31.

In an LCR circuit as shown in figure, both switches are open initially. Now switch S₁ is closed, S₂ kept open. (q is charge on the capacitor and τ = RC is capacitive time constant). Which of the following statement is correct?       

                                

• At t = $\frac{\mathrm{\tau }}{2}$, q = CV ( 1 $-$ e^-1 )

• Work done by the battery is half of the energy dissipated in the resistor

• At t = τ, q = CV/2

• At t = τ, q = CV ( 1 $-$ e^-2 )

As shown in figure, two vertical conducting rails separated by distance 1.0 m are placed parallel to z-axis. At z = 0, a capacitor of 0.15 F is connected between the rails and a metal rod of mass 100 gm placed across the rails slides down along the rails. If a constant magnetic fields of 2.0 T exists perpendicular to the plane of the rails, what is the acceleration of the rod? ( take g = 9.8 m/s²)

• 2.5 m/s²

• 1.4 m/s²

• 9.8 m/s²

• 0

For MRI, a patient is slowly pushed in a time of 10 s within the coils of the magnet where magnetic field is B = 2.0 T. If the patient's trunk is 0.8 m in circumference, the induced emf around the patient's trunk is

• 10.18 x 10^-2 V

• 9.66 × 10² V

• 10.18 × 10^-3 V

• 1.51 × 10^-2 V

The measurement of voltmeter in the following circuit is

• 2.25 V

• 4.25 V

• 2.75 V

• 6.25 V

A large solid sphere with uniformly distributed positive charge has a smooth narrow tunnel through its centre. A small particle with negative charge, initially at rest far from the sphere, approaches it along the line of the tunnel, reaches its surface with a speed v, and passes through the tunnel. Its speed at the centre of the sphere will be

• 0

• v

• $\sqrt{2}$v

• $\sqrt{1.5}$ v

A current I is flowing through the loop. The direction of the current and the shape of the loop are as shown in the figure. The magnetic field at the centre of the loop is μ₀I/R times ( MA = R, MB = 2R, ∠DMA = 90^o )

• 5/16, but out of the plane of the paper.

• 5/16, but into the plane of the paper.

• 7/16, but out of the plane of the paper.

• 7/16, but into the plane of the paper.

In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is however not constant, but increases uniformly along the positive z-direction at the rate of 10⁵ N C^-1m^-1 The force experienced by the system having a total dipole moment equal to 10^-7 C m in the negative z-direction is

• $-$10^-2 N

• 10^-2 N

• 10^-4 N

• $-$ 10^-4 N

A meter bridge is set-up as shown, to determine an unknown resistance X using a standard 10 ohm resistor. The galvanometer shows null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of

• 10.2 ohm

• 10.6 ohm

• 10.8 ohm

• 11.1 ohm

An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge upto a radius R. The atom as a whole is neutral. The electric field at a distance r from the nucleus is (r < R)

• $\frac{\mathrm{Ze}}{4{\mathrm{\pi \epsilon }}_0} \left[\frac{1}{{\mathrm{r}}^2} - \frac{\mathrm{r}}{{\mathrm{R}}^3}\right]$

• $\frac{\mathrm{Ze}}{4{\mathrm{\pi \epsilon }}_0} \left[\frac{1}{{\mathrm{r}}^3} - \frac{\mathrm{r}}{{\mathrm{R}}^2}\right]$

• $\frac{\mathrm{Ze}}{4{\mathrm{\pi \epsilon }}_0} \left[\frac{\mathrm{r}}{{\mathrm{R}}^3} - \frac{1}{{\mathrm{r}}^2}\right]$

• $\frac{\mathrm{Ze}}{4{\mathrm{\pi \epsilon }}_0} \left[\frac{\mathrm{r}}{{\mathrm{R}}^3} + \frac{1}{{\mathrm{r}}^2}\right]$

A deflection magnetometer is adjusted in the usual way. When a magnet is introduced, the deflection observed is θ, and the period of oscillation of the needle in the magnetometer is T. When the magnet is removed, the period of oscillation is T₀. The relation between T and T₀ is

• T² = ${\mathrm{T}}_0^2$ cosθ

• T = T₀ cosθ

• T = $\frac{{\mathrm{T}}_0}{\cos \mathrm{\theta }}$

• ${\mathrm{T}}^2 = \frac{{\mathrm{T}}_0^2}{\mathrm{cos\theta }}$