A particle moves in a circle of diameter
Question: A particle moves in a circle of diameter $1.0 \mathrm{~cm}$ under the action of a magnetic field of $0.40 T$. An electric field of $200 \mathrm{~V} / \mathrm{m}$ makes the path straight. Find the charge/mass ratio of the particle. Solution:...
Read More →A uniform magnetic field of magnitude 0.20T exists in space from east to west.
Question: A uniform magnetic field of magnitude 0.20T exists in space from east to west. With what speed should a particle of mass $0.010 \mathrm{~g}$ and having a charge $1.0 \times 10^{-5} \mathrm{C}$ be projected from south to north so that it moves with a uniform velocity? Solution:...
Read More →Find two positive number whose product is 49 and the sum is minimum.
Question: Find two positive number whose product is 49 and the sum is minimum. Solution:...
Read More →Two particles, each having mass m are placed at a separation d
Question: Two particles, each having mass $m$ are placed at a separation $d$ in a uniform magnetic field $B$ as shown in figure. They have opposite charges of equal magnitude q. At time $t=0$, the particles are projected towards each other, each with a speed $v$. Suppose the Coulomb force between the charges is switched off. (a) find the maximum value $v_{m}$ of the projection speed so that the two particles do not collide. (b) What would be the minimum and maximum separation between the particl...
Read More →Find the maximum and minimum values of
Question: Find the maximum and minimum values of $f(x)=(-x+2 \sin x)$ on $[0,2 \pi]$ Solution:...
Read More →A narrow-beam of singly-charged carbon ions,
Question: A narrow-beam of singly-charged carbon ions, moving at a constant velocity of $6.0 \times 10^{4} \mathrm{~m} / \mathrm{s}$, is sent perpendicularly in a rectangular region having uniform magnetic field $B=0.5 T$. It is found that two beams emerge from the field in the backward direction, the separations from the incident beam being $3.0 \mathrm{~cm}$ and $3.5 \mathrm{~cm}$. Identify the isotopes present in the ion beam. Take the mass of an ion $=A\left(1.6 \times 10^{-27}\right)_{\math...
Read More →An enemy jet is flying along the curve
Question: An enemy jet is flying along the curve $y=\left(x^{2}+2\right)$. A soldier is placed at the point $(3,2)$. Find the nearest point between the soldier and the jet. Solution:...
Read More →Find the maximum profit that a company can make,
Question: Find the maximum profit that a company can make, if the profit function is given by $\mathrm{p}(\mathrm{x})=41+24 \mathrm{x}-18 \mathrm{x}^{2}$ Solution:...
Read More →Solve this following
Question: Show that $\left(x+\frac{1}{x}\right)$ has a maximum and minimum, but the maximum value is less than the minimum value. Solution:...
Read More →A particle of mass m and charge q is projected into a region having
Question: A particle of mass $m$ and charge $q$ is projected into a region having a perpendicular magnetic field $B$. Find the angle of deviation of the particle as it comes out of the magnetic field if the width d of the region is very slightly smaller than (a) $\frac{m v}{q B}$ (b) $\frac{m v}{2 q B}$ (c) $\frac{2 m v}{q B}$ Solution:...
Read More →Show that the maximum value of
Question: Show that the maximum value of $\mathrm{X}^{1 / \mathrm{x}}$ is $\mathrm{e}^{1 / \mathrm{e}}$ Solution:...
Read More →A particle of mass m and positive charge q,
Question: A particle of mass $m$ and positive charge $q$, moving with a uniform velocity $v$, enters a magnetic field $B$ as shown in figure. (a) Find the radius of the circular arc it describes in the magnetic field. (b) Find the angle subtended by the arc at the center. (c) How long does the particle stay inside the magnetic field? (d) Solve the three parts of the above problem if the charge q on the particle is negative. Solution:...
Read More →Find the maximum and minimum of
Question: Find the maximum and minimum of $f(x)=\left(\sin x+\frac{1}{2} \cos x\right)$ in $0 \leq x \leq \frac{\pi}{2}$ Solution: max. value is $\frac{3}{4}$ at $x=\frac{\pi}{6}$ and min. value is $\frac{1}{2}$ at $x=\frac{\pi}{2}$...
Read More →Find the maximum and minimum values of
Question: Find the maximum and minimum values of $3 x^{4}-8 x^{3}+12 x^{2}-48 x+1$ on the interval $[1,4]$ Solution:...
Read More →(a) An electron moves along a circle of radius 1m
Question: (a) An electron moves along a circle of radius $1 \mathrm{~m}$ in a perpendicular magnetic field of strength $0.50 \mathrm{~T}$. What would be its speed? Is it reasonable? (b) If a proton moves along a circle of same radius in the same magnetic field, what would be its speed? Solution:...
Read More →Find the maximum and minimum values of
Question: Find the maximum and minimum values of $2 \mathrm{x}^{3}-24 \mathrm{x}+107$ on the interval $[-3,3]$. Solution:...
Read More →A proton is projected with a velocity of
Question: A proton is projected with a velocity of $3 \times 10^{6} \mathrm{~m} / \mathrm{s}$ perpendicular to a uniform magnetic field of $0.6 \mathrm{~T}$. find the acceleration of the proton. Solution:...
Read More →Doubly-ionized helium ions are projected with a speed of
Question: Doubly-ionized helium ions are projected with a speed of $10 \mathrm{~km} / \mathrm{s}$ in a direction perpendicular to a uniform magnetic field of magnitude 0.1T. Find (a) the force acting on an ion, (b) the radius of the circle in which it circulates and (c) the time taken by an ion to complete the circle. Solution:...
Read More →Find the point of local maxima or local minima or local minima
Question: Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions: $f(x)=\frac{x}{2}+\frac{2}{x}, x0$ Solution:...
Read More →A charged particle is accelerated through a potential difference
Question: A charged particle is accelerated through a potential difference of $12 \mathrm{kV}$ and acquires a speed of $1.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$. It is then injected perpendicularly into a magnetic field of strength $0.2 \mathrm{~T}$. Find the radius of the circle described by it. Solution:...
Read More →Find the point of local maxima or local minima or local minima and the corresponding
Question: Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions: $f(x)=-(x-1)^{3}(x+1)^{2}$ Solution:...
Read More →Electrons are emitted with negligible speed from an electron gun
Question: Electrons are emitted with negligible speed from an electron gun are accelerated through a potential difference $V$ along the $X$-axis. These electrons emerge from a narrow hole into a uniform magnetic field B directed along this axis. However, some of the electrons emerging from the hole make slightly divergent angles as shown in figure. Show that these paraxial electrons are focused on the $X$-axis at a distance $\sqrt{\frac{8 \pi^{2} m V}{e B^{2}}}$ Solution:...
Read More →Find the point of local maxima or local minima or local minima and the corresponding
Question: Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions: $f(x)=(x-1)(x+2)^{2}$ Solution: local max. value is 0 at $x=-2$ and local $\min$. value is $-4$ at $x=0$...
Read More →Find the point of local maxima or local minima or local minima and the corresponding
Question: Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions: $f(x)=(x-1)(x+2)^{2}$ Solution: local max. value is 0 at $x=-2$ and local $\min$. value is $-4$ at $x=0$...
Read More →Figure shows a convex lens of focal length 12cm lying
Question: Figure shows a convex lens of focal length $12 \mathrm{~cm}$ lying in a uniform magnetic field B of magnitude $1.2 \mathrm{~T}$ parallel to its principal axis. A particle having a charge $2.0 \times 10^{-3} \mathrm{C}$ and mass $2.0 \times 10^{-5} \mathrm{~kg}$ is projected perpendicular to the plane of the diagram with a speed of $4.8 \mathrm{~m} / \mathrm{s}$. The particle moves along a circle with its center on the principal axis at a distance of $18 \mathrm{~cm}$ from the lens. Sho...
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