \( s \)
Given that \( \frac{dI}{dt} = I \) and \( s = \frac{e}{m} \) (specific charge), we can express the force per unit length:
\[ \frac{F}{l} = \frac{l \cdot m}{e} \cdot \frac{dI}{dt} = \frac{l \cdot m}{e} \cdot I \]
Rearranging gives:
\[ \frac{F}{l} = \frac{I}{s} \]

### Final Answer
Thus, the total force experienced by all the charge carriers per unit length of the conductor is:
\[ \frac{F}{l} = \frac{I}{s} \]