Obtain the resonant frequency and Q-factor of a series LCR circuit with
Question: Obtain the resonant frequency andQ-factor of a seriesLCRcircuit withL= 3.0 H,C= 27 F, andR= 7.4 Ω. It is desired to improve the sharpness of the resonance of the circuit by reducing its full width at half maximum by a factor of 2. Suggest a suitable way. Solution: Inductance,L= 3.0 H Capacitance,C= 27 F = 27 106F Resistance,R= 7.4 Ω At resonance, angular frequency of the source for the givenLCRseries circuit is given as: $\omega_{r}=\frac{1}{\sqrt{L C}}$ $=\frac{1}{\sqrt{3 \times 27 \t...
Read More →Find (a + b)4 – (a – b)4. Hence, evaluate.
Question: Find $(a+b)^{4}-(a-b)^{4}$. Hence, evaluate $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}$. Solution: Using Binomial Theorem, the expressions, $(a+b)^{4}$ and $(a-b)^{4}$, can be expanded as $(a+b)^{4}={ }^{4} C_{0} a^{4}+{ }^{4} C_{1} a^{3} b+{ }^{4} C_{2} a^{2} b^{2}+{ }^{4} C_{3} a b^{3}+{ }^{4} C_{4} b^{4}$ $(a-b)^{4}={ }^{4} C_{0} a^{4}-{ }^{4} C_{1} a^{3} b+{ }^{4} C_{2} a^{2} b^{2}-{ }^{4} C_{3} a b^{3}+{ }^{4} C_{4} b^{4}$ $\therefore(a+b)^{4}-(a-b)^{4}={ }^{4} C_{1} a^{4}+{...
Read More →A series LCR circuit with L = 0.12 H,
Question: A seriesLCRcircuit withL= 0.12 H,C= 480 nF,R= 23 Ω is connected to a 230 V variable frequency supply. (a)What is the source frequency for which current amplitude is maximum. Obtain this maximum value. (b)What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power. (c)For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these f...
Read More →Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Question: Using Binomial Theorem, indicate which number is larger $(1.1)^{10000}$ or 1000 . Solution: By splitting 1.1 and then applying Binomial Theorem, the first few terms of (1.1)10000can be obtained as $(1.1)^{10000}=(1+0.1)^{10000}$ $={ }^{10000} \mathrm{C}_{0}+{ }^{10000} \mathrm{C}_{1}(1.1)+$ Other positive terms $=1+10000 \times 1.1+$ Other positive terms $=1+11000+$ Other positive terms $1000$ Hence, $(1.1)^{10000}1000$...
Read More →Compare the solubility and thermal stability of the following compounds of the alkali metals with those of the alkaline earth metals.
Question: Compare the solubility and thermal stability of the following compounds of the alkali metals with those of the alkaline earth metals. (a) Nitrates (b) Carbonates (c) Sulphates. Solution: (i)Nitrates Thermal stability Nitrates of alkali metals, except $\mathrm{LiNO}_{3}$, decompose on strong heating to form nitrites. $2 \mathrm{KNO}_{3(s)} \longrightarrow 2 \mathrm{KNO}_{2(s)}+\mathrm{O}_{2(g)}$ $\mathrm{LiNO}_{3}$, on decomposition, gives oxide. Similar to lithium nitrate, alkaline ear...
Read More →Using Binomial Theorem, evaluate (99)5
Question: Using Binomial Theorem, evaluate $(99)^{5}$ Solution: 99can be written as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied. It can be writtenthat, 99 = 100 1 $\therefore(99)^{5}=(100-1)^{5}$ $={ }^{5} \mathrm{C}_{0}(100)^{5}-{ }^{5} \mathrm{C}_{1}(100)^{4}(1)+{ }^{5} \mathrm{C}_{2}(100)^{3}(1)^{2}-{ }^{5} \mathrm{C}_{3}(100)^{2}(1)^{3}$ $+{ }^{5} \mathrm{C}_{4}(100)(1)^{4}-{ }^{5} \mathrm{C}_{5}(1)^{5}$ $=(100)^{5}-5(10...
Read More →Suppose the circuit in Exercise 7.18 has a resistance of 15 Ω.
Question: Suppose the circuit in Exercise 7.18 has a resistance of 15 Ω. Obtain the average power transferred to each element of the circuit, and the total power absorbed. Solution: Average power transferred to the resistor = 788.44 W Average power transferred to the capacitor = 0 W Total power absorbed by the circuit = 788.44 W Inductance of inductor,L= 80 mH = 80 103H Capacitance of capacitor,C= 60F = 60 106F Resistance of resistor,R= 15 Ω Potential of voltage supply,V= 230 V Frequency of sign...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{lr}6 -3 \\ -2 1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{lr}6 -3 \\ -2 1\end{array}\right]$ We know that $A=I A$ $\therefore\left[\begin{array}{cr}6 -3 \\ -2 1\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{cr}1 -\frac{1}{2} \\ -2 1\end{array}\right]=\left[\begin{array}{cc}\frac{1}{6} 0 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1}...
Read More →Using Binomial Theorem, evaluate (101)4
Question: Using Binomial Theorem, evaluate $(101)^{4}$ Solution: 101can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied. It can be writtenthat, 101 = 100 + 1 $\therefore(101)^{4}=(100+1)^{4}$ $={ }^{4} \mathrm{C}_{0}(100)^{4}+{ }^{4} \mathrm{C}_{1}(100)^{3}(1)+{ }^{4} \mathrm{C}_{2}(100)^{2}(1)^{2}+{ }^{4} \mathrm{C}_{3}(100)(1)^{3}+{ }^{4} \mathrm{C}_{4}(1)^{4}$ $=(100)^{4}+4(100)^{3}+6(100)^{2}+4(100)+(1)^{4}$ ...
Read More →A circuit containing a 80 mH inductor and a 60 μF capacitor in series is connected to a 230 V, 50 Hz supply.
Question: A circuit containing a 80 mH inductor and a 60 F capacitor in series is connected to a 230 V, 50 Hz supply. The resistance of the circuit is negligible. (a)Obtain the current amplitude and rms values. (b)Obtain the rms values of potential drops across each element. (c)What is the average power transferred to the inductor? (d)What is the average power transferred to the capacitor? (e)What is the total average power absorbed by the circuit? [Average implies averaged over one cycle.] Solu...
Read More →Using Binomial Theorem, evaluate (102)5
Question: Using Binomial Theorem, evaluate $(102)^{5}$ Solution: 102can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied. It can be writtenthat, 102 = 100 + 2 $\therefore(102)^{5}=(100+2)^{5}$ $={ }^{5} \mathrm{C}_{0}(100)^{5}+{ }^{5} \mathrm{C}_{1}(100)^{4}(2)+{ }^{5} \mathrm{C}_{2}(100)^{3}(2)^{2}+{ }^{5} \mathrm{C}_{3}(100)^{2}(2)^{3}$ $+{ }^{5} \mathrm{C}_{4}(100)(2)^{4}+{ }^{5} \mathrm{C}_{5}(2)^{5}$ $=(100)^...
Read More →Find the inverse of each of the matrices, if it exists
Question: Find the inverse of each of the matrices, if it exists $\left[\begin{array}{ll}2 -6 \\ 1 -2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 -6 \\ 1 -2\end{array}\right]$ We know thatA=AI $\therefore\left[\begin{array}{ll}2 -6 \\ 1 -2\end{array}\right]=A\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ $\Rightarrow\left[\begin{array}{ll}2 0 \\ 1 1\end{array}\right]=A\left[\begin{array}{ll}1 3 \\ 0 1\end{array}\right] \quad\left(\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}+3...
Read More →Why is Li2CO3 decomposed at a lower temperature whereas
Question: Why is $\mathrm{Li}_{2} \mathrm{CO}_{3}$ decomposed at a lower temperature whereas $\mathrm{Na}_{2} \mathrm{CO}_{3}$ at higher temperature? Solution: As we move down the alkali metal group, the electropositive character increases. This causes an increase in the stability of alkali carbonates. However, lithium carbonate is not so stable to heat. This is because lithium carbonate is covalent. Lithium ion, being very small in size, polarizes a large carbonate ion, leading to the formation...
Read More →Keeping the source frequency equal to the resonating frequency of the series LCR circuit,
Question: Keeping the source frequency equal to the resonating frequency of the seriesLCRcircuit, if the three elements,L,CandRare arranged in parallel, show that the total current in the parallelLCRcircuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified in Exercise 7.11 for this frequency. Solution: An inductor (L), a capacitor (C), and a resistor (R) is connected in parallel with each other in a circuit where, L= 5....
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{lr}3 -1 \\ -4 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{lr}3 -1 \\ -4 2\end{array}\right]$ We know that $A=A l$ $\therefore\left[\begin{array}{lr}3 -1 \\ -4 2\end{array}\right]=A\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ $\Rightarrow\left[\begin{array}{rr}1 -1 \\ 0 2\end{array}\right]=A\left[\begin{array}{ll}1 0 \\ 2 1\end{array}\right] \quad\left(\mathrm{C}_{1} \rightarrow \mathrm{C}...
Read More →Potassium carbonate cannot be prepared by Solvay process.
Question: Potassium carbonate cannot be prepared by Solvay process. Why? Solution: Solvay process cannot be used to prepare potassium carbonate. This is because unlike sodium bicarbonate, potassium bicarbonate is fairly soluble in water and does not precipitate out....
Read More →Using Binomial Theorem, evaluate (96)3
Question: Using Binomial Theorem, evaluate (96) $^{3}$ Solution: 96can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, binomial theorem can be applied. It can be writtenthat, 96 = 100 4 $\therefore(96)^{3}=(100-4)^{3}$ $={ }^{3} \mathrm{C}_{0}(100)^{3}-{ }^{3} \mathrm{C}_{1}(100)^{2}(4)+{ }^{3} \mathrm{C}_{2}(100)(4)^{2}-{ }^{3} \mathrm{C}_{3}(4)^{3}$ $=(100)^{3}-3(100)^{2}(4)+3(100)(4)^{2}-(4)^{3}$ $=1000000-120000+4800-64$ $=884736$...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{rr}3 10 \\ 2 7\end{array}\right]$ Solution: Let $A=\left[\begin{array}{rr}3 10 \\ 2 7\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{rr}3 10 \\ 2 7\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ll}1 3 \\ 2 7\end{array}\right]=\left[\begin{array}{cc}1 -1 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-...
Read More →Expand
Question: Expand $\left(x+\frac{1}{x}\right)^{6}$ Solution: By using Binomial Theorem, the expression $\left(x+\frac{1}{x}\right)^{6}$ can be expanded as $\left(x+\frac{1}{x}\right)^{6}={ }^{6} C_{0}(x)^{6}+{ }^{6} C_{1}(x)^{5}\left(\frac{1}{x}\right)+{ }^{6} C_{2}(x)^{4}\left(\frac{1}{x}\right)^{2}$ $+{ }^{6} \mathrm{C}_{3}(\mathrm{x})^{3}\left(\frac{1}{\mathrm{x}}\right)^{3}+{ }^{6} \mathrm{C}_{4}(\mathrm{x})^{2}\left(\frac{1}{\mathrm{x}}\right)^{4}+{ }^{6} \mathrm{C}_{5}(\mathrm{x})\left(\fra...
Read More →Obtain the answers to
Question: Obtain the answers to (a) and (b) in Exercise 7.15 if the circuit is connected to a 110 V, 12 kHz supply? Hence, explain the statement that a capacitor is a conductor at very high frequencies. Compare this behaviour with that of a capacitor in a dc circuit after the steady state. Solution: Capacitance of the capacitor,C= 100 F = 100 106F Resistance of the resistor,R= 40 Ω Supply voltage,V= 110 V Frequency of the supply,= 12 kHz = 12 103Hz Angular Frequency,= 2 = 2 12 10303 = 24 103rad/...
Read More →Discuss the various reactions that occur in the Solvay process.
Question: Discuss the various reactions that occur in the Solvay process. Solution: Solvay process is used to prepare sodium carbonate. When carbon dioxide gasis bubbled through a brine solution saturated with ammonia, sodium hydrogen carbonate is formed. This sodium hydrogen carbonate is then converted to sodium carbonate. Step 1:Brine solution is saturated with ammonia. $2 \mathrm{NH}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \longrightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}$...
Read More →Expand the expression
Question: Expand the expression $\left(\frac{x}{3}+\frac{1}{x}\right)^{5}$ Solution: By using Binomial Theorem, the expression $\left(\frac{x}{3}+\frac{1}{x}\right)^{5}$ can be expanded as $\left(\frac{x}{3}+\frac{1}{x}\right)^{5}={ }^{5} C_{0}\left(\frac{x}{3}\right)^{5}+{ }^{5} C_{1}\left(\frac{x}{3}\right)^{4}\left(\frac{1}{x}\right)+{ }^{5} C_{2}\left(\frac{x}{3}\right)^{3}\left(\frac{1}{x}\right)^{2}$ $+{ }^{5} \mathrm{C}_{3}\left(\frac{\mathrm{x}}{3}\right)^{2}\left(\frac{1}{\mathrm{x}}\ri...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{ll}4 5 \\ 3 4\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}4 5 \\ 3 4\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{ll}4 5 \\ 3 4\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ll}1 1 \\ 3 4\end{array}\right]=\left[\begin{array}{cc}1 -1 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\ma...
Read More →Expand the expression (2x – 3)6
Question: Expand the expression $(2 x-3)^{6}$ Solution: By using Binomial Theorem, the expression $(2 x-3)^{6}$ can be expanded as $(2 x-3)^{6}={ }^{6} \mathrm{C}_{0}(2 x)^{6}-{ }^{6} \mathrm{C}_{1}(2 x)^{5}(3)+{ }^{6} \mathrm{C}_{2}(2 x)^{4}(3)^{2}-{ }^{6} \mathrm{C}_{3}(2 x)^{3}(3)^{3}$ $\begin{aligned} +{ }^{6} \mathrm{C}_{+}(2 x)^{2}(3)^{4}-{ }^{6} \mathrm{C}_{5}(2 x)(3)^{5}+{ }^{6} \mathrm{C}_{6}(3)^{6} \\= 64 x^{6}-6\left(32 x^{5}\right)(3)+15\left(16 x^{4}\right)(9)-20\left(8 x^{3}\right)...
Read More →A 100 μF capacitor in series with a 40 Ω resistance is connected to a 110 V,
Question: A 100 F capacitor in series with a 40 Ω resistance is connected to a 110 V, 60 Hz supply. (a)What is the maximum current in the circuit? (b)What is the time lag between the current maximum and the voltage maximum? Solution: Capacitance of the capacitor, $C=100 \mu \mathrm{F}=100 \times 10^{-6} \mathrm{~F}$ Resistance of the resistor, $R=40 \Omega$ Supply voltage,V= 110 V (a)Frequency of oscillations,= 60 Hz Angular frequency, $\omega=2 \pi \nu=2 \pi \times 60 \mathrm{rad} / \mathrm{s}$...
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