A transformer is used to light a 140 W, 24 V lamp from 240 V AC mains. The current in mains cable is 0.7 A, find the efficiency of transformer.

Answer

To find the efficiency of the transformer used to light a 140 W, 24 V lamp from a 240 V AC mains, we will follow these steps:

### Step 1: Identify the given values
- Power of the lamp (P2) = 140 W
- Voltage of the lamp (E2) = 24 V
- Voltage of the mains (E1) = 240 V
- Current in the mains (I1) = 0.7 A

### Step 2: Calculate the power in the primary coil (P1)
The power in the primary coil can be calculated using the formula:
\[ P1 = E1 \times I1 \]
Substituting the values:
\[ P1 = 240 \, \text{V} \times 0.7 \, \text{A} = 168 \, \text{W} \]

### Step 3: Calculate the current in the secondary coil (I2)
First, we need to calculate the resistance of the lamp (R2) using the formula:
\[ R2 = \frac{E2^2}{P2} \]
Substituting the values:
\[ R2 = \frac{24^2}{140} = \frac{576}{140} = \frac{144}{35} \, \Omega \]

Now, we can find the current in the secondary coil (I2) using Ohm's law:
\[ I2 = \frac{E2}{R2} \]
Substituting the values:
\[ I2 = \frac{24}{\frac{144}{35}} = 24 \times \frac{35}{144} = \frac{840}{144} = \frac{35}{6} \, \text{A} \]

### Step 4: Calculate the power in the secondary coil (P2)
The power in the secondary coil can be calculated using:
\[ P2 = E2 \times I2 \]
Substituting the values:
\[ P2 = 24 \, \text{V} \times \frac{35}{6} \, \text{A} = 140 \, \text{W} \]

### Step 5: Calculate the efficiency of the transformer
The efficiency (η) of the transformer can be calculated using the formula:
\[ \eta = \left( \frac{P2}{P1} \right) \times 100 \]
Substituting the values:
\[ \eta = \left( \frac{140}{168} \right) \times 100 = 83.33\% \]

### Final Answer
The efficiency of the transformer is **83.33%**.

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