The magnetic induction and the intensity of the magnetic field inside an iron core of an electromagnet are $1 \ Wb \ m^{-2}$ and $150 \ A \ m^{-1}$,respectively. The relative permeability of iron is $\left(\mu_0 = 4 \pi \times 10^{-7} \ H \ m^{-1}\right)$.

Let $g(x)$ be an antiderivative for $f(x)$. Then $\ln(1 + (g(x))^2)$ is an antiderivative for

The figure shows a system of logic gates. From the study of the truth table,it can be found that to produce a high output $(1)$ at $R$,we must have

$A$ paramagnetic sample shows a net magnetization of $8 \ Am^{-1}$ when placed in an external magnetic field of $0.6 \ T$ at a temperature of $4 \ K$. When the same sample is placed in an external magnetic field of $0.2 \ T$ at a temperature of $16 \ K$,the magnetization will be

The set of all points for which $f(x) = x^2 \cdot e^{-x}$ strictly increases is

$A$ cylinder of radius $R$ made of a material of thermal conductivity $K_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of material of thermal conductivity $K_2$. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is