CBSE 2024, Set 1

Question

An electric lamp is designed to operate at $110	ext{ V}$ dc and $11	ext{ A}$ current. If the lamp is operated on $220	ext{ V}$, $50	ext{ Hz}$ ac source with a coil in series, then find the inductance of the coil.

Edvaya · Accepted Answer

The correct answer is Subjective. **SOLUTION** **Step 1: Find the resistance of the lamp.** Using the DC parameters: $$R = \frac{V_{dc}}{I} = \frac{110	ext{ V}}{11	ext{ A}} = 10	ext{ \Omega}$$ **Step 2: Find the required impedance for the AC circuit.** The lamp must still draw $11	ext{ A}$ (RMS) to operate correctly. The AC source is $V_{rms} = 220	ext{ V}$. The required impedance ($Z$) of the circuit is: $$Z = \frac{V_{rms}}{I_{rms}} = \frac{220}{11} = 20	ext{ \Omega}$$ **Step 3: Calculate the inductive reactance of the coil.** The circuit consists of the lamp ($R$) and the coil ($L$) in series. $$Z^2 = R^2 + X_L^2$$ $$20^2 = 10^2 + X_L^2$$ $$400 = 100 + X_L^2$$ $$X_L^2 = 300 \implies X_L = 10\sqrt{3}	ext{ \Omega}$$ **Step 4: Find the Inductance ($L$).** $$X_L = 2\pi f L$$ $$10\sqrt{3} = 2\pi (50) L$$ $$10\sqrt{3} = 100\pi L$$ $$L = \frac{10\sqrt{3}}{100\pi} = \frac{\sqrt{3}}{10\pi}	ext{ H}$$ The required inductance is **$\frac{\sqrt{3}}{10\pi}	ext{ H}$**.

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Physics > Magnetism Induction And Alternating Current > Alternating Current

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