In the transmission of a.c. power through transmission lines, when the voltage is stepped up n times, the power loss in transmission

Options

A. Increases n times
B. Decreases n times
C. Increases `n^(2)` times
D. Decreases `n^(2) ` times

Explanation

To solve the problem of power loss in AC transmission lines when the voltage is stepped up by a factor of \( n \), we can follow these steps:

### Step 1: Understand Power Loss in Transmission
The power loss \( P_{\text{loss}} \) in transmission lines is given by the formula:
\[ P_{\text{loss}} = I^2 R \]
where \( I \) is the current flowing through the transmission line and \( R \) is the resistance of the transmission line.

### Step 2: Relate Power, Voltage, and Current
The power \( P \) transmitted through the line is related to voltage \( V \) and current \( I \) by the formula:
\[ P = V I \]
From this equation, we can express the current \( I \) as:
\[ I = \frac{P}{V} \]

### Step 3: Substitute Current into Power Loss Formula
Substituting the expression for \( I \) into the power loss formula gives:
\[ P_{\text{loss}} = \left(\frac{P}{V}\right)^2 R \]
This simplifies to:
\[ P_{\text{loss}} = \frac{P^2 R}{V^2} \]

### Step 4: Analyze the Effect of Stepping Up Voltage
When the voltage is stepped up by a factor of \( n \), the new voltage \( V' \) is:
\[ V' = nV \]
Substituting this new voltage into the power loss formula, we get:
\[ P'_{\text{loss}} = \frac{P^2 R}{(nV)^2} = \frac{P^2 R}{n^2 V^2} \]

### Step 5: Compare Power Loss Before and After Stepping Up Voltage
Now we can compare the new power loss \( P'_{\text{loss}} \) with the original power loss \( P_{\text{loss}} \):
\[ P'_{\text{loss}} = \frac{P_{\text{loss}}}{n^2} \]
This shows that the power loss decreases by a factor of \( n^2 \) when the voltage is stepped up by \( n \).

### Conclusion
Thus, when the voltage is stepped up by \( n \) times, the power loss in transmission decreases by a factor of \( n^2 \).

### Final Answer
The power loss in transmission decreases by \( n^2 \) times.

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