The current of a conductor following through a conductor in terms of the drift speed of electrons is (the symbols have their usual meanings)

Options

A. `I=sqrt(nAe)vd`
B. `I=Asqrt("ne")vd`
C. `I=("ne")/(Avd)`
D. `I="ne"Avd`

Answer

To derive the expression for the current \( I \) in a conductor in terms of the drift speed \( v_d \) of the electrons, we can follow these steps:

### Step 1: Understand the definition of current
Current \( I \) is defined as the amount of charge \( Q \) flowing through a cross-section of a conductor per unit time \( t \). Mathematically, this is expressed as:
\[ I = \frac{Q}{t} \]

### Step 2: Relate charge to the number of electrons
The charge \( Q \) can be expressed in terms of the number of electrons \( n_e \) that pass through the conductor. The charge of a single electron is denoted by \( e \). Therefore, the total charge can be expressed as:
\[ Q = n_e \cdot e \]

### Step 3: Determine the number of electrons passing through a section
To find \( n_e \), we need to consider the volume of the conductor that the electrons occupy. If \( A \) is the cross-sectional area of the conductor and \( v_d \) is the drift speed of the electrons, then the volume \( V \) of the conductor that passes through a section in time \( t \) is given by:
\[ V = A \cdot v_d \cdot t \]

### Step 4: Calculate the number of electrons in that volume
If \( n \) is the electron density (number of electrons per unit volume), then the number of electrons \( n_e \) that pass through the cross-section in time \( t \) is:
\[ n_e = n \cdot V = n \cdot (A \cdot v_d \cdot t) \]

### Step 5: Substitute \( n_e \) into the expression for current
Now substitute \( n_e \) back into the expression for current:
\[ I = \frac{Q}{t} = \frac{n_e \cdot e}{t} = \frac{(n \cdot A \cdot v_d \cdot t) \cdot e}{t} \]

### Step 6: Simplify the expression
The \( t \) in the numerator and denominator cancels out, giving us:
\[ I = n \cdot A \cdot v_d \cdot e \]

### Final Expression
Thus, the expression for the current \( I \) in terms of the drift speed \( v_d \) is:
\[ I = n \cdot A \cdot e \cdot v_d \]