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MHT-CET Physics · Formula sheet

AC Circuits formulas

17 formulas and 17 common traps for MHT-CET Physics AC Circuits, grouped by subtopic.

Full notes with worked examples

Alternating Current: Instantaneous, Peak and RMS Values

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Reading a Sinusoid: Peak Times, Zeros and Phase

Instantaneous value

i=I0sin⁡(ωt+ϕ),ω=2πfi = I_0\sin(\omega t + \phi), \qquad \omega = 2\pi f

Peak and R.M.S. Values

R.m.s. value

Irms=I02≈0.707 I0I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \approx 0.707\,I_0

Common traps

Forgetting the starting phase

With a phase ϕ\phi already in the equation, the peak comes EARLIER: solve ωt+ϕ=π2\omega t + \phi = \frac{\pi}{2}, not ωt=π2\omega t = \frac{\pi}{2}. T4\frac{T}{4} is the planted answer.

Using the peak where the r.m.s. is meant

The number in front of sin⁡\sin is the PEAK. A meter reading, a power, or a '220 V supply' all mean r.m.s.; divide by 2\sqrt{2} first.

Reactance and Single-Element AC Circuits

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How Reactance Depends on Frequency, L and C

Reactances

XL=ωL,XC=1ωCX_L = \omega L, \qquad X_C = \frac{1}{\omega C}

Current and Phase With a Single L or C

CIVIL

C: I leads V by 90∘;L: V leads I by 90∘\text{C: } I \text{ leads } V \text{ by } 90^\circ; \qquad \text{L: } V \text{ leads } I \text{ by } 90^\circ

A Bulb in Series With a Coil or a Capacitor

Current falls as reactance rises

I=VR2+X2I = \frac{V}{\sqrt{R^2 + X^2}}

Common traps

Treating X_C like X_L

XCX_C goes DOWN as frequency or capacitance goes up. Doubling both divides it by 4; the options offer 4X4X for the student who multiplied.

Swapping lead and lag

Every phase question offers both. CIVIL settles it: Capacitor — I before V; inducto-L — V before I.

Reversing the capacitor's rule

For a coil, more L or more frequency means dimmer; for a capacitor, more C or more frequency means BRIGHTER. The two rules run in opposite directions.

Series LR, RC and LCR: Impedance, Phase and Phasors

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Impedance and Current

Impedance

Z=R2+(XL−XC)2Z = \sqrt{R^2 + (X_L - X_C)^2}

The Phase Angle

Phase angle

tan⁡ϕ=XL−XCR\tan\phi = \frac{X_L - X_C}{R}

Voltages Add as Phasors

Series voltages

V=VR2+(VL−VC)2V = \sqrt{V_R^2 + (V_L - V_C)^2}

Common traps

Adding R and X directly

Z=R+XLZ = R + X_L is the tempting shortcut and is always wrong: 30 Ω and 40 Ω make 50 Ω, not 70 Ω.

Inverting the tangent

tan⁡ϕ=XR\tan\phi = \frac{X}{R}, reactance on top. XL=400X_L = 400, R=300R = 300 gives tan⁡−143\tan^{-1}\frac{4}{3}; tan⁡−134\tan^{-1}\frac{3}{4} sits beside it.

Adding the voltages as numbers

50 V across a circuit can put 90 V on L and 60 V on C — more than the source — because they cancel in part. Only the phasor sum equals the supply.

Resonance in Series LCR Circuits

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The Resonant Frequency

Resonant frequency

f0=12πLCf_0 = \frac{1}{2\pi\sqrt{LC}}

What Happens at Resonance

At resonance

XL=XC,Z=R,ϕ=0X_L = X_C, \quad Z = R, \quad \phi = 0

Voltages Across L and C at Resonance, and the Quality Factor

Quality factor

Q=VLVR=1RLCQ = \frac{V_L}{V_R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Common traps

'Increased by 3C' is 4C

'C increased BY 3C' makes 4C; 'changed TO 3C' makes 3C. The two readings land on different options, f23\frac{f}{2\sqrt{3}} and f3\frac{f}{3}.

Impedance zero at resonance

The reactances cancel, the resistance does not: Z=RZ = R, never zero. 'At resonance, impedance is zero' is a planted false statement.

Capping V_L at the supply voltage

Series resonance MAGNIFIES the voltage on L and C. 0.1 V applied can put 25 V across the coil; the answer below the supply voltage is there for students who assume otherwise.

Power in AC Circuits: Average Power, Power Factor and Wattless Current

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Average power

P=VrmsIrmscos⁡ϕ=Irms2RP = V_{\text{rms}}I_{\text{rms}}\cos\phi = I_{\text{rms}}^2R

Power factor

cos⁡ϕ=RZ=RR2+(XL−XC)2\cos\phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}}

Wattless Current

Purely reactive circuit

ϕ=90∘  ⇒  P=VrmsIrmscos⁡90∘=0\phi = 90^\circ \;\Rightarrow\; P = V_{\text{rms}}I_{\text{rms}}\cos 90^\circ = 0

Common traps

Using peak values without the ½

P=V0I02cos⁡ϕP = \frac{V_0I_0}{2}\cos\phi: the ½ turns two peaks into r.m.s. values. Dropping it doubles the power and lands on a printed option.

Doubling the frequency of an RC circuit

Frequency UP lowers XCX_C and RAISES an RC circuit's power factor. To make XC=2RX_C = 2R the frequency must be halved — the RC and LR versions of the same question move in opposite directions.

Current means power

A pure coil can carry amperes and consume nothing. Multiplying VrmsIrmsV_{\text{rms}}I_{\text{rms}} without cos⁡ϕ\cos\phi gives the apparent power, not the power used.

LC Oscillations, the Transformer and the AC Generator

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LC Oscillations

LC circuit

T=2πLC,I0=VCLT = 2\pi\sqrt{LC}, \qquad I_0 = V\sqrt{\frac{C}{L}}

The Transformer

Transformer

VsVp=NsNp,η=VsIsVpIp\frac{V_s}{V_p} = \frac{N_s}{N_p}, \qquad \eta = \frac{V_sI_s}{V_pI_p}

The AC Generator

Generator e.m.f.

e0=NABω,Pˉ=(NABω)22Re_0 = NAB\omega, \qquad \bar P = \frac{(NAB\omega)^2}{2R}

Common traps

Taking the full period for 'current greatest'

From a full capacitor, the current is greatest a QUARTER period later. 2πLC2\pi\sqrt{LC} is the time to come back to a full capacitor of the same polarity.

Applying the efficiency to the primary current

The primary current comes from the INPUT power, 3000200=15\frac{3000}{200} = 15 A. The 90% applies to what comes out: Vs=27006=450V_s = \frac{2700}{6} = 450 V.

Squaring only part of e₀

Power goes as e02e_0^2, so as N2A2B2ω2N^2A^2B^2\omega^2 — every factor squared. Options with NABωNAB\omega to the first power are the e.m.f., not the power.

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