Problem 64 - Physics for Scientists and Engineers with Modern Physics

Question

Show that the power loss in transmission lines, \(P_{\mathrm{L}},\) is given by \(P_{\mathrm{L}}=\left(P_{\mathrm{T}}\right)^{2} R_{\mathrm{L}} / V^{2},\) where \(P_{\mathrm{T}}\) is the power transmitted to the user, \(V\) is the delivered voltage, and \(R_{\mathrm{L}}\) is the resistance of the power lines.

Step-by-Step Solution

Answer

The derived power loss formula is \( P_{\mathrm{L}} = \frac{(P_{\mathrm{T}})^{2} R_{\mathrm{L}}}{V^{2}} \).

Step 1: Understand the given parameters and formula

We are asked to prove the formula for power loss in transmission lines. The given parameters are: the power transmitted to the user, \( P_{\mathrm{T}} \), the delivered voltage, \( V \), and the resistance of the power lines, \( R_{\mathrm{L}} \). We have the target equation for power loss: \( P_{\mathrm{L}} = \frac{(P_{\mathrm{T}})^{2} R_{\mathrm{L}}}{V^{2}} \).

Step 2: Derive power transmitted using Ohm's Law

The transmitted power \( P_{\mathrm{T}} \) can be expressed using the formula \( P = IV \), where \( I \) is the current. Expressing current in terms of other quantities using Ohm's Law \( V = IR \), gives us \( I = \frac{V}{R_{\mathrm{L}}} \). Substituting \( I \) in the power formula, we get \( P_{\mathrm{T}} = I_{\mathrm{T}} \cdot V \).

Step 3: Express power loss in terms of current

The power loss in the line \( P_{\mathrm{L}} \) due to the resistance \( R_{\mathrm{L}} \) is given by \( P_{\mathrm{L}} = I^{2} R_{\mathrm{L}} \). By substituting \( I = \frac{P_{\mathrm{T}}}{V} \) into this expression, we aim to derive the given formula for \( P_{\mathrm{L}} \).

Step 4: Substitute current expression into power loss formula

Substitute \( I = \frac{P_{\mathrm{T}}}{V} \) into \( P_{\mathrm{L}} = I^{2} R_{\mathrm{L}} \). This gives \[ P_{\mathrm{L}} = \left(\frac{P_{\mathrm{T}}}{V}\right)^{2} R_{\mathrm{L}}. \]

Step 5: Simplify the expression to match target formula

Simplifying, we have \[ P_{\mathrm{L}} = \frac{(P_{\mathrm{T}})^{2}}{V^{2}} R_{\mathrm{L}}, \] which matches the given target equation for power loss in transmission lines.

Key Concepts

Power Transmission

Power transmission involves the transfer of electrical energy from power plants to end users. This process is vital for delivering electricity over long distances efficiently. Understanding how power is transmitted is critical for minimizing losses. In transmission lines, electrical energy travels from one point to another using wires or cables.

Efficiency in power transmission is measured by how much of the transmitted power reaches the end consumer. The main aim in designing transmission systems is to maintain high efficiency while minimizing power loss.

Key components of power transmission include:

• Generators: Convert mechanical energy into electrical energy.
• Substations: Step-up or step-down voltage levels for efficient transmission.
• Transmission lines: Carry electricity over distances, facing challenges like electrical resistance and power loss.

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. It is expressed by the formula:\[ V = IR \]where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance.

In the context of power transmission, Ohm's Law helps us determine the current in electrical circuits given known values of voltage and resistance. It plays a crucial role in calculating power loss in transmission lines.

By rearranging Ohm's Law, we can express the current as:\[ I = \frac{V}{R} \]This rearrangement is particularly useful for understanding how electrical resistance affects the flow of current and eventually the efficiency of power transmission.

Electrical Resistance

Electrical resistance is a measure of how strongly a material opposes the flow of electric current. This concept is critical in understanding power loss in transmission lines because it directly influences how much power is lost as energy travels through conductors.

The greater the resistance of a wire or cable, the more power is converted to heat and lost from the system. This is why materials with low resistance, such as copper or aluminum, are typically used in power lines to improve efficiency.

Key factors affecting electrical resistance include:

• Material: Different materials have different inherent resistances.
• Length: Longer wires have greater resistance.
• Cross-sectional area: Wider wires have less resistance.
• Temperature: Resistance can increase with temperature, affecting the performance of transmission lines.

Voltage

Voltage is the electrical potential difference between two points in a circuit. In power transmission, voltage is crucial because it determines how efficiently power can be transported. Higher voltages allow for more efficient power transmission over long distances by reducing current for the same amount of power.

Here's how voltage impacts power transmission:

• Efficiency: At higher voltages, the current is lower for the same power level, reducing resistive power loss.
• Safety and equipment limits: While high voltage helps with efficiency, transmission systems must be carefully designed to handle these levels safely.
• Transformation: Substations transform voltages up for transmission and down for distribution to make power usable by consumers.

Understanding voltage's role helps comprehend why power loss can be minimized by maintaining high voltages in transmission lines. Lower current levels at higher voltages mean less resistance to flow and therefore less power is lost as heat.