A time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current I(t) = I0 cos(ωt) with I0 =1A and ω = 500 rad/s starts flowing in it with the initial direction shown in the figure. At t=7π6ω, the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If C = 20μF, R = 10Ω and the battery is ideal with emf of 50 V, identify the correct statements.

If q represents the charge on capacitor’s upper plateI(t)=I0cos⁡(ωt)=dqdt⇒q(t)=I0ωsin⁡(ωt)Max. Charge =1A500rads−1=2×10−3CCharge on upper plat at t=7π6ω=I0ωsin⁡7π6=−I02ωWhen capacitor is fully charged, charge on upper plate50V×20μF=1×10−3C∴Q=1×10−2C−−I02ω=2×10−3CVoltage across capacitor when A and D are connected=1×10−3C20×10−6μF=50VTotal voltage across resistor = 100 V⇒current =100V10Ω=10ACurrent is anticlockwise.

A time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current I(t) = I0 cos(ωt) with I0 =1A and ω = 500 rad/s starts flowing in it with the initial direction shown in the figure. At t=7π6ω, the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If C = 20μF, R = 10Ω and the battery is ideal with emf of 50 V, identify the correct statements.

If q represents the charge on capacitor’s upper plateI(t)=I0cos⁡(ωt)=dqdt⇒q(t)=I0ωsin⁡(ωt)Max. Charge =1A500rads−1=2×10−3CCharge on upper plat at t=7π6ω=I0ωsin⁡7π6=−I02ωWhen capacitor is fully charged, charge on upper plate50V×20μF=1×10−3C∴Q=1×10−2C−−I02ω=2×10−3CVoltage across capacitor when A and D are connected=1×10−3C20×10−6μF=50VTotal voltage across resistor = 100 V⇒current =100V10Ω=10ACurrent is anticlockwise.

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A time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current I(t) = I₀ cos( $ω$ t) with I₀ =1A and $ω$ = 500 rad/s starts flowing in it with the initial direction shown in the figure. At $t = \frac{7 π}{6 ω}$ , the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If C = 20 $μ$ F, R = 10 $Ω$ and the battery is ideal with emf of 50 V, identify the correct statements.

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Magnitude of the maximum charge on the capacitor before $t = \frac{7 π}{6 ω}$ is 1 x 10^-3 C

The current in the left part of the circuit just before $t = \frac{7 π}{6 ω}$ is clockwise.

Immediately after A is connected to D, the current in R is 10 A.

Q = 2 x 10^-3 C

answer is C, D.

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Detailed Solution

If q represents the charge on capacitor’s upper plate
$\begin{array}{l} I (t) = I_{0} \cos (ωt) = \frac{dq}{dt} \\ \Rightarrow q (t) = \frac{I_{0}}{ω} \sin (ωt) \end{array}$
Max. Charge $= \frac{1 A}{500 {rads}^{- 1}} = 2 \times 10^{- 3} C$
Charge on upper plat at $t = \frac{7 π}{6 ω} = \frac{I_{0}}{ω} \sin (\frac{7 π}{6}) = - \frac{I_{0}}{2 ω}$
When capacitor is fully charged, charge on upper plate
$\begin{array}{l} 50 V \times 20 μF = 1 \times 10^{- 3} C \\ ∴ Q = 1 \times 10^{- 2} C - (- \frac{I_{0}}{2 ω}) = 2 \times 10^{- 3} C \end{array}$
Voltage across capacitor when A and D are connected
$= \frac{1 \times 10^{- 3} C}{20 \times 10^{- 6} μF} = 50 V$
Total voltage across resistor = 100 V
$\Rightarrow current = \frac{100 V}{10 Ω} = 10 A$
Current is anticlockwise.