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Two thin dielectric slabs of dielectric constants $K_1$ and $K_2(K_1<K_2)$ are inserted between the plates of a parallel plate capacitor, as shown in the figure below. The variation of electric field $E$ between the plates with distance $d$ as measured from plate $P$ is correctly shown by -

Figure shows an ampereian path $\text{ABCDA}$. Part $\text{ABC}$ lies in vertical plane $\text{PSTU}$ while part $\text{CDA}$ lies in horizontal plane $\text{PQRS}$. $\oint \overrightarrow{B}.\overrightarrow{dl}$ for this path according to the Ampere's circuital law will be:

Speed of sound wave is v. If a reflector moves towards a stationary source emitting waves of frequency f with speed u, the wavelength of reflected waves will be

1. 3.2

The variation of magnetic susceptibility $\left(\chi \right)$ with temperature $\left(T\right)$ for diamagnetic substances is best represented by,

A liquid of mass $M$ and specific heat $c$ is at temperature $2t$. If another liquid of equal mass and thermal capacitiy $1.5$ that of the first at a temperature of $\frac{t}{3}$ is added to it, the resultant temperature will be

If $\underset{0}{\overset{\alpha }{\int }}\frac{dx}{1-\cos \alpha \cos x}=\frac{A}{\sin \alpha }+B(\alpha \ne 0),$ then value of $A$ and $B$ is

The moment of inertia of a solid sphere about an axis passing through its centre is $0.4{\text{kg-m}}^2$. If mass of the sphere is kept same, but its density is increased eight times, find the new moment of inertia of the sphere about the same axis.

A ball of mass $1\text{kg}$ is tied to one end of a string of length $1\text{m}$, pivoting the other end of string, ball is rotated at an angular speed of $2\text{rev/min}$. The tension in the string is

The distance between the upper surface of water and treasure box is 0.054 km. Find the distance in mm. (1 km = ${10}^6$ mm)

Two particle A and B move on a circle. Initially Particle A and B are diagonally opposite to each other. Particle A moves with angular velocity $\pi \frac{rad}{sec}$, angular acceleration $\frac{\pi }{2}\frac{rad}{se{c}^2}$ and particle B moves with constant angular velocity $2\pi \frac{rad}{sec}$. Find the time after which both the particle A and B will collide.

A block of mass 30.0 kg is being brought down by a chain. If the block acquires a speed of 40.0 cm/s in dropping down 2.00 m, find the work done by the chain during the process.

A battery has an internal resistance $4\Omega$ is connected to the network of resistance as shown. What must be the value of $R$ (in $\Omega$) so that maximum power is delivered to the network ?

In one frame of reference the distance between two points is $d$, then the distance between the same two points in other frame of reference will be

If LHS is $=4\left({\cos }^3\left(10°\right)+{\sin }^3\left(20°\right)\right)$, then RHS will be.

(a) A
monoenergetic electron beam with electron speed of 5.20 × 10⁶
m s⁻¹
is subject to a magnetic field of 1.30 × 10⁻⁴
T normal to the beam velocity. What is the radius of the circle
traced by the beam, given e/m for
electron equals 1.76 × 10¹¹
C kg⁻¹.

(b) Is the
formula you employ in (a) valid for calculating radius of the path of
a 20 MeV electron beam? If not, in what way is it modified?

[Note:
Exercises 11.20(b) and 11.21(b) take
you to relativistic mechanics which is beyond the scope of this book.
They have been inserted here simply to emphasise the point that the
formulas you use in part (a) of the exercises are not valid at very
high speeds or energies. See answers at the end to know what ‘very
high speed or energy’ means.]

Two particles of same mass $2\text{kg}$ are rigidly attached to the ends of a massless rod of length $10\text{m}$. The rod is clamped at the centre such that the system rotates about the perpendicular bisector of the rod at an angular speed $10\text{rad/s}$. Calculate the magnitude of angular momentum of the system about the axis of rotation [in ${\text{kg m}}^2\text{/s}$]

A uniform metal rod of length $L$ and mass $M$ is rotating with an angular speed $\omega$ about an axis passing through one of the ends and perpendicular to the rod. If the temperature increases by $t^{\circ }C,$ then the change in its angular speed is proportional to

A Gaussian message signal $m\left(t\right)=e^{–t^2}V$ is applied to a phase modulator. If the phase sensitivity of the modulator is $k_p=8000\pi rad/V$, then the maximum frequency deviation of the resultant PM signal will be _____ kHz.

1. 3.43

An electron of mass $m$ is accelerated through a potential difference of $V$ and then it enters a magnetic field of induction $B$ normal to the lines. Then, the radius of the circular path is

The acceleration $\frac{d^{2}x}{d{t}^2}$ of a particle varies with displacement x as $\frac{d^{2}x}{d{t}^2}=-kx$, where k is a constant of the motion.The time period T of the motion is equal to

Which of the following combinations should be selected for better tuning of an $\text{LCR}$ circuit used for communications ?

A simply supported beam of length 2L is subjected to a moment M at the mid - point x = 0 as shown in the figure. The vertical deflection in the domain $0\le x\le L$ is given by $y=\frac{-Mx}{12EIL}(L-x)(x+C)$ Where E is Young's modulus, I is the area moment of inertia and C is a constant (to be determined).

The slope at the centre x = 0 is