Current (I_n) at a point n unit lengths down the infinite line is given by:

Explanation:

Understanding the Current (In) in an Infinite Line:

The problem addresses how the current at a point \( n \) unit lengths down an infinite line is expressed mathematically. Among the given options, the correct formula representing this relationship is:

Correct Option: \(\rm I_{n}=I_{S} e^{-n\gamma}\)

This formula accurately describes how the current diminishes exponentially as we move down the infinite line. Let’s explore the reasoning behind this, step-by-step.

Mathematical Explanation:

The current \( I_n \) at a distance \( n \) down the infinite line is determined by an exponential decay function. The general form of an exponential decay is:

\(\rm I_{n}=I_{S} e^{-n\gamma}\)

Where:

• \( \rm I_{S} \): Initial current or the source current at the starting point (n = 0).
• \( n \): Distance down the infinite line measured in unit lengths.
• \( \gamma \): A decay constant that governs the rate at which the current diminishes as the distance increases.

The exponential term \(\rm e^{-n\gamma}\) reflects the physical phenomenon of energy dissipation or attenuation along the line, which is common in electrical or signal transmission systems. This decay happens due to resistance, impedance, or other factors that cause a reduction in the current as it travels further down the line.

Why Option 3 is Correct:

The exponential decay function, \(\rm I_{n}=I_{S} e^{-n\gamma}\), precisely models the behavior of the current on an infinite line. As \( n \) increases, the term \(\rm e^{-n\gamma}\) decreases exponentially, leading to a corresponding reduction in the current \( I_n \). This aligns with the laws of physics governing attenuation in electrical or signal transmission systems.

Key Insights:

• The negative exponent \( -n\gamma \) ensures that the current decreases rather than increases as \( n \) grows.
• The decay constant \( \gamma \) is a critical parameter that quantifies how quickly the current diminishes over distance.

Additional Information

Analysis of Other Options:

Let’s evaluate why the other options are incorrect:

Option 1: \(\rm I_{n}=I_{S} e^{n\gamma}\)

This option suggests that the current increases exponentially as we move down the infinite line. However, this contradicts the physical behavior of current attenuation. In real systems, resistance or impedance causes the current to decrease rather than increase. Therefore, this option is incorrect.

Option 2: \(\rm I_{n}=I_{S} e^{-\frac{n}{\gamma}}\)

While this expression involves exponential decay, the term \( -\frac{n}{\gamma} \) implies an incorrect relationship between \( n \) and \( \gamma \). The decay constant \( \gamma \) should directly multiply \( n \) in the exponent (as seen in the correct option). The division by \( \gamma \) is not physically valid in this context, making this option incorrect.

Option 4: \(\rm I_{n}=I_{S} e^{\frac{n}{\gamma}}\)

This option suggests exponential growth rather than decay, as indicated by the positive exponent \( \frac{n}{\gamma} \). Similar to Option 1, this contradicts the behavior of current attenuation in practical systems, where the current diminishes as the distance increases. Hence, this option is incorrect.

Conclusion:

The correct formula for the current at a distance \( n \) down an infinite line is \(\rm I_{n}=I_{S} e^{-n\gamma}\). This expression accurately captures the phenomenon of exponential decay, reflecting how the current diminishes due to factors like resistance or impedance. By understanding the behavior of exponential functions and their applications in physical systems, we can correctly identify the relationship governing current attenuation.