A bulb with rating `250V, 100W` is connected to a power supply of `220 V` situatd `10 m` away using a copper wire of area of cross section `5 mm^2`. How much power will be consumed by the connecting wire? Resistivity of copper `= 1.7 xx 10^(-8) Omegam`.

Explanation

To solve the problem step by step, we will calculate the resistance of the bulb, the resistance of the copper wire, the total resistance in the circuit, and finally the power consumed by the connecting wire.

### Step 1: Calculate the resistance of the bulb (Rb)

The resistance of the bulb can be calculated using the formula:
\[ R_b = \frac{V^2}{P} \]
Where:
- \(V = 250 \, \text{V}\) (voltage rating of the bulb)
- \(P = 100 \, \text{W}\) (power rating of the bulb)

Substituting the values:
\[ R_b = \frac{(250)^2}{100} = \frac{62500}{100} = 625 \, \Omega \]

### Step 2: Calculate the resistance of the copper wire (Rw)

The resistance of the wire can be calculated using the formula:
\[ R_w = \frac{\rho \cdot L}{A} \]
Where:
- \(\rho = 1.7 \times 10^{-8} \, \Omega \cdot m\) (resistivity of copper)
- \(L = 10 \, m\) (length of one wire)
- \(A = 5 \, mm^2 = 5 \times 10^{-6} \, m^2\) (cross-sectional area)

Substituting the values:
\[ R_w = \frac{1.7 \times 10^{-8} \cdot 10}{5 \times 10^{-6}} = \frac{1.7 \times 10^{-7}}{5 \times 10^{-6}} = 0.034 \, \Omega \]

Since there are two wires, the total resistance of the wires (R_total) will be:
\[ R_{\text{total}} = 2 \cdot R_w = 2 \cdot 0.034 = 0.068 \, \Omega \]

### Step 3: Calculate the total resistance in the circuit (R_total_circuit)

The total resistance in the circuit is the sum of the resistance of the bulb and the total resistance of the wires:
\[ R_{\text{total circuit}} = R_b + R_{\text{total}} = 625 + 0.068 = 625.068 \, \Omega \]

### Step 4: Calculate the current (I) in the circuit

Using Ohm's law, the current can be calculated as:
\[ I = \frac{V}{R_{\text{total circuit}}} \]
Where \(V = 220 \, V\) (voltage from the power supply).

Substituting the values:
\[ I = \frac{220}{625.068} \approx 0.352 \, A \]

### Step 5: Calculate the power consumed by the connecting wire (P_wire)

The power consumed by the wire can be calculated using the formula:
\[ P_{\text{wire}} = I^2 \cdot R_{\text{total}} \]

Substituting the values:
\[ P_{\text{wire}} = (0.352)^2 \cdot 0.068 \approx 0.0084 \, W = 8.4 \, mW \]

### Final Answer:
The power consumed by the connecting wire is approximately **8.4 mW**.

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