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QUESTION: 1

What is the electric field at the center of a uniformly charged semicircular arc i.e., at point P in Fig.

Solution:

Let λ is the charge per unit length of arc of radius ‘a' splitting the arc in to small segment Δs_{1} and point charge on each is λΔs_{1},. From coulomb’s law, the electric field due to the Δs_{1} is

...(i)

From fig. it is clear that at point PE_{x} will be zero because i E, will be canceled by the contribution from a symmetrically placed Δ_{t} on the left half of the arc. So in order to find total E we need only compute E_{y} at point P. so.

...(ii)

-ve sign arises Because ΔE_{v} is in - y direction sum over the whole are

...(iii)

Along the circle of radius a. ds = adθ

So.

QUESTION: 2

Two electric dipoles p_{1} and p_{2} are placed at (0, 0, 0) and (2, 0, 0) respectively with both of them pointing in the +z direction. Without changing the orientations of the dipoles. P_{2} is moved to (0_{.} 4_{.} 0). The ratio of the electrostatic potential energy of the dipoles after moving to that before moving is.

Solution:

In first case

...(i)

...(ii)

In new case P_{2} is shifted to (0, 4, 0). then the electric field at the site of P_{2} is

...(iii)

So,

...(iv)

So from equation (2) & (4)

QUESTION: 3

A spherical piece of radius much less than the radius of a charged spherical shell (charge density σ) is removed from the shell itself then electric field intensity at the mid point of aperture is

Solution:

For a charged spherical shell.

...(i)

Let be the electric field due to small removed portion and the electric field due to remaining portion.

...(ii)

Comparing (i) and rii) we get

...(iii)

and (iv)

⇒ (from (iii) and (iv)

QUESTION: 4

A cubical box sits in a uniform electric field as shown in Fig. What is the net flux coming out of the box.

Solution:

Because the field lines simply skim the four sides of the box. the flux through them is zero. The flux through the top is since cosθ = cos 0^{0} = 1 in this case .

For the bottom. The negative sign arises because the flux is into the bottom area, not out of it.

Because A_{1} = A_{0}. the net flux from the box is

QUESTION: 5

A charge q is placed at the centre of the line joining two equal charges O. The system of three charges will be in equilibrium if q equal to

Solution:

Let two equal charges Q each, be held at A and B. where AB = 2x. C is the centre of AB. where charge q is held.

For the three charges to be in equilibrium, net force on each charge must be zero.

Now. total force on Q at B is

QUESTION: 6

Two point charges q_{1} = - 2 C and q_{2} = - 1 C are separa ted by a distance d. The position on the line joining the two charges where a third charge q = + 1C will be in equilibrium is at a distance

Solution:

When two charges are of same nature (both positive or both negative) then neutral point is always between the two charges on the line joining them. But when two charges are of opposite nature (one positive and other negative) then neutral point is always outside the two charges on the extended line joining the two charges.

So. we conclude that options (A) and (C) are absolutely ruled out.

QUESTION: 7

A long cylinder carries a volume charge density that is given by p = kr where k is constant and r is the distance from the axis. Then the electric field inside the cylinder is

Solution:

So,

Now,

QUESTION: 8

Does the potential function φ = q(x^{2} + y^{2} + z_{2})^{-1/2} satisfies the laplace’s equations.

Solution:

We have

...(i)

...(ii)

and ...(iii)

Adding (1), (2) and (3)

i.e. which is Laplace's equation. Thus the function

satisfies Laplace's equation.

QUESTION: 9

If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by

Solution:

Equivalent circuit is as shown in Figure

or

Since we get

∴

i.e.

QUESTION: 10

Two slabs of the same dimensions, having dielectric constants K_{1 }and K_{2}. completely fill die space between the plates of a parallel plate capacitor as shown in the figure. If C is the original capacitance of the capacitor, the new capacitance is

Solution:

The arrangement is equivalent to two capacitors, each of plate area A and separation d/2, connected in series having capacitances

∴

Original capacitance

∴

QUESTION: 11

An electron moving with a speed 'u‘ along the position x - axis at y = 0 enters a region of uniform magnetic field which exists to the right of y - axis . The electron exists from the region after sometimes with the speed V at coordinate y then

Solution:

Magnetic force does not change the speed of charged particle. Hence, v = u. Further magnetic field on the electron in the given condition is along negative y-axis in the starting. Or it describes a circular path in clockwise direction. Hence, when it exits from the field, y < 0.

QUESTION: 12

A long straight conductor, carrying a current L is bent into the shape shown in the figure. The radius of the circular loop is r. The magnetic field at the centre of the loop is

Solution:

Field due to straight part. out of the page

Field due to circular part. to the page

Net field B = into the page.

QUESTION: 13

A battery of emf V volt is connected across a coil of uniform wire as shown. The radius of the coil is 'a' metres and the total resistance of the coil is R ohm. The magnetic field at the centre O of the coil (in tesla) is

Solution:

QUESTION: 14

In terms of basic units of mass (M). Length (L). time (Tr and charge 0. the dimensions of magnetized field (H» are

Solution:

We know

and M^{1}T^{-1}Q^{-1}

So,

QUESTION: 15

A magnetic needle is kept in a non-uniform magnetic field. It experiences:-

Solution:

Option(A) is correct as both the magnetic force & torque will be applied on it. (torque):

*Multiple options can be correct

QUESTION: 16

A particle of charge +q and mass m moving under the influence of a uniform electric field E_{i } and a uniform magnetic field B_{ k} follows a trajectory from P to Q as shown in the figure. The velocities at P and Q are Which of the following statement is correct?

Solution:

Work done by tire electric force = Change of kinetic energy

or

Now. rate of work done = F.v

∴ Rate of work done by the electric field at P

Rate of work done by the electric field at Q

= qE 2v cos 90" = 0

Work done by the magnetic field is obviously zero at it points.

So C is not possible

*Multiple options can be correct

QUESTION: 17

A series LCR circuit containing a resistance of 120 Ω has angular lesonance frequency 4 * 10^{5} rad s^{-1}. At resonance the voltages across resistance and inductance are 60 V and 40 V respectively.

Solution:

Now in case of series LCR circuit,

So current will lag the applied voltage by 45^{0 }if

*Multiple options can be correct

QUESTION: 18

A positively charged thin metal ring of radius R is fixed in the xy plane with its centre at the origin 0. A negatively charged particle P is released from rest at the point (0. 0. Z_{0}) where Z_{0} > 0. Then the motion of P is :

Solution:

Let o be the charge on the ling, the negative charge -q is released from point P (0, 0, Z_{0}). The electric field at P due to the charged ring will be along positive z-axis and its magnitude will be

E = 0 at centre of the ring because Z_{0}= 0

Therefore, force on charge P will be towards centre as shown, and its magnitude is

Similarly, when it crosses the origin . the force is again towards centre O.

Thus tire motion of the particle is period ic for all values of Z_{0} lying between 0 and

Secondly

(From equation 1)

i.e. there storing force 0. Hence the motion of the particle will be simple harmonic. (Here negative sign implies that the force is towards its mean position.)

*Multiple options can be correct

QUESTION: 19

In the series circuit of Fig. suppose Then

Solution:

The inductive and capacitive reactances are

The impedance Z of the circuit is

Z = 500 Ω

Witfi source voitage amplitude V = 50 V the current amplitude is

The phase angle is

from fig. the voltage amplitudes V_{ R}, V_{ L}. and V_{c} across the resistor, inductor, and capacitor. respectively, are

*Multiple options can be correct

QUESTION: 20

In a nonmagnetic medium E = 4sin Then,

Solution:

we know electric field vector

Since the medium is not free space but a lossless medium.

(nonmagnetic).

Hence

or

On plane 2x + y = 5

Hence the total power is

*Answer can only contain numeric values

QUESTION: 21

Three point charges are placed at the following points on the x axis: + 2μC at x = 0, - 3μC at x = 40 cm. - 5μC at x = 120 cm. What is the force (in Newton) on the - 3μC chaige ?

Solution:

Figure is a diagram of the step with . The force on q_{2} is the vector sum of two contributions, the attractive force due to q_{1}(toward q_{1}) and the repulsive force due to q3 (also toward q_{1}). The sum of these two forces, taken algebraically, since they are along the same line, is

F = F_{1} + F_{2}

= - 0 . 55N or 0.55 N to left

*Answer can only contain numeric values

QUESTION: 22

Four capacitors, each of 25μF.are connected as shown in the following circuit. A d.c. voltmeter reads 200 volts. The charge (in coulomb) on each capacitor is _____

Solution:

All the capacitors are in parallel and P.D. across each is 200 V.

∴ O = CV = 25 * 10^{-6} * 200 = 5 * 10^{-3} C

*Answer can only contain numeric values

QUESTION: 23

The figure shows two identical parallel plate capacitors connected to a battery with switch S closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of constant 3. The ratio of the total energy stored in both capacitors before and after tire introduction of the dielectric is ____

Solution:

Initial energy

Final energy of A

Final p.d. across B. V . is given by (3C) V = C V or V = V/3

∴ Final energy of B

∴ Total final energy

*Answer can only contain numeric values

QUESTION: 24

The electric field intensity at a point on tine surface of a conductor is given by What is the surface charge density (in Pc/m^{2}) at the point.

Solution:

Supposing the conductor to be surrounded by free space.

The ambiguity in sign arises from that in the direction of the outer normal to the surface at the given point.

*Answer can only contain numeric values

QUESTION: 25

A copper rod of length 0.19 m is moving with uniform velocity 10 ms^{-1} parallel to a long straight wire carrying a current of 5A. The rod is perpendicular to the wire with its ends at distance 0.01 m and 0.2 m from it. Calculate the e.m.f. induced

Solution:

Here Due to motion of element dx in the field. B induced e.m. f. dE = Bvdx Total e.m. f.

= 30 x 10^{-6} V = 30 μV

*Answer can only contain numeric values

QUESTION: 26

In an LCR circuit, the capacitance is made one - fourth when in resonance, then how many times the inductance should be changed, so that the circuit remains in resonance?

Solution:

At resonance ; X_{L} = X_{c}

Now

Thus change in inductance should be 4 times.

*Answer can only contain numeric values

QUESTION: 27

A 50 Hz A.C. current of peak value 1A flows through the primary coil of a transformer. If the mutual inductance between primary and secondary is 1.5 H. the mean value of the induced voltage (in V) is ________ .

Solution:

Frequency of A.C. n = 50 Hz

time period T

Time in which current changes from peak

value (i_{i}) to zero (i_{i})

i.e.

then Δi- change in current

= i_{i} — i_{i} = 0 — 1 = - 1 A .

∴ mean induced e.m.f.

*Answer can only contain numeric values

QUESTION: 28

A long solenoid of diameter 0.1 m has 2 * 10^{4 }turns per meter. At the centre of, the solenoid a 100 turn coil of radius 0.01m is placed with its axis coinciding with that of the solenoid. The current in the solenoid is decreasing at a constant rate from +2A to -2A in 0.05 sec. Find the total charge (in μC ) flowing through the coil during this time if the resistance of the coil is 10 π^{2} ohm.

Solution:

As field inside, a solenoid is given by

so here.

And hence flux linked with the coil

Further as the coil has a resistance of 10π^{2} ohm. the current induced in it.

and as,

So,

*Answer can only contain numeric values

QUESTION: 29

Find the ratio of average and rms value for the saw-tooth voltage of peak value V_{0 }as shown in Fig.

Solution:

As the equation of the saw-tooth wave shown in Fig. will be.

so (a)

i.e.

and (b)

i.e.

i.e.

*Answer can only contain numeric values

QUESTION: 30

A 750 hertz, 20 V source is connected to a resistance of 100 ohm. an inductance of 0.1803 henry and a capacitance of 10 microfarad all in series. Calculate the time (in sec .) in which the resistance (thermal capacity 2J/^{o}C ) will get heated by 10^{o}C.

Solution:

As in this problem,

and hence.

but as in case of ac.

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