Alternating Current Test

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Alternating Current Test - 53

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Weekly Quiz Competition

Question 1
1 / -0

A circuit containing a 20 $$\Omega$$ resistor and 0.1$$\mu F$$ capacitor in series is connected to 230 V AC supply of angular frequency 100 rad $$s^{-1}$$. The impedance of the circuit is

A
$$10^5\Omega$$

B
$$10^4\Omega$$

C
$$10^6\Omega$$

D
$$10^{10}\Omega$$

Solution

Hence, $$R = 20 \Omega, C = 0.1 \mu F= 0.1 \times 10^{-6}F = 10^{-7}F$$

Impedance ,$$ Z = \sqrt{R^2 + \dfrac{1}{\omega^2C^2}}$$

$$\sqrt{20^2 + \dfrac{1}{(100)^2 \times (10^{-7})^2}} \, = \, \sqrt{400 + 10^{10}} = 10^5 \, \Omega$$
Question 2
1 / -0

An inductor of 30 mH is connected to a 220 V, 100 Hz ac source. The inductive reactance is then

A
$$10.58 \Omega$$

B
$$12.64 \Omega$$

C
$$18.85 \Omega$$

D
$$22.67 \Omega$$

Solution

Hence ,$$ L = 30 mH = 30 \, \times \, 10^{-3} \, H$$

$$V_{rms} \, = \, 220, \, \upsilon \, = \, 100 \, Hz$$

Inductive reactance,

$$X_L \, = \, 2 \pi \upsilon L \, = \, 2 \, \times \, 3.14 \, \times \, 100 \, \times \, 30 \, \times \, 10^{-3} \, = \, 18.85 \, \Omega$$
Question 3
1 / -0

In A.C circuit having only capacitor, the current _____

A
lags behind the voltage by $$\cfrac { \pi }{ 2 } $$ in phase

B
leads the voltage by $$\cfrac { \pi }{ 2 } $$ in phase

C
leads the voltage by $$\pi$$ in phase

D
lags behind the voltage by $$\pi$$ in phase

Solution

Current in the capacitive A.C circuit leads the voltage by $$\dfrac{\pi}{2}$$ in phase.
Correct answer is option B.
Question 4
1 / -0

A series LCR circuit with R = 22 $$\Omega$$, L = 1.5 H and C = 40 $$\mu$$ F is connected to a variable frequency 220 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?

A
2000 W

B
2200 W

C
2400 W

D
2500 W

Solution

When the frequency of the supply equals to the natural frequency of circuit, resonance occurs.
$$\therefore \, Z \, = \, R \, = 22 \, \Omega \, and \, I_{rms} \, = \, \dfrac{V_{rms}}{Z} \, = \dfrac{220}{22} \, = \, 10 \, A$$
Average power transferred per cycle,
$$P \, = \, V_{rms} \, I_{rms} \, cos \, 0^{\circ} \, = \, 220 \, \times \, 10 \, \times \, 1 \, = \, 2200 \, W$$
Question 5
1 / -0

An LC circuit contains a 20 mH inductor and a 50 .$$\mu$$ F capacitor with an initial charge of 10 mC. The resistance of the circuit is.negligible. Let the instant at which the circuit which is closed be t = 0. At what time the energy stored is completely magnetic?

A
t = 0

B
t = 1.54 ms

C
t = 3.14 ms

D
t = 6.28 ms

Solution

At time $$t \, = \, \dfrac{T}{4},$$ energy stored is completely mangnetic.

$$\therefore \, Time \, t \, = \, \dfrac{T}{4} \, = \, \dfrac{2\pi\sqrt{LC}}{4} \, \Rightarrow \, t \, = \, \dfrac{\pi\sqrt{20 \, \times \, 10^{-3} \, \times \, 50 \, \times \, 10^{-6}}}{2}$$

$$t \, = \, \dfrac{\pi\sqrt{1000 \, \times \, 10^{-9}}}{2} \, = \, \dfrac{\pi\sqrt{1 \, \times \, 10^{-6}}}{2} \, = \, 1.57 ms$$
Question 6
1 / -0

In the given circuit, initially $${ K }_{ 1 }$$ is closed and $${ K }_{ 2 }$$ is open. Then $${ K }_{ 1 }$$ is opened and $${ K }_{ 2 }$$ is closed. If $${ q }_{ 1 }^{ \prime }$$ and $${ q }_{ 2 }^{ \prime }$$ are charges on $${ C }_{ 1 }$$ and $${ C }_{ 2 }$$ and $${ V }_{ 1 }$$ and $${ V }_{ 1 }$$ are the voltage respectively, then

A
charge on $${ C}_{ 1 }$$ gets redistributed such that $${ V }_{ 1 }$$ = $${ V }_{ 2 }$$

B
charge on $${ C}_{ 1 }$$ gets redistributed such that $${ q }_{ 1 }^{ \prime }$$ = $${ q }_{ 2 }^{ \prime }$$

C
charge on $${ C}_{ 1 }$$ gets redistributed such that $${ C }_{ 1 }{ V }_{ 1 }$$ = $${ C }_{ 2 }{ V }_{ 2 }$$ = $${ C }_{ 1 }V$$

D
charge on $${ C}_{ 1 }$$ gets redistributed such that $${ q }_{ 1 }^{ \prime }$$ + $${ q }_{ 2 }^{ \prime }$$ = 2q

Solution

From the figure, when $${K}_{1}$$ is closed and $${K}_{2}$$ is open, $${C}_{1}$$ is charged to potential $$V$$ acquiring a total charge
$$q={C}_{1} V$$
When $${K}_{1}$$ is open and $${K}_{2}$$ is closed, battery is cut off $${C}_{1}$$ and $${C}_{2}$$ are in parallel. The charge on $${C}_{1}$$ is shared between $${C}_{1}$$ and $${C}_{2}$$ such that $${V}_{1}={V}_{2}$$
As there is no loss of charge $${q}_{1}'+{q}_{2}'=q$$
Question 7
1 / -0

A sinusoidal voltage of peak value 293 V and frequency 50 Hz is applied to a series LCR circuit in which R = 6 $$\Omega$$, L = 25 mH and C = 750 $$\mu$$F. The impedance of the circuit is:

A
7.0 $$\Omega$$

B
8.9 $$\Omega$$

C
9.9 $$\Omega$$

D
10.0 $$\Omega$$

Solution

Hence , $$R = 6\Omega, L = 25 mH = 25 \times10^{-3}H$$
C$$ = 750\mu F= 750\times10^{-6}F, \upsilon = 50 Hz$$
$$X_L = 2\pi \upsilon L = 2\times3.14\times50\times25\times10^{-3} = 7.85 \Omega$$

$$X_C = \dfrac{1}{2\pi \upsilon C} = \dfrac{1}{2 \times 3.14 \times 50\times 750 \times 10^{-6}}= 4.25 \Omega$$

$$\therefore X_L -X_C = 7.85 - 4.25 = 3.6\Omega$$

Impedence of the series LCR circuit is
$$Z = \sqrt {R^2 + ( X_L - X_C)^2}$$

$$\therefore Z = \sqrt{(6)^2+(3.6)^2} = \sqrt{36 + 12.96} = 7.0\Omega$$
Question 8
1 / -0

The resonant frequency of a series LCR circuit with L = 2.0H,C=32$$\mu$$F and R= 10 $$\Omega$$ is

A
$$20 Hz$$

B
$$30 Hz$$

C
$$40 Hz$$

D
$$50 Hz$$

Solution

Here, $$L = 2H$$
$$C = 32\mu F = 32 \times 10^{-6}F $$
$$ R = 10\Omega $$

$$\therefore \omega = \dfrac{1}{\sqrt{LC}} = \dfrac{1}{\sqrt{2\times32\times10{-6}}} =125\, rad s^{-1}$$

$$\nu_r = \dfrac{\omega_r}{2\pi} = \dfrac{125}{2\times3.14}= 20Hz$$
Question 9
1 / -0

A 0.2 $$\Omega$$ resistor and 15 $$\mu$$F capacitor are connected in series to a 220 V, 50 Hz ac source. The impedance of the circuit is

A
$$250 \Omega$$

B
$$268 \Omega$$

C
$$29.15 \Omega$$

D
$$291.5 \Omega$$

Solution

Here, $$R = 0.2 k \Omega = 200 \Omega$$
$$C = 15 \mu F = 15 \times 10^{-6}F$$
$$\nu= 50Hz$$

Capacitive reactance,
$$ X_C = \dfrac{1}{2 \pi \nu C} = \dfrac{1}{2 \times 3.14 \times 50 \times 15 \times 10^{-6}} = 212 \, \Omega$$

The impedance of the RC circuit is
$$ Z = \sqrt{R^2 + X^2_C} = \sqrt{(200)^2 + (212)^2} = 291.5 \, \Omega$$
Question 10
1 / -0

A series resonant LCR circuit has a quality factor (Q-factor) = 0.4. If R 2 k$$\Omega$$, C = 0.1 $$\mu$$F, then the value of Inductance is

A
0.1 H

B
0.064 H

C
2 H

D
5 H

Solution

Quality factor $$Q \, = \, \dfrac{1}{R} \, \sqrt{\dfrac{L}{C}} \, or \, \dfrac{L}{C} \, = \, (QR)^2$$
Here, $$Q \, = \, 0.4, \, R \, = \, 2 \, K \Omega \, = \, 2 \, \times \, 10^3 \, \omega$$

$$C \, = \, 0.1 \, \mu F \, = \, 0.1 \, \times \, 10^{-6} \, F$$

$$\therefore \, L \, = \, (QR)^2 \, C$$

$$\therefore \, L \, = \, (0.4 \, \times \, 2 \, \times \, 10^3)^2 \, \times \, 0.1 \, \times \, 10^{-6} \, = \, 0.064 \, H$$

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Alternating Current Test - 53

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Alternating Current Test - 53

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SHARING IS CARING

Question 1

$$10^5\Omega$$

$$10^4\Omega$$

$$10^6\Omega$$

$$10^{10}\Omega$$

Solution

Question 2

$$10.58 \Omega$$

$$12.64 \Omega$$

$$18.85 \Omega$$

$$22.67 \Omega$$

Solution

Question 3

lags behind the voltage by $$\cfrac { \pi }{ 2 } $$ in phase

leads the voltage by $$\cfrac { \pi }{ 2 } $$ in phase

leads the voltage by $$\pi$$ in phase

lags behind the voltage by $$\pi$$ in phase

Solution

Question 4

2000 W

2200 W

2400 W

2500 W

Solution

Question 5

t = 0

t = 1.54 ms

t = 3.14 ms

t = 6.28 ms

Solution

Question 6

charge on $${ C}_{ 1 }$$ gets redistributed such that $${ V }_{ 1 }$$ = $${ V }_{ 2 }$$

charge on $${ C}_{ 1 }$$ gets redistributed such that $${ q }_{ 1 }^{ \prime }$$ = $${ q }_{ 2 }^{ \prime }$$

charge on $${ C}_{ 1 }$$ gets redistributed such that $${ C }_{ 1 }{ V }_{ 1 }$$ = $${ C }_{ 2 }{ V }_{ 2 }$$ = $${ C }_{ 1 }V$$

charge on $${ C}_{ 1 }$$ gets redistributed such that $${ q }_{ 1 }^{ \prime }$$ + $${ q }_{ 2 }^{ \prime }$$ = 2q

Solution

Question 7

7.0 $$\Omega$$

8.9 $$\Omega$$

9.9 $$\Omega$$

10.0 $$\Omega$$

Solution

Question 8

$$20 Hz$$

$$30 Hz$$

$$40 Hz$$

$$50 Hz$$

Solution

Question 9

$$250 \Omega$$

$$268 \Omega$$

$$29.15 \Omega$$

$$291.5 \Omega$$

Solution

Question 10

0.1 H

0.064 H

2 H

5 H

Solution

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