To determine the momentum of all the free electrons per unit length of a straight conductor carrying a current \( I \), we need to consider the relationship between current, charge, and drift velocity. Let's break this down step by step.
Understanding Current and Charge
Current (\( I \)) is defined as the flow of electric charge per unit time. Mathematically, it can be expressed as:
I = n \cdot A \cdot q \cdot v_d
- n = number density of charge carriers (electrons) per unit volume
- A = cross-sectional area of the conductor
- q = charge of an electron (approximately \( 1.6 \times 10^{-19} \) coulombs)
- v_d = drift velocity of the electrons
Relating Charge Density to Specific Charge
The specific charge (\( S \)) of an electron is defined as the charge per unit mass. For electrons, it can be expressed as:
S = \frac{q}{m_e}
where \( m_e \) is the mass of an electron (approximately \( 9.11 \times 10^{-31} \) kg). This relationship allows us to express the number density of electrons in terms of the specific charge.
Calculating Momentum
The momentum (\( p \)) of a single electron is given by:
p = m_e \cdot v_d
Now, to find the total momentum of all the free electrons per unit length of the conductor, we need to consider the number of electrons in a unit length of the conductor. The number of electrons per unit length (\( N \)) can be calculated as:
N = n \cdot A
Substituting \( n \) from the current equation, we have:
N = \frac{I}{q \cdot v_d} \cdot A
Combining Everything
The total momentum per unit length (\( P \)) of all the electrons can now be expressed as:
P = N \cdot p = N \cdot (m_e \cdot v_d)
Substituting for \( N \), we get:
P = \left(\frac{I}{q \cdot v_d} \cdot A\right) \cdot (m_e \cdot v_d)
Notice that \( v_d \) cancels out:
P = \frac{I \cdot m_e \cdot A}{q}
Expressing in Terms of Specific Charge
Now, substituting \( q \) with \( S \cdot m_e \) gives:
P = \frac{I \cdot m_e \cdot A}{S \cdot m_e} = \frac{I \cdot A}{S}
Final Result
Thus, the momentum of all the free electrons per unit length of the conductor, expressed in terms of the current \( I \) and the specific charge \( S \), is:
P = \frac{I \cdot A}{S}
This equation shows how the momentum of the electrons is directly proportional to the current and the cross-sectional area of the conductor, while inversely proportional to the specific charge of the electrons.
