Alternating Current Test

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Alternating Current Test - 56

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Weekly Quiz Competition

Question 1
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A resistance of $$ 10\Omega $$, a
capacitance of $$ 0.1\mu F$$ and an inductance of $$2mH$$ are connected in
series across a source of alternating
emf of variab frequency. At what frequency does maximum current flow?

A
$$11.25kHz$$

B
$$23.76kHz$$

C
$$35.46kHz$$

D
$$46.72kHz$$

Solution
Question 2
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The frequancy of oscillation of current in the inductor is-

A
$$\dfrac { 1 }{ 3\sqrt { LC } } $$

B
$$\dfrac {1}{6\pi \sqrt {LC}}$$

C
$$\dfrac { 1 }{\sqrt { LC } } $$

D
$$\dfrac {1}{2\pi \sqrt {LC}}$$

Solution

Frequency of oscilation$$=\cfrac { 1 }{ \sqrt { 3C(3L) } } =\cfrac { 1 }{ 3\sqrt { LC } } $$
Question 3
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The equation of an alternating voltage is $$V=100\sqrt {2}\sin {100\pi t}$$ volt. The RMS value of voltage and frequency will be respectively

A
$$100V,50Hz$$

B
$$50V,100Hz$$

C
$$150V, 50Hz$$

D
$$200V, 50Hz$$

Solution

Angular frequency is nothing but $$coefficient$$ of $$t$$ in the equation,
so we have $$\omega =100\pi$$ or $$2\pi f=100\pi$$, we get $$f=50Hz$$
rms voltage is $$\dfrac{\text{peak voltage}}{\sqrt {2}}=100 \sqrt{2}/ \sqrt{2}=100 V$$.

Option A is correct.
Question 4
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An alternating power supply of $$220 V$$ is applied across a series circuit of resistance $$10\sqrt{3}\Omega$$, capacitive reactance $$40 \Omega$$ and inductive reactance $$30 \Omega$$. The respective current in the circuit for zero and infinite frequencies are

A
$$2A,\dfrac{1}{2}A$$

B
$$0A,10A$$

C
$$10 A,0A$$

D
$$0A,0A$$

Solution

Case I: When the frequency is zero.
$$f=0$$ implies $$\omega = 0$$
Therefore, the capacitive reactance becomes infinite
$$X_C= \dfrac{1}{\omega C} \to \infty$$
As a result the impedance(Z) of the circuit becomes infinite and the current becomes zero.

Case I: When the frequency is infinite.
$$f \to \infty$$ implies $$\omega \to \infty$$
Therefore, the inductive reactance becomes infinite
$$X_L= \omega L \to \infty$$
As a result the impedance(Z) of the circuit becomes infinite and the current becomes zero.
Question 5
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A coil has resistance $$30 \Omega $$ and inductive reactance $$20 \Omega $$ at $$50 Hz$$ frequency.If an ac source of $$200 V,100Hz$$ is connected across the coil ,the current in the coil in the coil will be

A
$$\dfrac{20}{\sqrt{13}}A$$

B
$$2.0 A$$

C
$$4.0A$$

D
$$8.0A$$

Solution

Given,
$$X_L=20\Omega$$
frequency $$ f=50Hz$$
So, $$2\pi fL=20$$
$$\Rightarrow 2\pi\times 50L=20\Rightarrow L=\dfrac{1}{5\pi}$$
When the frequency is $$100Hz$$
$$X_L=2\pi\times\dfrac{1}{5\pi}\times100=40\Omega$$
Impedance $$Z= \sqrt{R^2+X_L^2}=\sqrt{900+1600}=50\Omega$$
$$V=200volt$$
Thus the current $$I=\dfrac{200}{50}=4A$$
Question 6
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Figure shows a long straight conductor carrying current i. A square loop of side a is kept at a distance r from it. Which of the following is correct?

A
Mutual Inductance $$M = \dfrac{\mu_0 a}{2 \pi} ln(1 + \dfrac{a}{r})$$

B
If loop is pulled with constant speed v along x-axis, emf induced at the instant is $$\dfrac{\mu_0 i a^2 v}{2 \pi r (r + a)}$$

C
If loop is pulled with constant speed v, no emf is induced

D
If i increases with time, the loop is replied away from the straight conductor.

Solution
Question 7
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A L - C resonant circuit contains a 200 pF capacitor and a $$ 100 \mu $$ H inductor it is set into oscillation coupled to an antenna. The wavelength of the radiated electromagnetic waves is

A
377 mm

B
266 m

C
377 cm

D
3.77 cm

Solution

Given that,
$$ C=200pF $$

$$ C=200\times {{10}^{-12}}F $$

$$ L=100\mu H $$

$$ L=100\times {{10}^{-6}}H $$

We know that,

Frequency of wave = $$ \nu =\dfrac{1}{2\pi \sqrt{LC}} $$

$$ \nu =\dfrac{1}{2\times 3.14\sqrt{200\times {{10}^{-12}}\times 100\times {{10}^{-6}}}} $$

$$ \nu =\dfrac{1}{2\times 3.14\times 1.4142\times {{10}^{-7}}} $$

Now, the wavelength is

$$ \lambda =\dfrac{3\times {{10}^{8}}}{\nu } $$

$$ \lambda =\dfrac{3\times {{10}^{8}}}{\dfrac{1}{2\times 3.14\times 1.4142\times {{10}^{-7}}}} $$

$$ \lambda =3\times {{10}^{8}}\times 2\times 3.14\times 1.4142\times {{10}^{-7}} $$

$$ \lambda =26.643\times 10 $$

$$ \lambda =266.43\,m $$

Hence, the wavelength of radiated electromagnetic wave is $$266$$ m.
Question 8
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In L-C-R series circuit consists of a resistance of 10 $$\Omega$$ a capacitor of reactance 6.0 $$\Omega$$ and an inductor coil. The circuit is found to resonate when put across a 300 V, 100 Hz supply. The inductance of coil is (Take $$\pi$$ = 3)

A
0.1 H

B
0.01 H

C
0.2 H

D
0.02 H

Solution
Question 9
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The sinusoidal potential difference $$V_1$$ shown in figure applied across a resistor $$R$$ produces heat at a rate $$W$$. what is the rate of heat dissipation when the square waves potential different $$V_2$$ as shown in figure is applied across the resistor?

A
$$\dfrac{W}{2}$$

B
$$W$$

C
$$\sqrt{2}W$$

D
$$2W$$

Solution
Question 10
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In a series LCR circuit,the inductive reactance is twice the resistance and the capacitance reactance is $${\frac{1}{3}^{rd}}$$ the inductive reactance. The power factor of the circuit is:

A
$$0.5$$

B
$$0.6$$

C
$$0.8$$

D
$$1$$

Solution

$$\begin{array}{l}\omega L = 2R\\\frac{1}{{\omega C}} = \frac{1}{3}\left( {\omega L} \right)\\\omega L - \frac{1}{{\omega C}} = 2R - \frac{{2R}}{3} = \frac{{4R}}{3}\\\tan \phi = \frac{{4R}}{3} \times \frac{1}{R} = \frac{4}{3}\\\cos \phi = \frac{1}{{\sqrt {1 + {{\tan }^2}\phi } }} = \frac{1}{{\sqrt {1 + \frac{{{4^2}}}{{{3^2}}}} }} = \frac{3}{5} = 0.6\end{array}$$