Thermal speed v of free electrons in a conductor at absolute temperaure T is proportional to

Options

A. T
B. `sqrt(T)`
C. `(1)/(sqrt(T))`
D. `(1)/(T)`

Answer

To solve the question regarding the thermal speed \( v \) of free electrons in a conductor at absolute temperature \( T \), we can follow these steps:

### Step 1: Understand the relationship between energy and temperature
The energy of an electron at absolute temperature \( T \) is given by the equation:
\[ E = \frac{1}{2} k T \]
where \( k \) is the Boltzmann constant.

### Step 2: Write the expression for kinetic energy
The kinetic energy \( KE \) of an electron moving with speed \( v \) is given by:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the electron.

### Step 3: Set the two energy expressions equal
Since the thermal energy of the electron is converted into kinetic energy, we can equate the two expressions:
\[ \frac{1}{2} k T = \frac{1}{2} m v^2 \]

### Step 4: Simplify the equation
We can cancel the \( \frac{1}{2} \) from both sides:
\[ k T = m v^2 \]

### Step 5: Solve for \( v^2 \)
Rearranging the equation gives:
\[ v^2 = \frac{k T}{m} \]

### Step 6: Take the square root to find \( v \)
Taking the square root of both sides, we find:
\[ v = \sqrt{\frac{k T}{m}} \]

### Step 7: Identify the proportional relationship
Since \( k \) (Boltzmann constant) and \( m \) (mass of the electron) are constants, we can conclude that:
\[ v \propto \sqrt{T} \]

### Final Answer
Thus, the thermal speed \( v \) of free electrons in a conductor at absolute temperature \( T \) is proportional to \( \sqrt{T} \).

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