Figure shows a circular wire-loop of radius a,
Question: Figure shows a circular wire-loop of radius a, carrying a current i, placed in a perpendicular magnetic field B. (a) Consider a small part dl of the wire. Find the force on this part of the wire exerted by the magnetic field. (b) Find the force of compression in the wire. Solution:...
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Question: Find ' $c$ ' of Lagrange's mean-value theorem for $f(x)=\sqrt{x+2}$ on $[4,6]$ Solution:...
Read More →A straight wire of length I can slide on two parallel plastic rails kept in
Question: A straight wire of length I can slide on two parallel plastic rails kept in a horizontal plane with a separation d. The coefficient of friction between the wire and the rails is $\mu$. If the wire carries a current $i$, what minimum magnetic field should exist in the space in order to slide the wire on the rails. Solution:...
Read More →A metal wire PQ of mass 10g lies at rest on two horizontal metal rails separated by 4.90cm.
Question: A metal wire $\mathrm{PQ}$ of mass $10 \mathrm{~g}$ lies at rest on two horizontal metal rails separated by $4.90 \mathrm{~cm}$. A vertically downward magnetic field of magnitude $0.800 \mathrm{~T}$ exists in the space. The resistance of the circuit is slowly decreased and it is found that when the resistance goes below $20.0 \Omega$, the wire $\mathrm{PQ}$ starts sliding on the rails. Find the coefficient of friction. Solution:...
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Question: Find ' $c$ ' of Lagrange's mean-value theorem for $f(x)=\sqrt{25-x^{2}}$ on $[1,5]$ Solution:...
Read More →Two metal strips, each of length l,
Question: Two metal strips, each of length l, are clamped parallel to each other on a horizontal floor with a separation b between them. A wire of mass $m$ lies on them perpendicularly as shown in figure. A vertically upward magnetic field of strength B exists in the space. The metal strips are smooth but the coefficient of friction between the wire and the floor is $\mu$. A current i is established when the switch $S$ is closed at the instant $t=0$. Discuss the motion of the wire after the swit...
Read More →Figure shows a rod PQ of length 20.0cm and mass 200g
Question: Figure shows a rod PQ of length $20.0 \mathrm{~cm}$ and mass $200 \mathrm{~g}$ suspended through a fixed point 0 by two threads of lengths $20.0 \mathrm{~cm}$ each. A magnetic field of strength $0.500 \mathrm{~T}$ exists in the vicinity of the wire PQ as shown in figure. The wires connecting $\mathrm{PQ}$ with the battery are loose and exert no force on $\mathrm{PQ}$. (a) Find the tension in the threads when the switch $\mathrm{S}$ is open. (b) A current of $2.0 \mathrm{~A}$ is establi...
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Question: Find ' $c$ ' of Lagrange's mean-value theorem for $f(x)=\left(x^{3}-3 x^{2}+2 x\right)$ on $\left[0, \frac{1}{2}\right]$ Solution:...
Read More →A straight, horizontal wire of mass 10mg and length 1.0m
Question: A straight, horizontal wire of mass $10 \mathrm{mg}$ and length $1.0 \mathrm{~m}$ carries a current of $2.0 \mathrm{~A}$. What minimum magnetic field B should be applied in the region so that the magnetic force on the wire may balance its weight? Solution:...
Read More →A rigid wire consists of a semicircular portion of radius
Question: A rigid wire consists of a semicircular portion of radius $\mathrm{R}$ and two straight sections. The wire is partially immersed in a perpendicular magnetic field $B$ as shown in figure. Find the magnetic force on the wire if it carries a current i. Solution:...
Read More →A wire, carrying a current i,
Question: A wire, carrying a current $i$, is kept in the X-Y plane along the curve $y=A \sin \left(\frac{2 \pi}{\lambda} x\right)$ A magnetic field B exists in the $z-$ direction. Find the magnitude of the magnetic force on the portion of the wire between $x=0$ and $x=\lambda$. Solution:...
Read More →A semi-circular wire of radius 5.0cm and
Question: A semi-circular wire of radius $5.0 \mathrm{~cm}$ and carries a current of $5.0 \mathrm{~A}$. A magnetic field B of magnitude $0.50 \mathrm{~T}$ exists along the perpendicular to the plane of the wire. Find the magnitude of the magnetic force acting on the wire. Solution:...
Read More →Show that Lagrange's mean-value theorem is not applicable
Question: Show that Lagrange's mean-value theorem is not applicable to $f(\mathrm{x})=\frac{1}{\mathrm{x}}$ on $[-1,1]$ Solution: Given: Since the graph is discontinuous at $x=0$ as shown in the graph....
Read More →Prove that the force acting on a current-carrying wire,
Question: Prove that the force acting on a current-carrying wire, joining two fixed points a and $b$ in a uniform magnetic field, is independent of the shape of the wire. Solution: Let us consider a semi-circular wire of radius $\mathrm{R}$ is placed in uniform magnetic field directed perpendicular of wire....
Read More →A current loop of arbitrary shape lies in a uniform magnetic field B.
Question: A current loop of arbitrary shape lies in a uniform magnetic field B. Show that the net magnetic force acting on the loop is zero. Solution:...
Read More →Show that Lagrange's mean-value theorem is not applicable to
Question: Show that Lagrange's mean-value theorem is not applicable to $f(\mathrm{x})=|\mathrm{x}|$ on $[-1,1]$. Solution:...
Read More →A rectangular wire-loop of width a is suspended from the insulted pan of
Question: A rectangular wire-loop of width a is suspended from the insulted pan of a spring balance as shown in figure. A current i exists in the anticlockwise direction in the loop. A magnetic field B exists in the lower region. Find the change in the tension of the spring if the current in the loop is reversed. Solution:...
Read More →Verify Lagrange's mean-value theorem for the following function:
Question: Verify Lagrange's mean-value theorem for the following function: $f(x)=(\sin x+\cos x)$ on $\left[0, \frac{\pi}{2}\right]$ Solution:...
Read More →A hypothetical magnetic field existing in a region is given by
Question: A hypothetical magnetic field existing in a region is given by $\vec{B}=B_{0} \vec{e}_{r}$, where $\vec{e}_{r}$ denotes the unit vector along the radial direction. A circular loop of radius a, carrying a current $i$, is placed with its plane parallel to the X-Y plane and the center at ( 0 , $0, d)$. Find the magnitude of the magnetic force acting on the loop. Solution:...
Read More →A circular loop of radius a, carrying a current I,
Question: A circular loop of radius a, carrying a current $\mathrm{l}$, is placed in a two-dimensional magnetic field. The center of the loop coincides with the center of the field. The strength of the magnetic field at the periphery of the loop is B. Find the magnetic force on the wire. Solution:...
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Question: Verify Lagrange's mean-value theorem for the following function: $f(x)=\sin x$ on $\left[\frac{\pi}{2}, \frac{5 \pi}{2}\right]$ Solution: $\mathrm{x}=\frac{\mathrm{n} \pi}{2}, \mathrm{n} \in\{1,2,3,4,5\}$...
Read More →A current of 5.0A exists in the circuit shown in figure.
Question: A current of $5.0 \mathrm{~A}$ exists in the circuit shown in figure. The wire $P Q$ has a length of $50 \mathrm{~cm}$ and the magnetic field in which it is immersed has a magnitude of $0.20 \mathrm{~T}$. Find the magnetic force acting on the wire $\mathrm{PQ}$. Solution:...
Read More →Verify Lagrange's mean-value theorem for the following function:
Question: Verify Lagrange's mean-value theorem for the following function: $f(\mathrm{x})=\tan ^{-1} \mathrm{x}$ on $[0,1]$ Solution: $\Rightarrow c=\sqrt{\frac{4}{\pi}-1}$...
Read More →A wire of length I carries a current i along the X-axis.
Question: A wire of length I carries a current i along the X-axis. A magnetic field exists which is given as $\vec{B}=B_{0}(\hat{\imath}+\hat{\jmath}+\hat{k})_{\mathrm{T}}$. Find the magnitude of the magnetic force acting on the wire. Solution:...
Read More →A magnetic field of strength 1.0T is produced by a strong electromagnet
Question: A magnetic field of strength $1.0 \mathrm{~T}$ is produced by a strong electromagnet in a cylindrical region of radius $4.0 \mathrm{~cm}$ as shown in figure. A wire, carrying a current of $2.0 \mathrm{~A}$, is placed perpendicular to and intersecting the axis of the cylindrical region. Find the magnitude of the force acting on the wire. Solution:...
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