A 4 µF capacitor is charged by a 200 V supply.
Question: A 4 F capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 F capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation? Solution: Capacitance of a charged capacitor, $C_{1}=4 \mu \mathrm{F}=4 \times 0^{-6} \mathrm{~F}$ Supply voltage,V1= 200 V Electrostatic energy stored inC1is given by, $E_{1}=\frac{1}{2} C_{1} V_{1}^{2}$ $=\frac{1}{2} \times 4 \times 10^...
Read More →Convert the given complex number in polar form: – 1 + i
Question: Convert the given complex number in polar form: $-1+1$ Solution: $-1+i$ Let $r \cos \theta=-1$ and $r \sin \theta=1$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(-1)^{2}+1^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=1+1$ $\Rightarrow r^{2}=2$ $\Rightarrow r=\sqrt{2} \quad$ [Conventionally, $r0$ ] $\therefore \sqrt{2} \cos \theta=-1$ and $\sqrt{2} \sin \theta=1$ $\Rightarrow \cos \theta=-\frac{1}{\sqrt{2}}$ and $\sin \theta=...
Read More →The plates of a parallel plate capacitor have an area of
Question: The plates of a parallel plate capacitor have an area of 90 cm2each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply. (a)How much electrostatic energy is stored by the capacitor? (b)View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volumeu. Hence arrive at a relation betweenuand the magnitude of electric fieldEbetween the plates. Solution: Area of the plates of a parallel plate capacitor,A= 9...
Read More →Find the value of
Question: Find the value of $\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right],|x|1, y0$ and $x y1$ Solution: Let $x=\tan \theta$. Then, $\theta=\tan ^{-1} x$ $\therefore \sin ^{-1} \frac{2 x}{1+x^{2}}=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)=\sin ^{-1}(\sin 2 \theta)=2 \theta=2 \tan ^{-1} x$ Let $y=\tan \Phi$. Then, $\Phi=\tan ^{-1} y$. $\therefore \cos ^{-1} \frac{1-y^{2}}{1+y^{2}}=\cos ^{-1}\left(\frac{1-\tan ^{2} \phi}{1+\ta...
Read More →Convert the given complex number in polar form: 1 – i
Question: Convert the given complex number in polar form: $1-$ Solution: $1-i$ Let $r \cos \theta=1$ and $r \sin \theta=-1$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=1^{2}+(-1)^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=1+1$ $\Rightarrow r^{2}=2$ $\Rightarrow r=\sqrt{2} \quad[$ Conventionally, $r0]$ $\therefore \sqrt{2} \cos \theta=1$ and $\sqrt{2} \sin \theta=-1$ $\Rightarrow \cos \theta=\frac{1}{\sqrt{2}}$ and $\sin \theta=-\fra...
Read More →Find the modulus and the argument of the complex number
Question: Find the modulus and the argument of the complex number$z=-\sqrt{3}+i$ Solution: $z=-\sqrt{3}+i$ Let $r \cos \theta=-\sqrt{3}$ and $r \sin \theta=1$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(-\sqrt{3})^{2}+1^{2}$ $\Rightarrow r^{2}=3+1=4 \quad\left[\cos ^{2} \theta+\sin ^{2} \theta=1\right]$ $\Rightarrow r=\sqrt{4}=2 \quad$ [Conventionally, $r0$ ] $\therefore$ Modulus $=2$ $\therefore 2 \cos \theta=-\sqrt{3}$ and $2 \sin \theta=1$ $\Rightarrow \c...
Read More →Obtain the equivalent capacitance of the network in Fig. 2.35.
Question: Obtain the equivalent capacitance of the network in Fig. 2.35. For a $300 \mathrm{~V}$ supply, determine the charge and voltage across each capacitor. Solution: Capacitance of capacitor $C_{1}$ is $100 \mathrm{pF}$. Capacitance of capacitorC2is 200 pF. Capacitance of capacitorC3is 200 pF. Capacitance of capacitorC4is 100 pF. Supply potential,V= 300 V Capacitors $C_{2}$ and $C_{3}$ are connected in series. Let their equivalent capacitance be $C^{\prime}$. $\therefore \frac{1}{C^{\prime}...
Read More →Find the value of
Question: Find the value of $\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$ Solution: $\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$ $=\cot \left(\frac{\pi}{2}\right) \quad\left[\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right]$ $=0$...
Read More →Find the modulus and the argument of the complex number
Question: Find the modulus and the argument of the complex number$z=-1-i \sqrt{3}$ Solution: $z=-1-i \sqrt{3}$ Let $r \cos \theta=-1$ and $r \sin \theta=-\sqrt{3}$ On squaring and adding, we obtain $(r \cos \theta)^{2}+(r \sin \theta)^{2}=(-1)^{2}+(-\sqrt{3})^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=1+3$ $\Rightarrow \mathrm{r}^{2}=4 \quad\left[\cos ^{2} \theta+\sin ^{2} \theta=1\right]$ $\Rightarrow r=\sqrt{4}=2 \quad$ [Conventionally, $r0$ ] $\therefore$ Modulus $=...
Read More →Find the value of
Question: Find the value of $\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]$ Solution: Let $\sin ^{-1} \frac{1}{2}=x$. Then, $\sin x=\frac{1}{2}=\sin \left(\frac{\pi}{6}\right)$. $\therefore \sin ^{-1} \frac{1}{2}=\frac{\pi}{6}$ $\therefore \tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]=\tan ^{-1}\left[2 \cos \left(2 \times \frac{\pi}{6}\right)\right]$ $=\tan ^{-1}\left[2 \cos \frac{\pi}{3}\right]=\tan ^{-1}\left[2 \times \frac{1}{2}\right]$ $=\tan ^{-1}...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}$ Solution: $\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)$ Put $x=a \tan \theta \Rightarrow \frac{x}{a}=\tan \theta \Rightarrow \theta=\tan ^{-1} \frac{x}{a}$ $\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)=\tan ^{-1}\left(\frac{3 a^{2} \cdot a \tan \theta-a^{3} \tan ^{3} \theta}{a^{3}-3 a \cdo...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}},|x|a$ Solution: $\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$ Put $x=a \sin \theta \Rightarrow \frac{x}{a}=\sin \theta \Rightarrow \theta=\sin ^{-1}\left(\frac{x}{a}\right)$ $\therefore \tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}=\tan ^{-1}\left(\frac{a \sin \theta}{\sqrt{a^{2}-a^{2} \sin ^{2} \theta}}\right)$ $=\tan ^{-1}\left(\frac{a \sin \theta}{a \sqrt{1-\sin ^{2} \theta}}\right)=\tan ^{-1}\left(\frac{a \sin ...
Read More →Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply,
Question: Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor. Solution: Capacitance of capacitorC1is 100 pF. Capacitance of capacitorC2is 200 pF. Capacitance of capacitorC3is 200 pF. Capacitance of capacitorC4is 100 pF. Capacitors $C_{2}$ and $C_{3}$ are connected in series. Let their equivalent capacitance be $C^{\prime}$. $\therefore \frac{1}{C^{\prime}}=\frac{1}{200}+\frac{1}{200}=\frac{2}{200}$ $C^{\prime}...
Read More →The concentration of sulphide ion in 0.1M HCl solution saturated with hydrogen sulphide
Question: The concentration of sulphide ion in $0.1 \mathrm{M} \mathrm{HCl}$ solution saturated with hydrogen sulphide is $1.0 \times 10^{-19} \mathrm{M}$. If $10 \mathrm{~mL}$ of this is added to $5 \mathrm{~mL}$ of $0.04 \mathrm{M}$ solution of the following: $\mathrm{FeSO}_{4}, \mathrm{MnCl}_{2}, \mathrm{ZnCl}_{2}$ and $\mathrm{CdCl}_{2}$. in which of these solutions precipitation will take place? Given $K_{s p}$ for FeS $=6.3 \times 10^{-18}, \mathrm{MnS}=2.5 \times 10^{-13}, \mathrm{ZnS}=1....
Read More →Express the following expression in the form of a + ib.
Question: Express the following expression in the form ofa+ib. $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$ Solution: $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$ $=\frac{(3)^{2}-(i \sqrt{5})^{2}}{\sqrt{3}+\sqrt{2} i-\sqrt{3}+\sqrt{2} i} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$ $=\frac{9-5 i^{2}}{2 \sqrt{2} i}$ $=\frac{9-5(-1)}{2 \sqrt{2} i} \quad\left[i^{2}=-1\right]$ $=\frac{9+5}{2 \sqrt{2} i} \times \frac{i}{i}$ $=\f...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), 0x\pi$ Solution: $\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)$ $=\tan ^{-1}\left(\frac{1-\frac{\sin x}{\cos x}}{1+\frac{\sin x}{\cos x}}\right)$ $=\tan ^{-1}\left(\frac{1-\tan x}{1+\tan x}\right)$ $=\tan ^{-1}(1)-\tan ^{-1}(\tan x) \quad\left(\tan ^{-1} \frac{x-y}{1+x y}=\tan ^{-1} x-\tan ^{-1} y\right)$ $=\frac{\pi}{4}-x$...
Read More →Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply,
Question: Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor. Solution: Capacitance of capacitorC1is 100 pF. Capacitance of capacitorC2is 200 pF. Capacitance of capacitorC3is 200 pF. Capacitance of capacitorC4is 100 pF. Capacitors $C_{2}$ and $C_{3}$ are connected in series. Let their equivalent capacitance be $C^{\prime}$. $\therefore \frac{1}{C^{\prime}}=\frac{1}{200}+\frac{1}{200}=\frac{2}{200}$ $C^{\prime}...
Read More →Find the multiplicative inverse of the complex number –i
Question: Find the multiplicative inverse of the complex number $-i$ Solution: Let $z=-i$ Then, $\bar{z}=i$ and $|z|^{2}=1^{2}=1$ Therefore, the multiplicative inverse of $-i$ is given by $z^{-1}=\frac{\bar{z}}{|z|^{2}}=\frac{i}{1}=i$...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), 0x\pi$ Solution: $\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)$ $=\tan ^{-1}\left(\frac{1-\frac{\sin x}{\cos x}}{1+\frac{\sin x}{\cos x}}\right)$ $=\tan ^{-1}\left(\frac{1-\tan x}{1+\tan x}\right)$ $=\tan ^{-1}(1)-\tan ^{-1}(\tan x) \quad\left(\tan ^{-1} \frac{x-y}{1+x y}=\tan ^{-1} x-\tan ^{-1} y\right)$ $=\frac{\pi}{4}-x$...
Read More →Find the multiplicative inverse of the complex number
Question: Find the multiplicative inverse of the complex number $\sqrt{5}+3 i$ Solution: Let $z=\sqrt{5}+3 i$ Then, $\bar{z}=\sqrt{5}-3 i$ and $|z|^{2}=(\sqrt{5})^{2}+3^{2}=5+9=14$ Therefore, the multiplicative inverse of $\sqrt{5}+3 i$ is given by $z^{-1}=\frac{\bar{z}}{|z|^{2}}=\frac{\sqrt{5}-3 i}{14}=\frac{\sqrt{5}}{14}-\frac{3 i}{14}$...
Read More →What is the minimum volume of water required to dissolve 1g of calcium sulphate at 298 K?
Question: What is the minimum volume of water required to dissolve $1 \mathrm{~g}$ of calcium sulphate at $298 \mathrm{~K}$ ? $\left(\right.$ For calcium sulphate, $K_{\text {sp }}$ is $\left.9.1 \times 10^{-6}\right)$. Solution: $\mathrm{CaSO}_{4(s)} \longleftrightarrow \mathrm{Ca}^{2+}{ }_{(a q)}+\mathrm{SO}_{4}^{2-}(a q)$ $K_{s p}=\left[\mathrm{Ca}^{2+}\right]\left[\mathrm{SO}_{4}^{2-}\right]$ Let the solubility of $\mathrm{CaSO}_{4}$ be $s$. Then, $K_{s p}=s^{2}$ $9.1 \times 10^{-6}=s^{2}$ $...
Read More →Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply,
Question: Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor. Solution: Capacitance of capacitorC1is 100 pF. Capacitance of capacitorC2is 200 pF. Capacitance of capacitorC3is 200 pF. Capacitance of capacitorC4is 100 pF. Capacitors $C_{2}$ and $C_{3}$ are connected in series. Let their equivalent capacitance be $C^{\prime}$. $\therefore \frac{1}{C^{\prime}}=\frac{1}{200}+\frac{1}{200}=\frac{2}{200}$ $C^{\prime}...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right), x\pi$ Solution: $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right), x\pi$ $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)=\tan ^{-1}\left(\sqrt{\frac{2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}}\right)$ $=\tan ^{-1}\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right)=\tan ^{-1}\left(\tan \frac{x}{2}\right)$ $=\frac{x}{2}$...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right), x\pi$ Solution: $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right), x\pi$ $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)=\tan ^{-1}\left(\sqrt{\frac{2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}}\right)$ $=\tan ^{-1}\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right)=\tan ^{-1}\left(\tan \frac{x}{2}\right)$ $=\frac{x}{2}$...
Read More →Find the multiplicative inverse of the complex number 4 – 3i
Question: Find the multiplicative inverse of the complex number $4-3$, Solution: Let $z=4-3 i$ Then, $\bar{z}=4+3 i$ and $|z|^{2}=4^{2}+(-3)^{2}=16+9=25$ Therefore, the multiplicative inverse of $4-3 i$ is given by $z^{-1}=\frac{\bar{z}}{|z|^{2}}=\frac{4+3 i}{25}=\frac{4}{25}+\frac{3}{25} i$...
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