Suppose that the particle in Exercise in 1.33 is an electron projected
Question: Suppose that the particle in Exercise in $1.33$ is an electron projected with velocity $v_{x}=2.0 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1} .$ If $E$ between the plates separated by $0.5$ $\mathrm{cm}$ is $9.1 \times 10^{2} \mathrm{~N} / \mathrm{C}$, where will the electron strike the upper plate? $\left(|e|=1.6 \times 10^{-19} \mathrm{C}, m_{e}=9.1 \times 10^{-31} \mathrm{~kg} .\right)$ Solution: Velocity of the particle, $v_{x}=2.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$ Separation ...
Read More →Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is
Question: Let $A=\{1,2,3\}$. Then number of equivalence relations containing $(1,2)$ is (A) 1 (B) 2 (C) 3 (D) 4 Solution: It is given thatA= {1, 2, 3}. The smallest equivalence relation containing (1, 2) is given by, $R_{1}=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$ Now, we are left with only four pairs i.e., (2, 3), (3, 2), (1, 3), and (3, 1). If we odd any one pair [say $(2,3)]$ to $R_{1}$, then for symmetry we must add $(3,2)$. Also, for transitivity we are required to add $(1,3)$ and $(3,1)$. Hence,...
Read More →A particle of mass m and charge (−q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in Fig. 1.33).
Question: A particle of mass $m$ and charge $(-q)$ enters the region between the two charged plates initially moving along $x$-axis with speed $v x$ (like particle 1 in Fig. 1.33). The length of plate is $L$ and an uniform electric field $E$ is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is $q E L^{2} /\left(2 m v_{x}^{2}\right)$. Compare this motion with motion of a projectile in gravitational field discussed in Section $4.10$ of...
Read More →Prove the following by using the principle of mathematical induction for all $n in N: 1.2+2.3+3.4+ldots+n .(n+1)=$
Question: Prove the following by using the principle of mathematical induction for all $n \in N: 1.2+2.3+3.4+\ldots+n .(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right]$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1.2+2.3+3.4+\ldots+n \cdot(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right]$ Forn= 1, we have $P(1): 1.2=2=\frac{1(1+1)(1+2)}{3}=\frac{1.2 .3}{3}=2$, which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $1.2+2.3+3.4+\ldots . .+k \cdot(k+1)=\left[\frac{k(k+...
Read More →The ionization constant of dimethylamine is 5.4 × 10–4. Calculate its degree of ionization in its 0.02 M solution
Question: The ionization constant of dimethylamine is $5.4 \times 10^{-4}$. Calculate its degree of ionization in its $0.02 \mathrm{M}$ solution. What percentage of dimethylamine is ionized if the solution is also $0.1 \mathrm{M}$ in $\mathrm{NaOH}$ ? Solution: $K_{b}=5.4 \times 10^{-4}$ $c=0.02 \mathrm{M}$ Then, $\alpha=\sqrt{\frac{K_{b}}{c}}$ $=\sqrt{\frac{5.4 \times 10^{-4}}{0.02}}$ $=0.1643$ Now, if 0.1 M of NaOH is added to the solution, then NaOH (being a strong base) undergoes complete io...
Read More →Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3)
Question: Let $A=\{1,2,3\}$. Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is (A) 1 (B) 2 (C) 3 (D) 4 Solution: The given set is A = {1, 2, 3}. The smallest relation containing (1, 2) and (1, 3) which is reflexive and symmetric, but not transitive is given by: R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)} This is because relation $R$ is reflexive as $(1,1),(2,2),(3,3) \in R$. Relation $R$ is symmetric since $(1,2),(2,1...
Read More →(a) Consider an arbitrary electrostatic field configuration
Question: (a)Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., whereE= 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable. (b)Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart. Solution: (a)Let the equilibrium of the test charge be stable. If a test charge is in equilibrium and displaced from its position in any directi...
Read More →It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks.
Question: It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so called up quark (denoted by u) of charge (+2/3)e, and the down quark (denoted by d) of charge (1/3)e, together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.)...
Read More →Calculate the degree of ionization of 0.05M acetic acid if its pKa value is 4.74.
Question: Calculate the degree of ionization of $0.05 \mathrm{M}$ acetic acid if its $\mathrm{pK}_{\mathrm{a}}$ value is $4.74$. How is the degree of dissociation affected when its solution also contains (a) $0.01 \mathrm{M}$ (b) $0.1 \mathrm{M}$ in $\mathrm{HCl}$ ? Solution: $c=0.05 \mathrm{M}$ $p K_{a}=4.74$ $p K_{a}=-\log \left(K_{a}\right)$ $K_{a}=1.82 \times 10^{-5}$ $K_{a}=c \alpha^{2} \quad \alpha=\sqrt{\frac{K_{a}}{c}}$ $\alpha=\sqrt{\frac{1.82 \times 10^{-5}}{5 \times 10^{-2}}}=1.908 \t...
Read More →Prove the following by using the principle of mathematical induction for all $n in N: 1.3+2.3^{2}+3.3^{3}+ldots+n .3^{n}$
Question: Prove the following by using the principle of mathematical induction for allnN:$1.3+2.3^{2}+3.3^{3}+\ldots+n .3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1.3+2.3^{2}+3.3^{3}+\ldots+n 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4}$ For $n=1$, we have $P(1): 1.3=3=\frac{(2.1-1) 3^{1+1}+3}{4}=\frac{3^{2}+3}{4}=\frac{12}{4}=3$, which is true. Let $P(k)$ be true for some positive integer $k$, i.e., $1.3+2.3^{2}+3.3^{3}+\ldots+k 3^{k}=\frac{(2 k-1) ...
Read More →Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f,
Question: Let $A=\{-1,0,1,2\}, B=\{-4,-2,0,2\}$ and $t, g: A \rightarrow B$ be functions defined by $f(x)=x^{2}-x, x \in \mathrm{A}$ and $g(x)=2\left|x-\frac{1}{2}\right|-1, x \in A$. Are $f$ and $g$ equal? Justify your answer. (Hint: One may note that two function $f: A \rightarrow B$ and $g: A \rightarrow B$ such that $f(a)=g(a) \ m n F o r E ; a \in A$, are called equal functions). Solution: It is given thatA= {1, 0, 1, 2},B= {4, 2, 0, 2}. Also, it is given that $f, g: A \rightarrow B$ are de...
Read More →Obtain the formula for the electric field due to a long thin wire of uniform linear charge density λ without using Gauss’s law.
Question: Obtain the formula for the electric field due to a long thin wire of uniform linear charge densitywithout using Gausss law. [Hint:Use Coulombs law directly and evaluate the necessary integral.] Solution: Take a long thin wire $X Y$ (as shown in the figure) of uniform linear charge density $\lambda$. Consider a point A at a perpendicular distancelfrom the mid-point O of the wire, as shown in the following figure. LetEbe the electric field at point A due to the wire, XY. Consider a small...
Read More →Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as
Question: Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as Show that zero is the identity for this operation and each elementa 0 of the set is invertible with 6 abeing the inverse ofa. Solution: LetX= {0, 1, 2, 3, 4, 5}. The operation * on X is defined as: $a * b= \begin{cases}a+b \text { if } a+b6 \\ a+b-6 \text { if } a+b \geq 6\end{cases}$ An element $e \in X$ is the identity element for the operation *, if $a * e=a=e * a \forall a \in X$. For $a \in X$, we observed that: $a * 0=a+...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2)
Question: Prove the following by using the principle of mathematical induction for allnN:$1.2 .3+2.3 .4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1.2 .3+2.3 .4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4}$ Forn= 1, we have $P(1): 1.2 .3=6=\frac{1(1+1)(1+2)(1+3)}{4}=\frac{1.2 .3 .4}{4}=6$, which is true. Let P(k) be true for some positive integerk, i.e., $1.2 .3+2.3 .4+\ldots+k(k+1)(k+2)=\frac{k(k+1)(k+2)(k+3)}{4}$ $\ldots$ (i...
Read More →A hollow charged conductor has a tiny hole cut into its surface.
Question: A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is $\left(\frac{\sigma}{2 \in_{0}}\right) \hat{n}$, where $\hat{n}$ is the unit vector in the outward normal direction, and $\sigma$ is the surface charge density near the hole. Solution: Let us consider a conductor with a cavity or a hole. Electric field inside the cavity is zero. Let $E$ is the electric field just outside the conductor, $q$ is the electric charge, $\sigma$ is the...
Read More →What is the pH of 0.001 M aniline solution? The ionization constant of aniline can be taken from Table 7.7.
Question: What is the pH of 0.001 M aniline solution? The ionization constant of aniline can be taken from Table 7.7. Calculate the degree of ionization of aniline in the solution. Also calculate the ionization constant of the conjugate acid of aniline. Solution: $K_{b}=4.27 \times 10^{-10}$ $c=0.001 \mathrm{M}$ $\mathrm{pH}=?$ $\mathrm{a}=?$ $k_{b}=c \alpha^{2}$ $4.27 \times 10^{-10}=0.001 \times \alpha^{2}$ $4270 \times 10^{-10}=\alpha^{2}$ $65.34 \times 10^{-5}=\alpha=6.53 \times 10^{-4}$ The...
Read More →(a) A conductor A with a cavity as shown in Fig. 1.36(a) is given a charge Q.
Question: (a) A conductor A with a cavity as shown in Fig. 1.36(a) is given a chargeQ. Show that the entire charge must appear on the outer surface of the conductor. (b) Another conductor B with chargeqis inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A isQ+q[Fig. 1.36(b)]. (c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way Solution: (a)Let us consider a Gaussian s...
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 all $n \in N: 1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+\ldots+\frac{1}{(1+2+3+\ldots n)}=\frac{2 n}{(n+1)}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots n}=\frac{2 n}{n+1}$ For $n=1$, we have $P(1): 1=\frac{2.1}{1+1}=\frac{2}{2}=1$ which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $1+\frac{1}{1+2}+\ldots+\frac...
Read More →Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B − A), &mnForE;
Question: Given a non-empty set $X$, let $^{*}: P(X) \times P(X) \rightarrow P(X)$ be defined as $A$ * $B=(A-B) \cup(B-A), \ m n F o r E ; A, B \in P(X)$. Show that the empty set $\Phi$ is the identity for the operation * and all the elements $A$ of $P(X)$ are invertible with $A^{-1}=A$. $($ Hint: $(A-\Phi) \cup(\Phi-A)=A$ and $(A-A) \cup$ $\left.(A-A)=A^{*} A=\Phi\right)$. Solution: It is given that *: P(X) P(X) P(X) is defined as A*B= (AB) (BA) mnForE;A,B P(X). LetA P(X). Then, we have: $A^{*}...
Read More →The pH of 0.005M codeine (C18H21NO3) solution is 9.95.
Question: The $\mathrm{pH}$ of $0.005 \mathrm{M}$ codeine $\left(\mathrm{C}_{18} \mathrm{H}_{21} \mathrm{NO}_{3}\right)$ solution is $9.95$. Calculate its ionization constant and $\mathrm{pK}_{\mathrm{b}}$. Solution: c= 0.005 pH = 9.95 pOH = 4.05 pH = log (4.105) $4.05=-\log \left[\mathrm{OH}^{-}\right]$ $\left[\mathrm{OH}^{-}\right]=8.91 \times 10^{-5}$ $c \alpha=8.91 \times 10^{-5}$ $\alpha=\frac{8.91 \times 10^{-5}}{5 \times 10^{-3}}=1.782 \times 10^{-2}$ Thus, $K_{b}=c \alpha^{2}$ $=0.005 \t...
Read More →In a certain region of space, electric field is along the z-direction throughout.
Question: In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is, however, not constant but increases uniformly along the positivez-direction, at the rate of 105NC1per metre. What are the force and torque experienced by a system having a total dipole moment equal to 107Cm in the negativez-direction? Solution: Dipole moment of the system,p=q dl= 107C m Rate of increase of electric field per unit length, $\frac{d E}{d l}=10^{+5} \mathrm...
Read More →2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it?
Question: 2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it? Solution: A man can alone finish the work in $x$days and one boy alone can finish it in $y$days then One mans one days work =\frac{1}{x} One boys one days work= \frac{1}{y} 2men one day work=\frac{2}{x} 7boys one day work=$\frac{7}{y}$ Since 2 men and 7 boys can finish the work in 4 days $4\left(\frac{2}{x}+\frac{7}{y}\right)=1$ $\...
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 all $n \in N: 1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2}$ Solution: Let the given statement be $\mathrm{P}(n)$, i.e., $\mathrm{P}(n): 1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2}$ For $n=1$, we have $P(1): 1^{3}=1=\left(\frac{1(1+1)}{2}\right)^{2}=\left(\frac{1.2}{2}\right)^{2}=1^{2}=1$, which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $1^{3}...
Read More →Consider the binary operations*: R ×R → and o: R × R → R defined as and a o b = a,
Question: Consider the binary operations*: $\mathbf{R} \times \mathbf{R} \rightarrow$ and $0: \mathbf{R} \times \mathbf{R} \rightarrow \mathbf{R}$ defined as $a * b=|a-b|$ and $a \circ b=a, \ m n$ ForE; $a, b \in \mathbf{R}$. Show that * is commutative but not associative, o is associative but not commutative. Further, show that \mnForE; $a, b, c \in \mathbf{R}, a^{\star}(b$ o $c)=\left(a^{*} b\right) \circ\left(a^{*} c\right) .$ [If it is so, we say that the operation * distributes over the ope...
Read More →Which among the curves shown in Fig. 1.35 cannot possibly represent electrostatic field lines?
Question: Which among the curves shown in Fig. 1.35 cannot possibly represent electrostatic field lines? (a) (b) (c) (d) (e) Solution: (a)The field lines showed in (a) do not represent electrostatic field lines because field lines must be normal to the surface of the conductor. (b)The field lines showed in (b) do not represent electrostatic field lines because the field lines cannot emerge from a negative charge and cannot terminate at a positive charge. (c)The field lines showed in (c) represen...
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