The total momentum of electrons in a straight wire of length L = 1000m carrying a current I = 70A, will be (in N.S)

(a) 0.40 X 10^-6

(b) 0.20 X 10^-6

(c) 0.80 X 10^-6

(d) 0.16 X 10^-6

To determine the total momentum of electrons in a straight wire carrying a current, we need to understand a few key concepts related to electric current, charge, and momentum. Let's break this down step by step.

Understanding Current and Electron Flow

Current (I) is defined as the flow of electric charge. In a conductor like a wire, this charge is primarily carried by electrons. The relationship between current, charge, and time is given by the equation:

• I = Q/t

Where Q is the total charge and t is the time in seconds. For our case, we need to find the total charge of electrons flowing through the wire.

Calculating Charge

The total charge Q that flows through the wire in a given time can be calculated using the current and the time duration. However, since we are looking for momentum, we need to consider the number of electrons involved. The charge of a single electron is approximately 1.6 x 10^-19 C.

Finding the Number of Electrons

To find the total number of electrons flowing through the wire, we can use the formula:

• N = Q/e

Where N is the number of electrons, Q is the total charge, and e is the charge of a single electron. The total charge can be expressed as:

• Q = I × t

For a wire of length L = 1000 m carrying a current I = 70 A, we can consider a time duration of 1 second for simplicity:

• Q = 70 C (since Q = I × t = 70 A × 1 s)

Calculating the Number of Electrons

Now, substituting the values into the equation for N:

• N = 70 C / (1.6 x 10^-19 C/electron) ≈ 4.375 x 10^20 electrons

Momentum of Electrons

The momentum (p) of an individual electron can be calculated using the formula:

• p = m × v

Where m is the mass of an electron (approximately 9.11 x 10^-31 kg) and v is the drift velocity of the electrons. The drift velocity can be calculated using the formula:

• v = I / (n × A × e)

Here, n is the number density of electrons (approximately 8.5 x 10^28 electrons/m³ for copper), and A is the cross-sectional area of the wire. For simplicity, let's assume a typical cross-sectional area of 1 mm² (1 x 10^-6 m²):

• v = 70 A / (8.5 x 10^28 electrons/m³ × 1 x 10^-6 m² × 1.6 x 10^-19 C/electron) ≈ 5.2 x 10^-4 m/s

Total Momentum Calculation

Now we can calculate the total momentum of all the electrons:

• Total momentum = N × p
• p = m × v = (9.11 x 10^-31 kg) × (5.2 x 10^-4 m/s) ≈ 4.74 x 10^-34 kg·m/s
• Total momentum = (4.375 x 10^20 electrons) × (4.74 x 10^-34 kg·m/s) ≈ 0.20 x 10^-6 N·s

Final Answer

Based on the calculations, the total momentum of electrons in the wire is approximately 0.20 x 10^-6 N·s, which corresponds to option (b).