Figure 2.34 shows a charge array known as an electric quadrupole.
Question: Figure 2.34 shows a charge array known as anelectric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential onrforr/a 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge). Solution: Four charges of same magnitude are placed at points X, Y, Y, and Z respectively, as shown in the following figure. A point is located at P, which isrdistance away from point Y. The system of charges forms an ele...
Read More →Prove $3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$
Question: Prove $3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$ Solution: To prove: $3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$ Let $x=\sin \theta$. Then, $\sin ^{-1} x=\theta$. We have, R.H.S. $=\sin ^{-1}\left(3 x-4 x^{3}\right)=\sin ^{-1}\left(3 \sin \theta-4 \sin ^{3} \theta\right)$ $=\sin ^{-1}(\sin 3 \theta)$ $=3 \theta$ $=3 \sin ^{-1} x$ $=L . H . S .$...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: x2n – y2n is divisible by x + y.
Question: Prove the following by using the principle of mathematical induction for all $n \in N: x^{2 n}-y^{2 n}$ is divisible by $x+y$. Solution: Let the given statement be $P(n)$, i.e., $P(n): x^{2 n}-y^{2 n}$ is divisible by $x+y .$ It can be observed that $\mathrm{P}(n)$ is true for $n=1$. This is so because $x^{2} \times 1-y^{2} \times 1=x^{2}-y^{2}=(x+y)(x-y)$ is divisible by $(x+y)$. Let P(k) be true for some positive integerk, i.e., $x^{2 k}-y^{2 k}$ is divisible by $x+y$ $\therefore x^{...
Read More →Two charges −q and +q are located at points (0, 0, − a) and (0, 0, a), respectively.
Question: Two chargesqand+qare located at points (0, 0, a) and (0, 0,a), respectively. (a)What is the electrostatic potential at the points? (b) Obtain the dependence of potential on the distance $r$ of a point from the origin when $r / a \gg 1$. (c)How much work is done in moving a small test charge from the point (5, 0, 0) to (7, 0, 0) along thex-axis? Does the answer change if the path of the test charge between the same points is not along thex-axis? Solution: (a) Zero at both the points Cha...
Read More →Find the value of is equal to
Question: Find the value of $\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)$ is equal to (A) $\pi$ (B) $-\frac{\pi}{3}$ (C) $\frac{\pi}{3}$ (D) $\frac{2 \pi}{3}$ Solution: Let $\tan ^{-1} \sqrt{3}=x$. Then, $\tan x=\sqrt{3}=\tan \frac{\pi}{3}$. We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$. $\therefore \tan ^{-1} \sqrt{3}=\frac{\pi}{3}$ Let $\sec ^{-1}(-2)=y .$ Then, $\sec y=-2=-\sec \left(\frac{\pi}{3}\right)=\sec \left(\pi-\frac{\pi}{3...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11.
Question: Prove the following by using the principle of mathematical induction for all $n \in N: 10^{2 n-1}+1$ is divisible by 11 Solution: Let the given statement be $\mathrm{P}(n)$, i.e., $P(n): 10^{2 n-1}+1$ is divisible by $11 .$ It can be observed that $\mathrm{P}(n)$ is true for $n=1$ since $\mathrm{P}(1)=10^{2.1-1}+1=11$, which is divisible by 11 . Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $10^{2 k-1}+1$ is divisible by 11 $\therefore 10^{2 k-1}+1=11 m$, where $m \i...
Read More →Calculate the pH of the resultant mixtures:
Question: Calculate the pH of the resultant mixtures: a) $10 \mathrm{~mL}$ of $0.2 \mathrm{M} \mathrm{Ca}(\mathrm{OH})_{2}+25 \mathrm{~mL}$ of $0.1 \mathrm{M} \mathrm{HCl}$ b) $10 \mathrm{~mL}$ of $0.01 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}+10 \mathrm{~mL}$ of $0.01 \mathrm{M} \mathrm{Ca}(\mathrm{OH})_{2}$ c) $10 \mathrm{~mL}$ of $0.1 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}+10 \mathrm{~mL}$ of $0.1 \mathrm{M} \mathrm{KOH}$ Solution: (a) Moles of $\mathrm{H}_{3} \mathrm{O}^{+}=\frac{25 \tim...
Read More →Two charged conducting spheres of radii a and b are connected to each other by a wire.
Question: Two charged conducting spheres of radiiaandbare connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions. Solution: Let $a$ be the radius of a sphere $A, Q_{A}$ be the charge on the sphere, and $C_{A}$ be the capacitance of the sphere. Let $b$ be the radius of a sphere $B$, $Q_{B}$ be the charge on t...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3.
Question: Prove the following by using the principle of mathematical induction for all $n \in N \cdot n(n+1)(n+5)$ is a multiple of 3 . Solution: Let the given statement be $P(n)$, i.e., $\mathrm{P}(n): n(n+1)(n+5)$, which is a multiple of $3 .$ It can be noted that $P(n)$ is true for $n=1$ since $1(1+1)(1+5)=12$, which is a multiple of 3 . Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $k(k+1)(k+5)$ is a multiple of 3 $\therefore k(k+1)(k+5)=3 m$, where $m \in \mathbf{N}$ We s...
Read More →If one of the two electrons of a
Question: If one of the two electrons of a $\mathrm{H}_{2}$ molecule is removed, we get a hydrogen molecular ion $\mathrm{H}_{2}^{+}$. In the ground state of an $\mathrm{H}_{2}^{+}$, the two protons are separated by roughly $1.5$Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy. Solution: The system of two protons and one electron is represented in the given figure. Charge on proton $1, q_{1}=1.6...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N:
Question: Prove the following by using the principle of mathematical induction for allnN:$1+2+3+\ldots+n\frac{1}{8}(2 n+1)^{2}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1+2+3+\ldots+n\frac{1}{8}(2 n+1)^{2}$ It can be noted that $\mathrm{P}(n)$ is true for $n=1$ since $1\frac{1}{8}(2.1+1)^{2}=\frac{9}{8}$. Let P(k) be true for some positive integerk, i.e., $1+2+\ldots+k\frac{1}{8}(2 k+1)^{2}$$\ldots(1)$ We shall now prove that P(k+ 1) is true whenever P(k) is true. Conside...
Read More →If one of the two electrons of a
Question: If one of the two electrons of a H2molecule is removed, we get a hydrogen molecular ion. In the ground state of an, the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy. Solution: The system of two protons and one electron is represented in the given figure. Charge on proton $1, q_{1}=1.6 \times 10^{-19} \mathrm{C}$ Charge on proton 2,q2= 1.6 10...
Read More →If one of the two electrons of a
Question: If one of the two electrons of a $\mathrm{H}_{2}$ molecule is removed, we get a hydrogen molecular ion $\mathrm{H}_{2}^{+}$. In the ground state of an $\mathrm{H}_{2}^{+}$, the two protons are separated by roughly $1.5 \AA$, and the electron is roughly $1 \AA$ from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy. Solution: The system of two protons and one electron is represented in the given figure. Charge on proton $1, q_...
Read More →Ionic product of water at 310 K is 2.7 × 10–14.
Question: lonic product of water at $310 \mathrm{~K}$ is $2.7 \times 10^{-14}$. What is the $\mathrm{pH}$ of neutral water at this temperature? Solution: Ionic product, $K_{w}=\left[\mathrm{H}^{+}\right]\left[\mathrm{OH}^{-}\right]$ Let $\left[\mathrm{H}^{+}\right]=x$ Since $\left[\mathrm{H}^{+}\right]=\left[\mathrm{OH}^{-}\right], K_{\mathrm{w}}=x^{2} .$ $\Rightarrow K_{w}$ at $310 \mathrm{~K}$ is $2.7 \times 10^{-14}$ $\therefore 2.7 \times 10^{-14}=x^{2}$ $\Rightarrow x=1.64 \times 10^{-7}$ $...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N:
Question: Prove the following by using the principle of mathematical induction for allnN:$\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\ldots+\frac{1}{(2 n+1)(2 n+3)}=\frac{n}{3(2 n+3)}$ Solution: Let the given statement be $\mathrm{P}(n)$, i.e., $\mathrm{P}(n): \frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\ldots+\frac{1}{(2 n+1)(2 n+3)}=\frac{n}{3(2 n+3)}$ Forn= 1, we have $P(1): \frac{1}{3.5}=\frac{1}{3(2.1+3)}=\frac{1}{3.5}$, which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$,...
Read More →In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
Question: In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å: (a)Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton. (b)What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)? (c)What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å se...
Read More →In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
Question: In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å: (a)Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton. (b)What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)? (c)What are the answers to (a) and (b) above if the zero of potential energy is taken at 1.06 Å se...
Read More →The ionization constant of chloroacetic acid is 1.35 × 10–3.
Question: The ionization constant of chloroacetic acid is $1.35 \times 10^{-3}$. What will be the $\mathrm{pH}$ of $0.1 \mathrm{M}$ acid and its $0.1 \mathrm{M}$ sodium salt solution? Solution: It is given that $\mathrm{K}_{a}$ for $\mathrm{ClCH}_{2} \mathrm{COOH}$ is $1.35 \times 10^{-3}$. $\mathrm{ClCH}_{2} \mathrm{COONa}$ is the salt of a weak acid i.e., $\mathrm{ClCH}_{2} \mathrm{COOH}$ and a strong base i.e., $\mathrm{NaOH}$. $\mathrm{ClCH}_{2} \mathrm{COO}^{-}+\mathrm{H}_{2} \mathrm{O} \lo...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N:
Question: Prove the following by using the principle of mathematical induction for allnN:$\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{(3 n+1)}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): \frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{(3 n+1)}$ For $n=1$, we have $P(1)=\frac{1}{1.4}=\frac{1}{3.1+1}=\frac{1}{4}=\frac{1}{1.4}$, which is true. Let P(k) be true for some positive integerk, i.e., $\mathrm...
Read More →A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder.
Question: A long charged cylinder of linear charged density is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders? Solution: Charge density of the long charged cylinder of lengthLand radiusris. Another cylinder of same length surrounds the pervious cylinder. The radius of this cylinder isR. LetEbe the electric field produced in the space between the two cylinders. Electric flux through the Gaussian surface is given by Gausss the...
Read More →(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
Question: (a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by $\left(\overrightarrow{E_{2}}-\overrightarrow{E_{1}}\right) \cdot \hat{n}=\frac{\sigma}{\epsilon_{0}}$Where $\hat{n}$ is a unit vector normal to the surface at a point and $\sigma$ is the surface charge density at that point. (The direction of $\hat{n}$ is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is $\sigma \hat...
Read More →(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
Question: (a)Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by$\left(\overrightarrow{E_{2}}-\overrightarrow{E_{1}}\right) \cdot \hat{n}=\frac{\sigma}{\epsilon_{0}}$Where $\hat{n}$ is a unit vector normal to the surface at a point and $\sigma$ is the surface charge density at that point. (The direction of $\hat{n}$ is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is $\sigma \hat{n...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N:
Question: Prove the following by using the principle of mathematical induction for allnN:$1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n)=1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$ For $n=1$, we have $P(1)=1^{2}=1=\frac{1(2.1-1)(2.1+1)}{3}=\frac{1.1 .3}{3}=1$, which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $\mathrm{P}(k)=1^{2}+3^{2}+5^{2}+\ldots+(2 k-1)^{2}=\frac{...
Read More →Find the value of$\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)$ is equal to
Question: Find the value of $\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)$ is equal to (A) $\pi(\mathbf {B})-\frac{\pi}{3}$ (C) $\frac{\pi}{3}$ (D) $\frac{2 \pi}{3}$ Solution: Let $\tan ^{-1} \sqrt{3}=x$. Then, $\tan x=\sqrt{3}=\tan \frac{\pi}{3}$. We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$. $\therefore \tan ^{-1} \sqrt{3}=\frac{\pi}{3}$ Let $\sec ^{-1}(-2)=y$. Then, $\sec y=-2=-\sec \left(\frac{\pi}{3}\right)=\sec \left(\pi-\frac{\...
Read More →A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q.
Question: A spherical conducting shell of inner radiusr1 and outer radiusr2 has a chargeQ. (a)A chargeqis placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell? (b)Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain. Solution: (a)Charge placed at the centre of a shell is +q. Hence, a charge of magnitude qwill be induced to the inner surface of the shell. ...
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