A machine requires 10 kW of electrical power when operating at 100V. 

[2 marks] If power is delivered through a cable with resistance 100, how much power is lost to the cable’s resistance?

[2 marks] In order to reduce the power loss, a transformer is used between the cable and the machine. What ratio of turns is required on the transformer in order to reduce power loss on the cable by a factor of 104?


### Part 1: Power loss due to cable's resistance

The machine requires 10 kW of electrical power, and it operates at 100 V. First, we can calculate the current \( I \) flowing through the cable using the formula for power:

\[
P = V \times I
\]

Where:
- \( P \) is the power (10 kW = 10,000 W),
- \( V \) is the voltage (100 V),
- \( I \) is the current.

Rearranging the formula to find \( I \):

\[
I = \frac{P}{V} = \frac{10,000}{100} = 100 \, \text{A}
\]

Now, the power lost due to the resistance in the cable can be found using the formula for power loss in a resistor:

\[
P_{\text{loss}} = I^2 \times R
\]

Where:
- \( P_{\text{loss}} \) is the power lost in the cable,
- \( I \) is the current (100 A),
- \( R \) is the resistance of the cable (100 Ω).

Substituting the values:

\[
P_{\text{loss}} = (100)^2 \times 100 = 10,000 \times 100 = 1,000,000 \, \text{W} = 1,000 \, \text{kW}
\]

Thus, the power lost to the cable’s resistance is **1,000 kW**.

---

### Part 2: Transformer turns ratio

To reduce the power loss by a factor of 104, the voltage supplied to the cable must be increased (while reducing the current). A transformer is used for this purpose. The power loss is proportional to \( I^2 \), so to reduce the power loss by a factor of 104, the current must be reduced by a factor of:

\[
\sqrt{104} \approx 10.2
\]

The transformer increases the voltage and reduces the current proportionally. The turns ratio \( n \) of the transformer (primary to secondary) is related to the voltage and current as:

\[
n = \frac{V_{\text{secondary}}}{V_{\text{primary}}} = \frac{I_{\text{primary}}}{I_{\text{secondary}}}
\]

Since we want the current reduced by a factor of 10.2, the turns ratio of the transformer must also be **10.2:1**.

---

### Summary:
- The power lost to the cable’s resistance is **1,000 kW**.
- The required turns ratio for the transformer is **10.2:1**.

Would you like a further breakdown of these steps or more detailed explanations?

---

Here are 5 follow-up questions for further exploration:
1. How is power loss affected if the resistance of the cable is halved?
2. What would the power loss be without a transformer if the operating voltage was increased to 200 V?
3. How does increasing the transformer’s turns ratio affect the machine's operation?
4. If the transformer is 95% efficient, how does this affect the overall system power loss?
5. Can you explain how transformer efficiency impacts the power delivered to the machine?

**Tip:** Transformers allow voltage transformation to reduce current and minimize power loss in transmission systems, which is why they are essential in long-distance power delivery.

A machine requires 10 kW of electrical power when operating at 100V. If power is delivered through a cable with resistance 100, how much power is lost to the cable’s resistance? In order to reduce the power loss, a transformer is used between the cable and the machine. What ratio of turns is required on the transformer in order to reduce power loss on the cable by a factor of 104?

Solve a physics problem calculating power loss in a cable with 100 Ohms resistance and determining the transformer turns ratio needed to reduce power loss by a factor of 104. Explore electrical power concepts, Ohm’s Law, and transformer equations with detailed step-by-step solutions.

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