A bicycle is resting on its stand in the east-west direction and
Question: A bicycle is resting on its stand in the east-west direction and the rear wheel is rotated at an angular speed of 100 revolutions per minute. It the length of each spoke is $30.0 \mathrm{~cm}$ and the horizontal component of earth's magnetic field is $2.0 \times 10^{-5}$ T, find the emf induced between the axis and the outer end of the spoke. Neglect centripetal force acting on the free electrons of the spoke. Solution:...
Read More →A conducting wire ab of length l,
Question: A conducting wire ab of length $\mathrm{l}$, resistance $\mathrm{r}$ and mass $\mathrm{m}$ starts sliding at $\mathrm{t}=0$ down a smooth, vertical, thick pair of connected rails as shown in figure. A uniform magnetic field $B$ exists in the space in a direction perpendicular to the plane of the rails. (a) Write the the induced emf in the loop at an instant $t$ when the speed of the wire is v. (b) What would be the magnitude and direction of the induced current in the wire? (c) Find th...
Read More →Figure shows a smooth pair of thick metallic rails connected
Question: Figure shows a smooth pair of thick metallic rails connected across a battery of emf $\varepsilon$ having a negligible internal resistance. A wire ab of length I and resistance $\mathrm{r}$ can slide smoothly on the rails. The entire system lies in a horizontal plane and is immersed in a uniform vertical magnetic field $B$. At and instant $t$, the wire is given a small velocity $v$ towards right. (a) Find the current in it at this instant. What is the direction of the current? (b) What...
Read More →The rectangular wire-frame,
Question: The rectangular wire-frame, shown in figure has a width $\mathrm{d}$, mass $\mathrm{m}$, resistance $\mathrm{R}$ and a large length. A uniform magnetic field B exists to the left of the frame. A constant force $F$ starts pushing the frame into the magnetic field at $t=0$. (a) Find the acceleration of the frame when its speed has increased to $v$. (b) Show that after some time the frame will move with a constant velocity till the whole frame enters into the magnetic field. Find this vel...
Read More →The system containing the rails and the wire of the previous problem is kept vertically
Question: The system containing the rails and the wire of the previous problem is kept vertically in a uniform horizontal magnetic field $\mathrm{B}$ that is perpendicular to the rails. It is found that the wire stays in equilibrium. If the wire ab is replaced by another wire of double its mass, how long will it take in falling through a distance equal to its length? Solution: $t=\sqrt{\frac{2 l}{g}}$...
Read More →The current generator Ig, shown in figure, sends a constant current
Question: The current generator $I_{g}$, shown in figure, sends a constant current i through the circuit. The wire ab has a length I and mass $\mathrm{m}$ and can slide on the smooth, horizontal rails connected to $I_{g}$. The entire system lies in a vertical magnetic field $B$. Find the velocity of the wire as a function of time. Solution:...
Read More →The current generator 1g, shown in figure,
Question: The current generator $I_{g}$, shown in figure, sends a constant current i through the circuit. The wire cd is fixed and ab is made to slide on the smooth, thick rails with a constant velocity v towards right. Each of these wires has resistance r. Find the current through the wire cd. Solution:...
Read More →Solve this following
Question: Mark $(\sqrt{ })$ against the correct answer in the following: If $y=\log _{10} x$ then $\frac{d y}{d x}=?$ A. $\frac{1}{\mathrm{x}}$ B. $\frac{1}{\mathrm{x}}(\log 10)$ C. $\frac{1}{x(\log 10)}$ D. none of these Solution:...
Read More →Solve this following
Question: Mark $(\sqrt{ })$ against the correct answer in the following: If $y=2^{x}$ then $\frac{d y}{d x}=$ ? A. $\mathrm{x}\left(2^{\mathrm{x}-1}\right)$ B. $\frac{2^{x}}{(\log 2)}$ C. $2^{x}(\log 2)$ D. none of these Solution:...
Read More →Find the equation of the tangent at
Question: Find the equation of the tangent at $t=\frac{\pi}{4}$ for the curve $x=\sin 3 t, y=\cos 2 t$ Solution:...
Read More →Find the equation of tangent to the curve
Question: Find the equation of tangent to the curve $\mathrm{x}=(\theta+\sin \theta), \mathrm{y}=(1+\cos \theta)$ at $\theta=\frac{\pi}{4}$ Solution:...
Read More →Show that the curves
Question: Show that the curves $x^{3}-3 x y^{2}+2=0$ and $3 x^{2} y-y^{3}-2=0$ cut orthogonally. Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. The wire PQ has a negligible resistance and is made to slide on the three rails with a constant speed of $5 \mathrm{~cm} / \mathrm{s}$. Find the current in the $10 \Omega$ resistor when the switch $\mathrm{S}$ is thrown to (a) the middle rail (b) the bottom rail. Solution:...
Read More →Solve this following
Question: Show that the curves $x y=a^{2}$ and $x^{2}+y^{2}=2 a^{2}$ touch each other. Solution:...
Read More →Suppose the 19Ω resistor of the previous problem is disconnected.
Question: Suppose the $19 \Omega$ resistor of the previous problem is disconnected. Find the current through $P_{2} Q_{2}$ in the two situations (a) and (b) of that problem. Solution:...
Read More →Solve this following
Question: Show that the curves $\mathrm{x}=\mathrm{y}^{2}$ and $\mathrm{xy}=\mathrm{k}$ cut at right angles if $8 \mathrm{k}^{2}=1$. Solution:...
Read More →Solve this following
Question: Find the point on the parabola $y=(x-3)^{2}$, where the tangent is parallel to the chord joining the point $(3,0)$ and $(4,1)$. Solution:...
Read More →Find the point on the curve
Question: Find the point on the curve $y=2 x^{2}-6 x-4$ at which the tangent is parallel to the $x$-axis. Solution:...
Read More →Consider the situation given in the figure.
Question: Consider the situation given in the figure. The wires $P_{1} Q_{1}$ and $P_{2} Q_{2}$ are made to slide on the rails with the same speed 5 $\mathrm{cm} / \mathrm{s}$. Find the electric current in the $19 \Omega$ resistor if (a) both the wires move towards right and (b) if $P_{1} Q_{1}$ moves towards left but $P_{2} Q_{2}$ moves towards right. Solution:...
Read More →Find the equation of the tangent to the curve
Question: Find the equation of the tangent to the curve $x^{2}+2 y=8$, which is perpendicular to the line $x-2 y+1=0$. Solution:...
Read More →A wire ab of length I, mass m and resistance R slides on a smooth,
Question: A wire ab of length I, mass $m$ and resistance $R$ slides on a smooth, thick pair of metallic rails joined at the bottom as shown in figure. The plane of the rails makes an angle $\theta$ with the horizontal. A vertical magnetic field B exists in the region. If the wire slides on the rails at a constant speed v, show that $B=\sqrt{\frac{m g \sin \theta}{v l^{2} \cos ^{2} \theta}}$ Solution:...
Read More →Find the equation of the tangents to the curve
Question: Find the equation of the tangents to the curve $2 x^{2}+3 y^{2}=14$, parallel to the line $x=3 y=4$. Solution:...
Read More →Figure shows a metallic wire of resistance
Question: Figure shows a metallic wire of resistance $0.20 \Omega$ sliding on a horizontal, U-shaped metallic rail. The separation between the parallel arms is $20 \mathrm{~cm}$. An electric current of $2.0 \mu \mathrm{A}$ passes through the wire when it is slid at the rate of $20 \mathrm{~cm} / \mathrm{s}$. If the horizontal component of the earth's magnetic field is $3.0 \times 10^{-5} \mathrm{~T}$, calculate the dip at the place. Solution:...
Read More →A rectangular frame of wire abcd has dimensions
Question: A rectangular frame of wire abcd has dimensions $32 \mathrm{~cm} \times 8.0 \mathrm{~cm}$ and a total resistance of $2.0 \Omega$. It is pulled out of a magnetic field $B=0.020 \mathrm{~T}$ by applying a force of $3.2 \times 10^{-5} \mathrm{~N}$. It is found that the frame moves with constant speed. Find (a) this constant speed, (b) the emf induced in the loop, (c) the potential difference between the points a and b and (d) the potential difference between the points $c$ and $d$. Soluti...
Read More →Find the point on the curve
Question: Find the point on the curve $y=x^{3}-11 x+5$ at which the equation of tangent is $y=x-11$. Solution:...
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