What are the oxidation numbers of the underlined elements in each of the following and how do you rationalise your results?
Question: What are the oxidation numbers of the underlined elements in each of the following and how do you rationalise your results? (a) $\mathrm{K} \underline{\mathrm{I}}_{3}$ (b) $\mathrm{H}_{2} \mathrm{~S}_{4} \mathrm{O}_{6}$ (c) $\mathrm{Fe}_{3} \mathrm{O}_{4}$ (d) $\underline{\mathrm{C}} \mathrm{H}_{3} \underline{\mathrm{C}} \mathrm{H}_{2} \mathrm{OH}$ (e) $\underline{\mathrm{C}} \mathrm{H}_{3} \underline{\mathrm{C}} \mathrm{O} \mathrm{OH}$ Solution: (a)KI3 In $\mathrm{KI}_{3}$, the oxidat...
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Question: (a)The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is about 100 Vm1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage so there is no field inside!) (b)A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluminium sheet of ...
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Question: Find the values of $\tan \left(\sin ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{2}\right)$ Solution: Let $\sin ^{-1} \frac{3}{5}=x$. Then, $\sin x=\frac{3}{5} \Rightarrow \cos x=\sqrt{1-\sin ^{2} x}=\frac{4}{5} \Rightarrow \sec x=\frac{5}{4}$. $\therefore \tan x=\sqrt{\sec ^{2} x-1}=\sqrt{\frac{25}{16}-1}=\frac{3}{4}$ $\therefore x=\tan ^{-1} \frac{3}{4}$ $\therefore \sin ^{-1} \frac{3}{5}=\tan ^{-1} \frac{3}{4}$....(1) Now, $\cot ^{-1} \frac{3}{2}=\tan ^{-1} \frac{2}{3}$...(2) $\left[\tan ^...
Read More →Solve the equation –x2 + x – 2 = 0
Question: Solve the equation $-x^{2}+x-2=0$ Solution: The given quadratic equation is $-x^{2}+x-2=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=-1, b=1$, and $c=-2$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=1^{2}-4 \times(-1) \times(-2)=1-8=-7$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \times(-1)}=\frac{-1 \pm \sqrt{7} i}{-2}$$[\sqrt{-1}=i]$...
Read More →A small sphere of radius
Question: A small sphere of radiusr1and chargeq1is enclosed by a spherical shell of radiusr2and chargeq2. Show that ifq1is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the chargeq2on the shell is. Solution: According to Gausss law, the electric field between a sphere and a shell is determined by the chargeq1on a small sphere. Hence, the potential difference,V, between the sphere and the shell is independent of chargeq2....
Read More →In a Van de Graaff type generator a spherical metal shell is to be a
Question: In a Van de Graaff type generator a spherical metal shell is to be a $15 \times 10^{6} \mathrm{~V}$ electrode. The dielectric strength of the gas surrounding the electrode is $5 \times 10^{7}$ $\mathrm{Vm}^{-1}$. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.) Solution: Potential difference, $V=15 \times 10^{...
Read More →Solve the equation x2 + 3x + 9 = 0
Question: Solve the equation $x^{2}+3 x+9=0$ Solution: The given quadratic equation is $x^{2}+3 x+9=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=1, b=3$, and $c=9$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=3^{2}-4 \times 1 \times 9=9-36=-27$ $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-3 \pm \sqrt{-27}}{2(1)}=\frac{-3 \pm 3 \sqrt{-3}}{2}=\frac{-3 \pm 3 \sqrt{3} i}{2}$ $[\sqrt{-1}=i]$...
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Question: Find the values of $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ Solution: $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ We know that $\tan ^{-1}(\tan x)=x$ if $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, which is the principal value branch of $\tan ^{-1} x$. Here, $\frac{3 \pi}{4} \notin\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ Now, $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ can be written as: $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\tan ^{-1}\left[-\tan \left(\frac{-3...
Read More →Solve the equation 2x2 + x + 1 = 0
Question: Solve the equation $2 x^{2}+x+1=0$ Solution: The given quadratic equation is $2 x^{2}+x+1=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=2, b=1$, and $c=1$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=1^{2}-4 \times 2 \times 1=1-8=-7$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \times 2}=\frac{-1 \pm \sqrt{7} i}{4}$ $[\sqrt{-1}=i]$...
Read More →Describe schematically the equipotential surfaces corresponding to
Question: Describe schematically the equipotential surfaces corresponding to (a)a constant electric field in thez-direction, (b)a field that uniformly increases in magnitude but remains in a constant (say,z) direction, (c)a single positive charge at the origin, and (d)a uniform grid consisting of long equally spaced parallel charged wires in a plane. Solution: (a)Equidistant planes parallel to thex-yplane are the equipotential surfaces. (b)Planes parallel to thex-yplane are the equipotential sur...
Read More →A parallel plate capacitor is to be designed with a voltage rating 1 kV,
Question: A parallel plate capacitor is to be designed with a voltage rating 1 kV, using a material of dielectric constant 3 and dielectric strength about 107Vm1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say 10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pF? So...
Read More →Solve the equation x2 + 3 = 0
Question: Solve the equation $x^{2}+3=0$ Solution: Therefore, the required solutions are The given quadratic equation is $x^{2}+3=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=1, b=0$, and $c=3$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=0^{2}-4 \times 1 \times 3=-12$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{\pm \sqrt{-12}}{2 \times 1}=\frac{\pm \sqrt{12} i}{2} \quad[\sqrt{-1}=i]$ $=\frac{\pm 2 \sqrt{3} i}{2}=\pm ...
Read More →Find the values of
Question: Find the values of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ Solution: $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ We know that $\sin ^{-1}(\sin x)=x$ if $x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, which is the principal value branch of $\sin ^{-1} x .$ Here, $\frac{2 \pi}{3} \notin\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ Now, $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ can be written as: $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\sin ^{-1}\left[\sin \left(\pi-\fra...
Read More →A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm.
Question: A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 C. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends). Solution: Length of a co-axial cylinder,l= 15 cm = 0.15 m Radius of outer cylinder,r1= 1.5 cm = 0.015 m Radius of inner cylinder,r2= 1.4 cm = 0.014 m Charge on the inne...
Read More →Answer carefully:
Question: (a) Two large conducting spheres carrying charges $Q_{1}$ and $Q_{2}$ are brought close to each other. Is the magnitude of electrostatic force between them exactly given by $Q_{1} Q_{2} / 4 \pi \epsilon_{0} r^{2}$, where $r$ is the distance between their centres? (b)If Coulombs law involved 1/r3dependence (instead of 1/r2), would Gausss law be still true? (c)A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line p...
Read More →If
Question: If $\tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1 \pi}{x+2}=\frac{\pi}{4}$, then find the value of $x$. Solution: $\tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}$ $\Rightarrow \tan ^{-1}\left[\frac{\frac{x-1}{x-2}+\frac{x+1}{x+2}}{1-\left(\frac{x-1}{x-2}\right)\left(\frac{x+1}{x+2}\right)}\right]=\frac{\pi}{4}$ $\left[\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}\right]$ $\Rightarrow \tan ^{-1}\left[\frac{(x-1)(x+2)+(x+1)(x-2)}{(x+2)(x-2)-(x-1)(x+1)}\rig...
Read More →Convert the given complex number in polar form: i
Question: Convert the given complex number in polar form: $i$ Solution: $i$ Let $r \cos \theta=0$ and $r \sin \theta=1$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=0^{2}+1^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=1$ $\Rightarrow r^{2}=1$ $\Rightarrow r=\sqrt{1}=1$ [Conventionally, $r0$ ] $\therefore \cos \theta=0$ and $\sin \theta=1$ $\therefore \theta=\frac{\pi}{2}$ $\therefore i=r \cos \theta+i r \sin \theta=\cos \frac{\pi}{2}+i...
Read More →A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm.
Question: A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 C. The space between the concentric spheres is filled with a liquid of dielectric constant 32. (a)Determine the capacitance of the capacitor. (b)What is the potential of the inner sphere? (c)Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller....
Read More →Convert the given complex number in polar form:
Question: Convert the given complex number in polar form:$\sqrt{3}+i$ Solution: $\sqrt{3}+i$ Let $r \cos \theta=\sqrt{3}$ and $r \sin \theta=1$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(\sqrt{3})^{2}+1^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=3+1$ $\Rightarrow r^{2}=4$ $\Rightarrow r=\sqrt{4}=2 \quad[$ Conventionally, $r0]$ $\therefore 2 \cos \theta=\sqrt{3}$ and $2 \sin \theta=1$ $\Rightarrow \cos \theta=\frac{\sqrt{3}}{2}$ an...
Read More →A spherical capacitor consists of two concentric spherical conductors,
Question: A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show that the capacitance of a spherical capacitor is given by $C=\frac{4 \pi \in_{0} r_{1} r_{2}}{r_{1}-r_{2}}$ where $r_{1}$ and $r_{2}$ are the radii of outer and inner spheres, respectively. Solution: Radius of the outer shell =r1 Radius of the inner shell =r2 The inner surface of the outer shell has charge +Q. The outer surface of the inner shell has...
Read More →Convert the given complex number in polar form: –3
Question: Convert the given complex number in polar form: $-3$ Solution: $-3$ Let $r \cos \theta=-3$ and $r \sin \theta=0$ On squaring and adding, we obtain $r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(-3)^{2}$ $\Rightarrow r^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=9$ $\Rightarrow r^{2}=9$ $\Rightarrow r=\sqrt{9}=3 \quad[$ Conventionally, $r0]$ $\therefore 3 \cos \theta=-3$ and $3 \sin \theta=0$ $\Rightarrow \cos \theta=-1$ and $\sin \theta=0$ $\therefore \theta=\pi$ $\therefore-3=r...
Read More →If, then find the value of x.
Question: If $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$, then find the value of $x$. Solution: $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$ $\Rightarrow \sin \left(\sin ^{-1} \frac{1}{5}\right) \cos \left(\cos ^{-1} x\right)+\cos \left(\sin ^{-1} \frac{1}{5}\right) \sin \left(\cos ^{-1} x\right)=1$ $[\sin (A+B)=\sin A \cos B+\cos A \sin B]$ $\Rightarrow \frac{1}{5} \times x+\cos \left(\sin ^{-1} \frac{1}{5}\right) \sin \left(\cos ^{-1} x\right)=1$ $\Rightarrow \frac{x...
Read More →Show that the force on each plate of a parallel plate capacitor has a magnitude equal
Question: Show that the force on each plate of a parallel plate capacitor has a magnitude equal to $(1 / 2) Q E$, where $Q$ is the charge on the capacitor, and $E$ is the magnitude of electric field between the plates. Explain the origin of the factor $1 / 2 .$ Solution: LetFbe the force applied to separate the plates of a parallel plate capacitor by a distance of Δx. Hence, work done by the force to do so =FΔx As a result, the potential energy of the capacitor increases by an amount given asuAΔ...
Read More →Convert the given complex number in polar form: – 1 – i
Question: Convert the given complex number in polar form: $-1-i$ 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 \thet...
Read More →Assign oxidation numbers to the underlined elements in each of the following species:
Question: Assign oxidation numbers to the underlined elements in each of the following species: (a) $\mathrm{NaH}_{2} \mathrm{PO}_{4}$ (b) $\mathrm{NaHS} \mathrm{SO}_{4}$ (c) $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{7}$ (d) $\mathrm{K}_{2} \mathrm{MnO}_{4}$ (e) $\mathrm{Ca} \underline{\mathrm{O}}_{2}$ (f) $\mathrm{NaB} \mathrm{H}_{4}$ (g) $\mathrm{H}_{2} \underline{\mathrm{S}}_{2} \mathrm{O}_{7}$ (h) $\mathrm{KAl}\left(\underline{\mathrm{S}} \mathrm{O}_{4}\right)_{2} \cdot 12 \mathrm{H}_{2} \m...
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