### Related Formula Power consumed in a purely resistive AC device (like a lightbulb) is: P = V_textrms I_textrms$P = V_{\text{rms}} I_{\text{rms}}$ The peak current I_0$I_0$ is related to the rms current I_textrms$I_{\text{rms}}$ by: I_0 = sqrt2 I_textrms$I_0 = \sqrt{2} I_{\text{rms}}$ ### Core Logic Given parameters: - Power P = 100mathrm~W$P = 100\mathrm{~W}$ - rms Voltage V_textrms = 220mathrm~V$V_{\text{rms}} = 220\mathrm{~V}$ ### Step 1: Calculate RMS Current (I_textrms$I_{\text{rms}}$) I_textrms = fracPV_textrms = frac100220 = frac511mathrm~A approx 0.455mathrm~A$I_{\text{rms}} = \frac{P}{V_{\text{rms}}} = \frac{100}{220} = \frac{5}{11}\mathrm{~A} \approx 0.455\mathrm{~A}$ ### Step 2: Calculate Peak Current ($ ### Step 2: Calculate Peak Current ($I_0) $) $I_0 = sqrt2 I_textrms = 1.414 times frac511 = frac7.0711 approx 0.64mathrm~A$I_0 = \sqrt{2} I_{\text{rms}} = 1.414 \times \frac{5}{11} = \frac{7.07}{11} \approx 0.64\mathrm{~A}$ ### Pattern Recognition Always remember that rated values specify the RMS limits. Since the source voltage matches the bulb's rated voltage exactly, the actual power equals the rated power ($ ### Pattern Recognition Always remember that rated values specify the RMS limits. Since the source voltage matches the bulb's rated voltage exactly, the actual power equals the rated power ($100\mathrm{~W}). Do not confuse $). Do not confuse $I_{\text{rms}} ($ ($0.45\mathrm{~A}) with the peak current $) with the peak current $I_0 ($ ($0.64\mathrm{~A}$). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Alternating Current

### Related Formula For a series RC branch, the voltage phase angle theta$\theta$ is: tantheta = fracX_CR = frac1omega C R$\tan\theta = \frac{X_C}{R} = \frac{1}{\omega C R}$ If the phase difference between V_textin$V_{\text{in}}$ and (V_B - V_A)$(V_B - V_A)$ is 90^circ$90^\circ$: theta = 45^circ implies tantheta = 1$\theta = 45^\circ \implies \tan\theta = 1$ ### Core Logic

From the phasor representation of the symmetric RC divider bridge: - The phase angle of V_A$V_A$ is theta$\theta$ behind the input voltage. - The phase angle of V_B$V_B$ is 90^circ - theta$90^\circ - \theta$ ahead of the input. - If their combined relative phase difference is 90^circ$90^\circ$, this symmetry dictates:

theta + theta = 90^circ implies theta = 45^circ$\theta + \theta = 90^\circ \implies \theta = 45^\circ$ Therefore, we have the condition: X_C = R implies frac1omega C = R$X_C = R \implies \frac{1}{\omega C} = R$ ### Step 1: Solve for Frequency Rearrange to solve for omega$\omega$: omega = frac1R C$\omega = \frac{1}{R C}$ Substitute the given values: - R = 100 mathrm~kOmega = 10^5 Omega$R = 100 \mathrm{~k}\Omega = 10^5 \Omega$ - C = 100 mathrm~pF = 100 times 10^-12 mathrm~F = 10^-10 mathrm~F$C = 100 \mathrm{~pF} = 100 \times 10^{-12} \mathrm{~F} = 10^{-10} \mathrm{~F}$ omega = frac110^5 times 10^-10 = frac110^-5 = 10^5 mathrm~rad/sec$\omega = \frac{1}{10^5 \times 10^{-10}} = \frac{1}{10^{-5}} = 10^5 \mathrm{~rad/sec}$ Since omega = 10^x$\omega = 10^x$, we get x = 5$x = 5$. ### Pattern Recognition Sees: Phase shift of 90^circ$90^\circ$ across a symmetric RC bridge. Shortcut: Clamping phase shift at 90^circ$90^\circ$ implies the reactive impedance equals the resistive impedance (X_C = R$X_C = R$). The frequency is simply the characteristic time constant frequency omega = 1/tau = 1/(RC)$\omega = 1/\tau = 1/(RC)$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Alternating Current

### Related Formula For a current having both DC and AC components i = I_textdc + I_0 cos(omega t + phi)$i = I_{\text{dc}} + I_0 \cos(\omega t + \phi)$, the mean-square value is: langle i^2 rangle = I_textdc^2 + fracI_0^22$\langle i^2 \rangle = I_{\text{dc}}^2 + \frac{I_0^2}{2}$ The RMS current I_textrms$I_{\text{rms}}$ is: I_textrms = sqrtlangle i^2 rangle = sqrtI_textdc^2 + fracI_0^22$I_{\text{rms}} = \sqrt{\langle i^2 \rangle} = \sqrt{I_{\text{dc}}^2 + \frac{I_0^2}{2}}$ ### Core Logic Identify the parameters from the given equation: - I_textdc = 5sqrt2 mathrm~A$I_{\text{dc}} = 5\sqrt{2} \mathrm{~A}$ - I_0 = 10 mathrm~A$I_0 = 10 \mathrm{~A}$ Calculate the square of the components: I_textdc^2 = (5sqrt2)^2 = 50$I_{\text{dc}}^2 = (5\sqrt{2})^2 = 50$ fracI_0^22 = frac1002 = 50$\frac{I_0^2}{2} = \frac{100}{2} = 50$ ### Step 1: Compute Resultant RMS Substitute back into the RMS formula: I_textrms = sqrt50 + 50 = sqrt100 = 10 mathrm~Amp$I_{\text{rms}} = \sqrt{50 + 50} = \sqrt{100} = 10 \mathrm{~Amp}$ ### Pattern Recognition Sees: Superposition of a DC current I_textdc$I_{\text{dc}}$ and a pure AC cosine wave with amplitude I_textac$I_{\text{ac}}$. Shortcut: Use the orthogonal component RMS formula: I_textrms = sqrtI_textdc^2 + I_textac,rms^2$I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_{\text{ac,rms}}^2}$. Here I_textdc = 5sqrt2$I_{\text{dc}} = 5\sqrt{2}$ and I_textac,rms = 10/sqrt2 = 5sqrt2$I_{\text{ac,rms}} = 10/\sqrt{2} = 5\sqrt{2}$. Thus, I_textrms = sqrt50 + 50 = 10 mathrm~A$I_{\text{rms}} = \sqrt{50 + 50} = 10 \mathrm{~A}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Alternating Current

### Related Formula Inductive Reactance: X_L = omega L = 2pi f L$X_L = \omega L = 2\pi f L$ Impedance of RL series circuit: Z = sqrtR^2 + X_L^2$Z = \sqrt{R^2 + X_L^2}$ Maximum current: I_textmax = sqrt2 I_textrms = fracV_textmaxZ quad textor quad I_textmax = sqrt2 fracV_textrmsZ$I_{\text{max}} = \sqrt{2} I_{\text{rms}} = \frac{V_{\text{max}}}{Z} \quad \text{or} \quad I_{\text{max}} = \sqrt{2} \frac{V_{\text{rms}}}{Z}$ ### Core Logic Calculate inductive reactance X_L$X_L$: X_L = 2pi times 50 times 1 = 100pi\ Omega$X_L = 2\pi \times 50 \times 1 = 100\pi\ \Omega$ Given resistance R = 100pi\ Omega$R = 100\pi\ \Omega$. Compute total impedance Z$Z$: Z = sqrt(100pi)^2 + (100pi)^2 = 100pisqrt2\ Omega$Z = \sqrt{(100\pi)^2 + (100\pi)^2} = 100\pi\sqrt{2}\ \Omega$ ### Step 1: Calculate Maximum Current Assuming the given supply voltage (100pitext V$100\pi\text{ V}$) is standard RMS voltage: I_textrms = fracV_textrmsZ = frac100pi100pisqrt2 = frac1sqrt2text A$I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{100\pi}{100\pi\sqrt{2}} = \frac{1}{\sqrt{2}}\text{ A}$ Then, peak/maximum current is: I_textmax = sqrt2 cdot I_textrms = sqrt2 times frac1sqrt2 = 1text A$I_{\text{max}} = \sqrt{2} \cdot I_{\text{rms}} = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1\text{ A}$ Hence, the maximum current is **1**. ### Pattern Recognition When inductive reactance equals resistance (X_L = R$X_L = R$), the impedance is exactly Rsqrt2$R\sqrt{2}$. The factor of sqrt2$\sqrt{2}$ in the denominator cancels perfectly with the peak current conversion multiplier. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Alternating Current

### Related Formula Standard alternating current expression: i = i_0 sin(omega t) $ i = i_0 \sin(\omega t) $ RMS Current value: i_textrms = fraci_0sqrt2 $ i_{\text{rms}} = \frac{i_0}{\sqrt{2}} $ Frequency calculation: f = fracomega2pi $ f = \frac{\omega}{2\pi} $ ### Core Logic Compare the given equation i = 100sqrt2sin(100pi t)$i = 100\sqrt{2}\sin(100\pi t)$ with the standard form: * Peak current value, i_0 = 100sqrt2mathrm~A$i_0 = 100\sqrt{2}\mathrm{~A}$ * Angular frequency, omega = 100pimathrm~rad/s$\omega = 100pimathrm{~rad/s}$ ### Step 1: Calculate RMS current and Frequency Substituting the extracted parameters: i_textrms = frac100sqrt2sqrt2 = 100mathrm~A $ i_{\text{rms}} = \frac{100\sqrt{2}}{\sqrt{2}} = 100\mathrm{~A} $ f = frac100pi2pi = 50mathrm~Hz $ f = \frac{100\pi}{2\pi} = 50\mathrm{~Hz} $ ### Pattern Recognition The multiplier of sin$\sin$ is i_0$i_0$. Divide by sqrt2$\sqrt{2}$ to get i_textrms$i_{\text{rms}}$. The coefficient of t$t$ is omega = 2pi f$\omega = 2\pi f$, so dividing the coefficient of pi t$\pi t$ by 2 directly delivers f$f$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Alternating Current