The primary origin(s) of magnetism lies in
Question: The primary origin(s) of magnetism lies in (a) atomic currents (b) Pauli exclusion principle (c) polar nature of molecules (d) intrinsic spin of electron Solution: (a) atomic currents (d) intrinsic spin of electron...
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : 34, 23, 46, 37, 40, 28, 32, 50, 35, 44 Solution: Here the number of observations is 10 which is odd.Arranging the data into ascending order, we have 23, 28, 32, 34, 35, 37, 40, 44, 46, 50 Now, $\quad$ Median $(M)=\left(\frac{5^{\text {th }} \text { observation }+6^{\text {th }} \text { observation }}{2}\right)=\frac{35+37}{2}=36$ The respective absolute values of the deviations from median, i.e., $\left|\mathrm{x}_{\math...
Read More →S is the surface of a lump of magnetic material
Question: S is the surface of a lump of magnetic material (a) lines of B are necessarily continuous across S (b) some lines of B must be discontinuous across S (c) lines of H are necessarily continuous across S (d) lines of H cannot all be continuous across S Solution: (a) lines of B are necessarily continuous across S (d) lines of H cannot all be continuous across S...
Read More →A paramagnetic sample shows a net magnetisation
Question: A paramagnetic sample shows a net magnetisation of 8 Am-1when placed in an external magnetic field of 0.6T at a temperature of 4K. When the same sample is placed in an external magnetic field of 0.2T at a temperature of 16K, the magnetisation will be (a) 32/3 Am-1 (b) 2/3 Am-1 (c) 6 Am-1 (d) 2.4 Am-1 Solution: (b) 2/3 Am-1...
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : 4, 15, 9, 7, 19, 13, 6, 21, 8, 25, 11 Solution: Here the number of observations is 11 which is odd. Arranging the data into ascending order, we have 4, 6, 7, 8, 9, 11, 13, 15, 19, 21, 25 Now, Median $(M)=\left(\frac{11+1}{2}\right)^{\text {th }}$ or $6^{\text {th }}$ observation $=11$ The respective absolute values of the deviations from median , i.e. $\left|\mathrm{x}_{\mathrm{i}}-\mathrm{M}\right|$ are 7, 5, 4, 3, 2, 0...
Read More →Consider the two idealized systems: (i) a parallel plate
Question: Consider the two idealized systems: (i) a parallel plate capacitor with large plates and small separation and (ii) a long solenoid of length L R, the radius of the cross-section. In (i) E is ideally treated as a constant between plates and zero outside. In (ii) magnetic field is constant inside the solenoid and zero outside. These idealised assumptions, however, contradict fundamental laws as below: (a) case (i) contradicts Gausss law for electrostatic fields (b) case (ii) contradicts ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: Solution: Let $t=\sin ^{2} x$ $d t=2 \sin x \cos x d x$ we know $\sin 2 x=2 \sin 2 x \cos 2 x$ therefore, $d t=\sin 2 x d x$ $\int \frac{\sin 2 x}{\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}} d x=\int \frac{d t}{\sqrt{t^{2}+4 t-2}}$ Add and subtract $2^{2}$ in denominator $=\int \frac{d t}{\sqrt{t^{2}+4 t-2}}=\int \frac{d t}{\sqrt{t^{2}+2 \times 2 t+2^{2}-2^{2}-2}}$ Let $\mathrm{t}+2=\mathrm{u}$ $\mathrm{dt}=\mathrm{du}$ $\left.\left.=\int \mathrm{dt} / \sqrt{(...
Read More →Find the mean deviation about the median for the following data :
Question: Find the mean deviation about the median for the following data : 12, 5, 14, 6, 11, 13, 17, 8, 10 Solution: Here the number of observations is 9 which is odd. Arranging the data into ascending order, we have 5, 6, 8, 10, 11, 12, 13, 14, 17 Now, Median $(M)=\left(\frac{9+1}{2}\right)^{\text {th }}$ or $5^{\text {th }}$ observation $=11$ The respective absolute values of the deviations from median, i.e.' $\left|\mathrm{x}_{\mathrm{i}}-\mathrm{M}\right|$ are 6, 5, 3, 1, 0, 1, 2, 3, 6 Thus...
Read More →In a permanent magnet at room temperature
Question: In a permanent magnet at room temperature (a) magnetic moment of each molecule is zero (b) the individual molecules have a non-zero magnetic moment which is all perfectly aligned (c) domains are partially aligned (d) domains are all perfectly aligned Solution: (d) domains are all perfectly aligned...
Read More →The magnetic field of the earth can be modelled
Question: The magnetic field of the earth can be modelled by that of a point dipole placed at the centre of the earth. The dipole axis makes an angle of 11.3owith the axis of the earth. At Mumbai, declination is nearly zero. Then, (a) the declination varies between 11.3oW to 11.3oE (b) the least declination is 0o (c) the plane defined by dipole axis and the earth axis passes through Greenwich (d) declination average over the earth must be always negative Solution: (a) the declination varies betw...
Read More →A toroid of n turns, mean radius R
Question: A toroid of n turns, mean radius R and cross-sectional radius a carries current I. It is placed on a horizontal table taken as an x-y plane. Its magnetic moment m (a) is non-zero and points in the z-direction by symmetry (b) points along the axis of the toroid $m=m \hat{\phi}$ (c) is zero, otherwise, there would be a field falling as 1/r3at large distances outside the toroid (d) is pointing radially outwards Solution: (c) is zero, otherwise, there would be a field falling as 1/r3at lar...
Read More →Find the mean deviation about the mean for the following data :
Question: Find the mean deviation about the mean for the following data : $17,20,12,13,15,16,12,18,15,19,12,11$ Solution: We have, 17, 20, 12, 13, 15, 16, 12, 18, 15, 19, 12, 11 Mean of the given data is $\overline{\mathrm{x}}=\frac{17+20+12+13+15+16+12+18+15+19+12+11}{12}$ $\overline{\mathrm{x}}=\frac{180}{12}=15$ The respective absolute values of the deviations from the mean, i.e.' $\left|\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right|$ are 2, 5, 3, 2, 0, 1, 3, 3, 0, 4, 3, 4 Thus, the req...
Read More →Find the mean deviation about the mean for the following data :
Question: Find the mean deviation about the mean for the following data : 39, 72, 48, 41, 43, 55, 60, 45, 54, 43 Solution: We have, 39, 72, 48, 41, 43, 55, 60, 45, 54, 43 Mean of the given data is $\overline{\mathrm{x}}=\frac{39+72+48+41+43+55+60+45+54+43}{10}=\frac{500}{10}=50$ The respective absolute values of the deviations from mean, i.e' $\left|\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right|$ are $11,22,2,9,7,5,10,5,4,7$ Thus, the required mean deviation about the mean is M. D. $(\over...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos 2 x}{\sqrt{\sin ^{2} 2 x+8}} d x$ Solution: Let $=\sin 2 x$ $d t=2 \cos 2 x d x$ $\cos 2 x d x=d t / 2$ $\int \frac{\cos 2 \mathrm{x}}{\sqrt{\sin ^{2} 2 \mathrm{x}+8}} \mathrm{dx}=\frac{1}{2} \int \mathrm{dt} / \sqrt{\left(\mathrm{t}^{2}+(2 \sqrt{2})^{2}\right.}$ Since we have, $\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left.\left(x^{2}+a^{2}\right)\right]+c}\right.$ $=\frac{1}{2} \int \mathrm{dt} / \sqrt{\...
Read More →Five long wires A, B, C, D, and E each carrying I are arranged
Question: Five long wires A, B, C, D, and E each carrying I are arranged to form edges of a pentagonal prism as shown in the figure. Each carries current out of the plane of paper. (a) what will be magnetic induction at a point on the axis O Axis is at a distance R from each wire (b) what will be the field if current in one of the wires is switched off (c) what if current in one of the wire A is reversed Solution: (a) The magnetic induction at a point on the axis will be 0 which is represented b...
Read More →Find the mean deviation about the mean for the following data
Question: Find the mean deviation about the mean for the following data 7, 8, 4, 13, 9, 5, 16, 18 Solution: We have, 7, 8, 4, 13, 9, 5, 16, 18 Mean of the given data is $\overline{\mathrm{x}}=\frac{7+8+4+13+9+5+16+18}{8}=\frac{80}{8}=10$ The respective absolute values of the deviations from the mean, i.e.' $\left|\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{X}}\right|$ are 3, 2, 6, 3, 1, 5, 6, 8 Thus, the required mean deviation about the mean is M. D. $(\overline{\mathrm{x}})=\frac{\sum_{\mathrm{i...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin 8 x}{\sqrt{9+\sin ^{4} 4 x}} d x$ Solution: Let $t=\sin ^{2} 4 x$ $d t=2 \sin 4 x \cos 4 x \times 4 d x$ we know $\sin 2 x=2 \sin 2 x \cos 2 x$ therefore, $d t=4 \sin 8 x d x$ or, $\sin 8 x d x=d t / 4$ $\int \frac{\sin 8 x}{\sqrt{9+\sin ^{4} x}} d x=\frac{1}{4} \int \frac{d t}{\sqrt{3^{2}+t^{2}}}$ Since we have, $\left.\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}\right.}+a^{2}\right)\right]+c$ $=\f...
Read More →A multirange current meter can be constructed
Question: A multirange current meter can be constructed by using a galvanometer circuit shown in the figure. We want a current meter that can measure 10 mA, 100 mA, and 1 A using a galvanometer of resistance 10Ω and that produces maximum deflection for a current of 1 mA. Find S1, S2, and S3 that have to be used. Solution: I1is measured as = 10 mA = IGG = (I1 IG)(S1+ S2+ S3) I2is measured as = 100 mA = IG(G+S1)=(I2-IG)(S2-S3) I3is measured as = 1 A = IG(G+S1+S2)=(I3-IG)(S3) S1= 1 Ω S2= 0.1 Ω S3= ...
Read More →Consider a circular current-carrying
Question: Consider a circular current-carrying loop of radius R in the x-y plane with centre at the origin. Consider the line integral $\Im(L)=\left|\int_{-L}^{L} B \cdot d l\right|$ taken along z-axis. (a) show that $\Im(L)$ monotonically increases with $\mathrm{L}$ (b) use an appropriate Amperian loop to that (c) verify directly the above result (d) suppose we replace the circular coil by a square coil of sides R carrying the same current I. What can you say about Solution: (a) Magnetic field ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x \sqrt{4-9(\log x)^{2}}} d x$ Solution: Put $3 \log x=t$ We have $\mathrm{d}(\log \mathrm{x})=1 / \mathrm{x}$ Hence, $\mathrm{d}(3 \log \mathrm{x})=\mathrm{dt}=3 / \mathrm{x} \mathrm{dx}$ Or $1 / x d x=d t / 3$ Hence, $\int \frac{1}{x \sqrt{4-9(\log x)^{2}}} d x=\int \frac{1}{3} \frac{d t}{\sqrt{2^{2}-t^{2}}}$ Since we have, $\int \frac{1}{\sqrt{a^{2}-x^{2}}} d x=\sin ^{-1}\left(\frac{x}{a}\right)+c$ Hence, $\int \frac{1}{3} \frac{\math...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x}{\sqrt{4-x^{4}}} d x$ Solution: Let $x^{2}=t$ $2 x d x=d t$ or $x d x=d t / 2$ Hence, $\int \frac{x}{\sqrt{4-x^{4}}}=\int \frac{d t}{2\left(\sqrt{2^{2}-t^{2}}\right)}$ Since we have, $\int \frac{1}{\sqrt{a^{2}-x^{2}}} d x=\sin ^{-1}\left(\frac{x}{a}\right)+c$ So, $\int \frac{\mathrm{dt}}{2\left(\sqrt{2^{2}-\mathrm{t}^{2}}\right)}=\frac{1}{2} \sin ^{-1}\left(\frac{\mathrm{t}}{2}\right)+\mathrm{c}$ Put $t=x^{2}$ $=\frac{1}{2} \sin ^{-1}\lef...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: Let $2 \cos x=t$ Solution: Then $d t=-2 \sin x d x$ Or, $\sin \mathrm{x} \mathrm{dx}=-\frac{\mathrm{dt}}{2}$ Therefore, $\int \frac{\sin x}{\sqrt{4 \cos ^{2} x-1}} d x=\int-\frac{d t}{2 \sqrt{\left(t^{2}-1^{2}\right)}}$ Since, $\int \frac{1}{\sqrt{\left(x^{2}-a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}-a^{2}\right)}\right]+c$ Therefore, $\int-\frac{d t}{2 \sqrt{\left(t^{2}-1^{2}\right)}}=-\frac{1}{2} \operatorname{lod}\left[t+\sqrt{t^{2}-1}\right...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos x}{\sqrt{4+\sin ^{2} x}} d x$ Solution: Let $\sin x=t$ Then $\mathrm{dt}=\cos \mathrm{x} \mathrm{dx}$ Hence, $\int \frac{\cos x}{\sqrt{4+\sin ^{2} x}} d x=\int \frac{d t}{\sqrt{2}^{2}+t^{2}}$ Since we have, $\left.\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}\right.}+a^{2}\right)\right]+c$ Therefore, $\int \frac{\mathrm{dt}}{\sqrt{2^{2}+\mathrm{t}^{2}}}=\log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}+2^{2...
Read More →A uniform conducting wire of length 12a and resistance
Question: A uniform conducting wire of length 12a and resistance R is wound up as a current-carrying coil in the shape of (i) an equilateral triangle of side a; (ii) a square if sides a and (iii) a regular hexagon of sides a. The coil is connected to a voltage source V0. Find the magnetic moment of the coils in each case. Solution: (a) An equilateral triangle with side a No.of loops = 4 Area of the triangle A = 3/4 a2 Magnetic moment, m = Ia23 (b) For a square with sides a Area, A = a2 No.of loo...
Read More →By giving a counter-example, show that the following statement is false :
Question: By giving a counter-example, show that the following statement is false : p : If all the sides of a triangle are equal, then the triangle is obtuse angled. Solution: By the properties of triangles, if all the sides of a triangle are equal, then the each of the angle of the triangle will also be equal. By the question, All sides of the triangle are equal. $\therefore$ All angles of the triangle are also equal. Let each angle of the equilateral triangle be $x^{\circ} .$ We know that the ...
Read More →