The susceptibility of magnesium at 300 K is
Question: The susceptibility of magnesium at $300 \mathrm{~K}$ is $1.2 \times 10^{-5}$. At what temperature will the susceptibility increase to $1.8 \times 10^{-5}$ ? Solution:...
Read More →Solve this following
Question: Let I be an interval disjoint from $]-1,1\left[\right.$. Prove that the function $f(\mathrm{x})=\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)$ is strictly increasing on I. Solution:...
Read More →The magnetic field B and the magnetic intensity
Question: The magnetic field $B$ and the magnetic intensity $\mathrm{H}$ in a material are found to be $1.6 \mathrm{~T}$ and $1000 \mathrm{~A} \mathrm{~m}^{-1}$, respectively. Calculate the relative permeability $\mu_{r}$ and the susceptibility $X$ of the material. Solution:...
Read More →Solve this following
Question: Prove that the function $f(x)=\log (1+x)-\frac{2 x}{(x+2)}$ is increasing for all $x-1$. Solution:...
Read More →The susceptibility of annealed iron at saturation is 5500 .
Question: The susceptibility of annealed iron at saturation is 5500 . Find the permeability of annealed iron at saturation. Solution: Susceptibility of annealed iron is $X=5500$ The relation between permeability and susceptibility is given by Permeability, $\mu=$ $\mu_{0}(1+x) \mu=4 \pi \pi \times 10^{-7}(1+5500) \Rightarrow \mu=4 \times 3.14 \times 10^{-7} \times 5501 \Rightarrow \mu=69092.56 \times 10^{-7} \Rightarrow \mu=6.9 \times 10^{-3}$...
Read More →Solve this following
Question: Show that $f(x)=\frac{x}{\sin x}$ is increasing on $] 0, \frac{\pi}{2}[$. Solution:...
Read More →Solve this following
Question: Show that $f(x)=\left(x^{3}+\frac{1}{x^{3}}\right)$ is decreasing on $]-1,1[$. Solution:...
Read More →Solve this following
Question: Show that $f(\mathrm{x})=\frac{1}{\left(1+\mathrm{x}^{2}\right)}$ is decreasing for all $\mathrm{x} \geq 0$ Solution:...
Read More →Solve this following
Question: Show that $f(\mathrm{x})=\left(\frac{1}{\mathrm{x}}+5\right)$ is decreasing for all $\mathrm{x} \in \mathrm{R}$, where $\mathrm{x} \neq 0$. Solution:...
Read More →A bar magnet of length 1 cm and cross-sectional area
Question: A bar magnet of length $1 \mathrm{~cm}$ and cross-sectional area $1.0 \mathrm{~cm}^{2}$ produces a magnetic field of $1.5 \times 10^{-4} \mathrm{~T}$ at a point in end-on position at a distance $15 \mathrm{~cm}$ away from the centre. (a) Find the magnetic moment $M$ of the magnet. (b) Find the magnetisation / of the magnet. (c) Find the magnetic field $B$ at the centre of the magnet. Solution:...
Read More →Solve this following
Question: Show that $f(\mathrm{x})=\left(\mathrm{x}-\frac{1}{\mathrm{x}}\right)$ is increasing all $\mathrm{x} \in \mathrm{R}$, where $\mathrm{x} \neq 0$ Solution:...
Read More →Solve this following
Question: Show that $f(\mathrm{x})=\mathrm{x}^{3}-15 \mathrm{x}^{2}+75 \mathrm{x}-50$ is increasing on $\mathrm{R}$. Solution:...
Read More →Solve this following
Question: Prove that $f(x)=3^{\mathrm{x}}$ is strictly increasing on $\mathrm{R}$. Solution:...
Read More →Solve this following
Question: Prove that the function $f(\mathrm{x})=\log _{\mathrm{a}} \mathrm{x}$ is strictly increasing on $] 0, \infty[$ when $\mathrm{a}1$ and strictly decreasing on ] $0, \infty[$ when $0a1$. Solution:...
Read More →The magnetic field inside a long solenoid of
Question: The magnetic field inside a long solenoid of 50 turns $\mathrm{cm}^{-1}$ is increased from $2.5 \times 10^{-3} \mathrm{~T}$ to $2.5 \mathrm{~T}$ when an iron core of crosssectional area $4 \mathrm{~cm}^{2}$ is inserted into it. Find (a) the current in the solenoid (b) the magnetisation / of the core and (c) the pole strength developed in the core. Solution:...
Read More →Prove that the function
Question: Prove that the function $f(\mathrm{x})=\log _{\mathrm{e}} \mathrm{x}$ is strictly increasing on $] 0, \infty[$. Solution:...
Read More →Solve this following
Question: Show that the function $f(\mathrm{x})=|\mathrm{x}|$ is a. strictly increasing on $] 0, \infty[$ b. strictly decreasing on] $-\infty, 0[$ Solution:...
Read More →A rod is inserted as the core in the current-carrying
Question: A rod is inserted as the core in the current-carrying solenoid of the previous problem. (a) What is the magnetic intensity $H$ at the centre? (b) If the magnetization / of the core is found to be $0.12 \mathrm{~A} \mathrm{~m}^{-1}$, find the susceptibility of the material of the rod. (c) Is the material paramagnetic, diamagnetic or ferromagnetic? Solution: Given: (a) Intensity of magnetisation $H=1500 \mathrm{~A} / \mathrm{m}$ As the solenoid and the rod are long and we are interested ...
Read More →The magnetic intensity H at the centre of a long solenoid carrying
Question: The magnetic intensity $\mathrm{H}$ at the centre of a long solenoid carrying a current of $2.0 \mathrm{~A}$, is found to be $1500 \mathrm{~A} \mathrm{~m}^{-1}$. Find the number of turns per centimetre of the solenoid. Solution: Here given in question, Current in the solenoid, $I=2$ A Magnetic intensity at the centre of long solenoid, $H=1500 \mathrm{Am}^{-1}$ Magnetic field produced by a solenoid(B) is given by $B=\mu_{0} n i . . .(T)$ where, $n=$ number of turns per unit length $i=$ ...
Read More →Solve this following
Question: Show that the function $f(x)=\mathrm{X}^{2}$ is a. strictly increasing on $[0, \infty[$ b. strictly decreasing on $[0, \infty[$ c. neither strictly increasing nor strictly decreasing on $R$ Solution:...
Read More →Solve this following
Question: Prove that the function $f(\mathrm{x})=\mathrm{e}^{2 \mathrm{x}}$ is strictly increasing on $\mathrm{R}$. Solution:...
Read More →Solve this following
Question: Prove that $f(\mathrm{x})=\mathrm{ax}+\mathrm{b}$, where $\mathrm{a}$ and $\mathrm{b}$ are constants and $\mathrm{a}0$, is a strictly increasing function on $\mathrm{R}$. Solution:...
Read More →Solve this following
Question: Show the function $f(\mathrm{x})=-2 \mathrm{x}+7$ is a strictly decreasing function on $\mathrm{R}$. Solution: Domain of the function is $R$ Finding derivative $f^{\prime}(x)=-2$ Which is less than 0 Means strictly decreasing in its domain i.e $R$...
Read More →A short magnet makes 40 oscillations per minute
Question: A short magnet makes 40 oscillations per minute when used in an oscillation magnetometer at a place where the earth's horizontal magnetic field is $25 \mu \mathrm{T}$. Another short magnet of magnetic moment $1.6 \mathrm{~A}-\mathrm{m}^{2}$ is placed $20 \mathrm{~cm}$ east of the oscillating magnet. Find the new frequency of oscillation if the magnet has its north pole (a) towards north and (b) towards south. Solution:...
Read More →Solve this following
Question: Show that the function $f(\mathrm{x})=5 \mathrm{x}-2$ is a strictly increasing function on $\mathrm{R}$. Solution:...
Read More →