(II) A power station delivers 750 kW of power at 12,000 V to a factory through wires with total resistance 3.0 Ω. How much less power is wasted if the electricity is delivered at 50,000 V rather than 12,000 V?

검증된 단계별 안내

Step 1: Calculate the current (I) in the circuit when the power is delivered at 12,000 V. Use the formula for power:

P = V I

, where P is the power (750 kW) and V is the voltage (12,000 V). Rearrange to find

I = P / V

Step 2: Calculate the power wasted in the wires when the voltage is 12,000 V. Use the formula for power loss due to resistance:

P_{loss} = I^{2} R

, where I is the current calculated in Step 1 and R is the total resistance of the wires (3.0 Ω).

Step 3: Repeat Step 1 to calculate the current (I) when the power is delivered at 50,000 V. Use the same formula:

I = P / V

, but substitute 50,000 V for the voltage.

Step 4: Calculate the power wasted in the wires when the voltage is 50,000 V. Again, use the formula

P_{loss} = I^{2} R

, but use the current calculated in Step 3.

Step 5: Find the difference in power wasted between the two cases. Subtract the power loss at 50,000 V from the power loss at 12,000 V to determine how much less power is wasted when the electricity is delivered at the higher voltage.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Power and Voltage Relationship

The relationship between power (P), voltage (V), and current (I) is defined by the equation P = VI. This means that for a given power output, increasing the voltage allows for a decrease in current, which is crucial for minimizing energy losses in transmission lines.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor, expressed as V = IR. This principle helps in understanding how resistance affects power loss in electrical systems.

Power Loss in Transmission Lines

Power loss in transmission lines is primarily due to the resistance of the wires, calculated using the formula P_loss = I²R. As current increases, the power lost as heat in the wires also increases, making high-voltage transmission more efficient by reducing current and thus minimizing losses.