Physics Mock Test - 10

The correct answer is option 1 i.e Mercury.

• The resistivity of a material is the resistance of a wire of that material of unit length and unit cross-sectional area.
  • Resistivity, commonly symbolized by the letter ρ (rho).
  • The unit for resistivity is the ohm-meter.
  • The resistivity of a material depends on its nature and the temperature of the conductor, but not on its shape and size
  • Materials that conduct electrical current easily are called conductors and have a low resistivity.
  • Materials that do not conduct electricity easily are called insulators and these materials have a high resistivity.

CONCEPT:

• X-Rays are powerful waves of electromagnetic energy.
  • Most of them have a wavelength ranging from 0.01 to 10 nanometres, corresponding to frequencies in the range of 30 petahertz to 30 exahertz.
  • The energies of X-Rays are in the range of 100 eV to 100 keV.

EXPLANATION:

• To emit energy, an electron must jump from a higher energy level to a low energy level.
• To emit X-Rays, the difference in the higher and the lower energy levels to be large enough to produce an X-ray.
• But in a hydrogen atom, energy levels are so close that even in the most energetic line emitted i.e., when an electron drops down from the second shell to the first, has only enough energy to be an ultraviolet photon. So hydrogen atoms do not emit X rays.
• Hence, option 4 is the answer

CONCEPT:

Electric dipole:

• When two equal and opposite charges are separated by a small distance then this combination of charges is called an electric dipole.
• The strength of an electric dipole is measured by a quantity known as a dipole moment i.e.

\(\vec P = q \times \overrightarrow {2a} \)

Where q = charge and 2a = distance between two charged particles

• The electric dipole moment is denoted by P and the SI unit of dipole moment is coulombmeter (Cm).

EXPLANATION:

• Let this dipole be placed in a uniform electric field E at an angle θ with the direction of E.
• Force on charge +q at A = qE, along the direction of E and force on a charge -q at B = qE, along the direction opposite to E.
• Since the electric field (E) is uniform, therefore, the net force on the dipole is 0.
• The net force on the electric dipole in a constant electric field is always zero. Therefore option 4 is correct

Additional Information

• However, as the forces are equal, unlike and parallel, acting at different points, therefore, they form a couple which rotates the dipole. Hence, the couple tends to align the dipole along the direction of the electric field (E).

⇒ τ = force × arms of a couple

⇒ τ = F × 2asinθ = (qE) × 2asinθ

⇒ τ = (q × 2a)E sinθ \(\vec{\tau }=\vec{p}\times ~\vec{E}\)

⇒ τ = pE sinθ    

CONCEPT:

• Rutherford’s Model of the Atom:
  • He bombarded the beam of alpha particles on a very thin gold foil.
  • While bombarding he observes the number of scattering of particles –
    • Most of the particles passed either un-deviated or with very small deviation.
    • Some deviated by large angles
    • 1 out of 8000 was deflected by more than 90°
  • Conclusion to this deflection:
    • Most of the space of an atom is empty.
    • At the center of an atom having a tiny positively charged particle called a nucleus.
    • The center nucleus has all the mass of an atom.
    • The amount of positive charge at the nucleus is equal to the total amount of negative charges on all the electrons of the atom.
    • All the electrons revolve around the nucleus and coulomb force provide centripetal force

EXPLANATION:

In the experiment Rutherford observed

• Most of the particles passed either un-deviated or with very small deviation.
• Some deviated by large angles
• 1 out of 8000 was deflected by more than 90° so option-3 is correct.
• Most of the space of an atom is empty.
• At the center of an atom having a tiny positively charged particle called a nucleus.
• The center nucleus has all the mass of an atom.
• The amount of positive charge at the nucleus is equal to the total amount of negative charges on all the electrons of the atom.

CONCEPT:

• Wavefront: The locus of all particles in a medium, vibrating in the same phase/constant phase is called waveFront.
  • The direction of propagation of light (ray of light) is perpendicular to the waveFront.
  • Every point on the given wavefront acts as a source of a new disturbance called secondary wavelets which travel in all directions with the velocity of light in the medium.
  • A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new wavefront at that instant. This is called a secondary wavefront.

EXPLANATION:

• A wavefront is defined as a surface of constant phase. So option 2 is correct.

The correct answer is option 1) i.e. P is a conductor and Q is a semiconductor

CONCEPT:

• Resistance: The hindrance to the flow of current in a material is called electrical resistance.
  • The movement of free electrons in material causes a current. 
  • Resistance is inversely proportional to the current flowing in a material. 

EXPLANATION:

• Different materials behave differently on heating.
• Heat is a form of energy, and on heating an object, the electrons in it will have increased kinetic energy. 
• In conductors, the electrons will vibrate vigorously and undergo collisions with an increase in their kinetic energy.
• Each collision causes the electrons to slow down and thus, resistance increases.
• The electrons in a semiconductor require very little energy to excite to the conduction band.
• At higher temperatures, the electrons will have more energy to jump into the conduction band, and therefore, resistance decreases.    
• Thus, on increasing the temperature, resistance increases in a conductor and decreases in a semiconductor.
• Therefore, wire P is a conductor and wire Q is a semiconductor.

CONCEPT:

There are two types of Kirchoff’s Laws:

Kirchoff’s first law:

• This law is also known as junction rule or current law (KCL). According to it the algebraic sum of currents meeting at a junction is zero i.e. Σ i = 0.

• In a circuit, at any junction, the sum of the currents entering the junction must be equal the sum of the currents leaving the junction i.e., i1 + i3 = i2 + i4
• This law is simply a statement of “conservation of charge” as if current reaching a junction is not equal to the current leaving the junction, the charge will not be conserved.

Kirchoff’s second law:

• This law is also known as loop rule or voltage law (KVL) and according to it “the algebraic sum of the changes in potential in a complete traversal of a mesh (closed-loop) is zero”, i.e. Σ V = 0.
• This law represents “conservation of energy” as if the sum of potential changes around a closed loop is not zero, unlimited energy could be gained by repeatedly carrying a charge around a loop.
• If there are n meshes in a circuit, the number of independent equations in accordance with the loop rule will be (n - 1).

EXPLANATION:

• From above it is clear that Kirchoff’s second law is also known as loop rule or voltage law (KVL). Therefore statement 1 is correct.
• From above it is clear that Kirchoff’s second law represents “conservation of energy” as if the sum of potential changes around a closed loop is not zero, unlimited energy could be gained by repeatedly carrying a charge around a loop. Therefore statement 2 is correct.

CONCEPT:

Van de Graaff generator

Principle:

• Static charges are accumulated on a large radius conducting shell using corona discharge.

• This is a machine that can build up high voltages of the order of a few million volts.
• The resulting large electric fields are used to accelerate charged particles (electrons, protons, ions) to high energies needed for experiments to probe the small-scale structure of matter.

Working

• Charges from a ground metal base are collected by an insulating belt.
• These charges are driven upward by the belt attached to the pulley
• The charges are collected on a small conducting sphere attached to the pulley with the help of a metal brush.
• A metallic wire is then connected between the small and the large conducting spheres.
• The charge transfers to the large sphere due to its low potential.
• The process is repeated to accumulate a high amount of static charge on the large sphere.

EXPLANATION:

• Potential inside conducting spherical shell of radius R carrying charge Q

\(\Rightarrow V = \frac {KQ}{R}\)

• A small sphere of radius r, carrying some charge q, is placed into the large one, at the center.
• The potential due to this new charge q is

\(\Rightarrow V = \frac {Kq}{r}\) at the surface of the small sphere

\(\Rightarrow V = \frac {Kq}{R}\) at the surface of the large sphere

• The net potential at a distance R from the center due to both the spheres is

\(\Rightarrow V(R) = \frac {KQ}{R}+\frac {Kq}{R} \)

• The net potential at a distance r from the center due to both the spheres is

\(\Rightarrow V(r) = \frac {KQ}{R}+\frac {Kq}{r} \)

• The potential difference between points at distance r and R from the center is

\(\Rightarrow V(r)-V(R) = \frac {KQ}{R}+\frac {Kq}{r} - \frac {KQ}{R}-\frac {Kq}{R} =Kq(\frac {1}{r}-\frac {1}{R})\)

• From the above equation, it is clear that the potential difference between the surfaces of the small and large spherical conductors is independent of charge at the larger sphere.​ Therefore option 4 is correct

Concept:

Myopia

• A normal human eye can see any object from 25 cm to infinity. 
• The nearest point at which the human eye can see (25 cm from the eye) is called the near point. 
• The far point is the farthest point where the human eye can see. It is infinity for the human eye. 
• In myopia, the person cannot see far objects. The far point is less than infinity. 
• Myopia can be caused by a longer-than-normal eyeball or by any condition that prevents light rays from focusing on the retina.
• This is corrected by a suitable concave lens.
• The concave lens diverges the ray of light such that the image of the far object can be formed on the retina. 

​

Conclusion:

• In myopia, nothing happens to the near point. It remains at 25 cm. But, the far point is affected. 
• The correct option remains unaffected. 

Additional Information

The correct answer is option 3) i.e. the product of the magnification of objective lens and eyepiece

CONCEPT:

• Compound microscope: A compound microscope is an optical instrument that is used to obtain highly magnified images. It uses a set of two lenses - the eyepiece lens and the objective lens.
  • Generally, the eyepiece used is of lower power than that of the objective lens.
  • A typical working of the compound microscope is as shown.

• Magnification of a compound microscope (m) is given by the formula

m = mo × me

Where mo is the magnifying power of the objective lens and me is the magnifying power of the eyepiece.

EXPLANATION:

Magnification of a compound microscope, m = mo × me

• Therefore, the magnification of a compound microscope is the product of the magnification of the objective lens and eyepiece.

Explanation:

Coulomb's law in Electrostatics:

• It state’s that force of interaction between two stationary point charges is directly proportional to the product of the charges, and inversely proportional to the square of the distance between them and acts along the straight line joining the two charges.

F ∝ q₁ × q₂

\(F \propto \frac{1}{{{r^2}}}\)

\(F = \frac{1}{{4\pi {\epsilon_0}}}\frac{{{q_1}{q_2}}}{{{r^2}}} = K\frac{{{q_1}{q_2}}}{{{r^2}}}\)

Where K = Coulomb constant called electrostatic force constant.

• The value of K depends on the nature of the medium between the two charges and the system of units chosen.

Hence the correct option is \(\frac{1}{{4\pi {\epsilon_0}}}\)

The correct answer is 55 J.

Given,

charge, Q= 5C

Potential difference, V= 11V

W= Q*V

   =  5*11

   = 55 J

Additional Information

• Potential Difference:
  • Work done in bringing a unit positive charge from one point to another point is the potential difference between the two points.
  • Its SI unit is volt.
  • It is a scalar quantity.

CONCEPT:

• NAND Gate: The logic Gate is obtained after adding NOT Gate after AND Gate.

\(Y=\overline{A.B}\)

EXPLANATION:

In this we can see that, if output is Low (0)  it means both input must be high (1)
Hence option 3 is correct among all

CONCEPT:

Faraday's Laws of Electromagnetic Induction:

• First law: Whenever the number of magnetic lines of force (magnetic flux) passing through a circuit changes an emf is produced in the circuit called induced emf. The induced emf persists only as long as there is change or cutting of flux.
• Second law: The induced emf is given by the rate of change of magnetic flux linked with the circuit i.e.

\(\Rightarrow ε = -N\frac{d\phi}{dt}=-N\frac{d(BA\, cos\theta)}{dt}\)

Where dΦ = change in magnetic flux and e = induced e.m.f.

EXPLANATION:

• When a magnet is brought near a conducting loop along its axis, the distance between the loop and the magnet will reduce so the intensity of the magnetic field associated with the loop also increases.
  • Because the magnetic field associated with the loop increases so the magnetic flux linked with the loop also increases.
  • So in this case the magnetic flux is changing, hence an electric current will generate.  
• From the above equation, it is cleared that the induced emf depends upon the magnetic field and number of turns and it is independent of the resistance of the coil.

Option 3 is correct, i.e. The current flowing through a wire is directly proportional to the potential difference across it’s two ends.

• Ohm's law:
  • A basic law regarding the flow of currents was discovered by G.S. Ohm in 1828.
  • It is defined as the amount of steady current through a material is directly proportional to the potential difference, or voltage, across the material.

let V be the potential difference between the ends of the conductor.

Then Ohm’s law states that

 V ∝ I

or, V = R I 

where the constant of proportionality R is called the resistance of the conductor.

• Limitations of ohms law:
  • Ohm’s law is not applicable to unilateral networks. Unilateral networks allow the current to flow in one direction. Such types of networks consist of elements like a diode, transistor, etc.
  • Ohm’s law is also not applicable to non – linear elements. Non-linear elements are those which do not have current exactly proportional to the applied voltage that means the resistance value of those elements’ changes for different values of voltage and current. An example of a non-linear element is a thyristor.
  • Ohm’s law is also not applicable to vacuum tubes.

CONCEPT:

Series RLC circuit:

• The ac circuit containing the capacitor, resistor, and the inductor is called an LCR circuit.
• For a series LCR circuit, the total potential difference of the circuit is given by:

\(V = \sqrt {{V_R^2} + {{\left( {{V_L} - {V_C}} \right)}^2}} \)    

Where VR = potential difference across R, VL =  potential difference across L and VC =  potential difference across C

• For a series LCR circuit, Impedance (Z) of the circuit is given by:

\(\)\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)     
Where R = resistance, XL =induvtive reactance and XC = capacitive reactive

• The resonant frequency of a series LCR circuit is given by

\(⇒ ν =\frac{1}{2π\sqrt{LC}}\)   

where L = inductance and C = capacitance

• The phase difference between the current and the voltage in the series RLC circuit is given as,

\(\Rightarrow tan\phi=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)

EXPLANATION:

• The phase difference between the current and the voltage in the series RLC circuit is given as,

\(\Rightarrow tan\phi=\frac{X_L-X_C}{R}=\frac{V_L-V_C}{V_R}\)     -----(1)

• The phasor diagram of the series RLC circuit is given as,

• From the phasor diagram, it is clear that if the current is leading the voltage, then the phase difference should be negative.
• From equation 1 it is clear that if X_C is greater than X_L in the circuit then the phase difference will be negative.
• So we can say that in the series RLC circuit if the current is leading the voltage, then XC is greater than XL. Hence, option 1 is correct.

CONCEPT:

• The process of increasing the volume of sound with the help of an amplifier is called amplifications.
• Extracting the original information-bearing signal from a carrier wave is called Demodulation.
• The process of transmitting the signal wave from one place to another is called transmission.
• The process of super imposing message signal on carrier wave is called modulation.

EXPLANATION:

• Since the process of super imposing the message signal on high frequency carrier wave is called modulation. So option 4 is correct.

CONCEPT:

• Photoelectric effect: When the light of a suitable wavelength is incident on the metal surface, electrons are ejected from the metal instantly. This phenomenon is called the photoelectric effect.
• Stopping potential (V): The minimum potential required to stop the electron emitted from metal so that its kinetic energy becomes zero is called stopping potential.
• Work function: It is the minimum amount of energy required so that metal emits an electron. It is represented by ϕ.

Albert Einstein equation for photoelectric emission:

​h ν = ϕ + e V

Where, h = planks constant, ν = incident frequency, ϕ = work function, e = charge on an electron and V = stopping potential.

EXPLANATION: 

• From the above discussion, In the photoelectric effect ejection of electrons from the surface of metal takes place when Light of suitable wavelength falls on it. So option 4 is correct.

CONCEPT:

• If the charge is carried by the free electrons passing through a section of the wire in t second be Q, then the strength of the current in the wire is

\(I = \frac{Q}{t}\)

• suppose, the area of the cross-section of the wire is A and the number of free electrons per unit volume of the wire is n.
• Then, the number of electrons passing per second through a cross-section of the wire will be nAV_d.
• Hence in t seconds, nAV_dt electrons will pass. The charge on an electron is e. Therefore, the charge passing through any cross-section of the wire in t seconds will be

\(Q = (nAV_dt) \times e\)

• substituting the value of Q in equation (i).

\(I = neAV_d\)

• This is the relation between electric current and drift velocity of free electrons.

​EXPLANATION:

\(I = neAV_d\)

• Hence option 1 is correct.

​Additional Information

• The strength of the electric current in a conductor is measured by the magnitude of electric charge flowing per second through a cross-section of the conductor.
• suppose the ends of a metallic wire are connected to a battery so that an electric field E is established in the wire and the free electrons begin to move with drift velocity. (Vd) in the direction opposite to te field (since electrons have a negative  charge)

CONCEPT:

• The potential energy of a magnetic dipole in a magnetic field is the energy possessed by the dipole due to its particular position in the field.
• The expression of the potential energy of the dipole is

⇒ U = W = -MB (cosθ2 – cosθ1)

Where M = Magnetic dipole of moments and B =  uniform magnetic field

EXPLANATION:

The total work done in rotating the dipole is

⇒ W = - MH cosα

When θ = 180° 

⇒ W = - MH cos180°                  [∵ cos180°  = -1]

⇒ W = MH

• When the magnetic moment of the magnetic dipole is antiparallel to the magnetic field i.e., when θ = 180°, potential energy is maximum. The correct option is 2.

CONCEPT:

Photodiode:

• A Photodiode is a special purpose p-n junction diode fabricated with a transparent window to allow light to fall on the diode.
• It is operated under reverse bias.
• When the photodiode is illuminated with light (photons) with energy (hν) greater than the energy gap (Eg) of the semiconductor, then electron-hole pairs are generated due to the absorption of photons.
• The diode is fabricated such that the generation of e-h pairs takes place in or near the depletion region of the diode.
• Due to the electric field of the junction, electrons and holes are separated before they recombine.
• The direction of the electric field is such that electrons reach the n-side and holes reach the p-side.
• Electrons are collected on the n-side and holes are collected on the p-side giving rise to an emf.
• When an external load is connected, current flows.
• The magnitude of the photocurrent depends on the intensity of incident light (photocurrent is proportional to incident light intensity).
• If a reverse bias is applied. Thus photodiode can be used as a photodetector to detect optical signals.

EXPLANATION:

• We know that photodiode can be used as a photodetector to detect optical signals. Hence, option 1 is correct.

Additional Information

Rectifier:

• A rectifier is an electronic device that converts an alternating current into a direct current by using one or more P-N junction diodes.
• A diode behaves as a one-way valve that allows current to flow in a single direction. This process is known as rectification.

• Polarization: When unpolarised (diffracted) light is passed through a filter, then we obtain polarised light, and this process is called polarization. 

​

• Polarisation tells about the wave nature of light, as light wave is polarised in a particular plane.
• The longitudinal waves cannot to be polarised.

EXPLANATION:

• Polarization does not work on the longitudinal waves. 
• Sound waves cannot be polarized. 
• Only transverse waves can be polarized.

So, by the phenomenon of polarization, electromagnetic wave shows transverse nature.

CONCEPT:

• The mirror which has a spherical reflection surface is called as spherical mirror.
• The spherical mirror whose reflecting surface is away from the center of curvature is called as convex mirror.
• Convex mirror always from a virtual image of any object.
• The mirror whose reflecting surface is inward is called as concave mirror.
• The concave mirror can form both real as well as virtual image of any object.

EXPLANATION:

Given: Focal length (f)  of concave mirror = 16 cm

• Since the object is shifted towards focus by 8 cm so object is between focus and center of curvature of the mirror.
• When the object is between focus and Centre of curvature the image formed by the concave mirror is real and magnified. Hence option 1 is correct.
  

• The real image is always inverted.
• Virtual image is always erect.

CONCEPT:

• Charge: The property of matter which is responsible for electromagnetic phenomenon is called charge.

• The space or region around the current carrying wire/moving electric charge or around the magnetic material in which force of magnetism can be experienced by other magnetic material is called as magnetic field/magnetic induction by that material/current.
• The space or region around the electric charge in which electrostatic force can be experienced by other charge particle is called as electric field by that electric charge.

EXPLANATION:

• When a charge particle is at rest then it only creates electric field around it. So option 1 is wrong.
• When a charge particle is moving with a constant velocity then it equivalent to an electric current which only creates a magnetic field around it. So option 2 is not correct.
• When a charge particle is moving with an acceleration then it produced an electromagnetic wave around it. So option 3 is correct.

CONCEPT:

• Reactance: It is basically the inertia against the motion of the electrons in an electrical circuit.
• The unit of reactance is the ohm. In an electrical circuit.

​\(⇒ Reactance= \frac{V}{I}\)

Where V = potential difference and I = current

• Impedance is a combination of resistance and reactance. It is essentially everything that resists the flow of electrons within an electrical circuit.

​CALCULATION:

Given - Z₁ = 100 ohm, f₁ = 50 Hz and f₂ = 150 Hz

• Since there is only a coil in the circuit so the resultant impedance will be equal to the reactance of the coil.

​Therefore,

⇒ Z = X_L = 2πfL     -----(1)

Where f = frequency of ac current and L = self-inductance of the coil

By equation 1,

⇒ Z₁ = 2πf₁L

⇒ 100 = 2π×50 L

\(\Rightarrow L=\frac{1}{\pi}H\)

So,

⇒ Z₂ = 2πf₂L

⇒ Z2 = 300 ohm

• Hence, option 3 is correct.

CONCEPT:

Capacitor:

• The capacitor is a device in which electrical energy can be stored.
  • In a capacitor two conducting plates are connected parallel to each other and carrying charges of equal magnitudes and opposite sign and separated by an insulating medium.
  • The space between the two plates can either be a vacuum or an electric insulator such as glass, paper, air, or semi-conductor called a dielectric.​

​Parallel plate capacitor:

• A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance.
  • The space between the two plates can either be a vacuum or an electric insulator such as glass, paper, air, or a semi-conductor called a dielectric.​
  • The electric field intensity at the outer region of the parallel plate capacitor is always zero whatever be the charge on the plate.
  • The electric field intensity in the inner region between the plates of a parallel plate capacitor remains the same at every point.
  • When the dielectric medium is filled in the space between the plates of the parallel plate capacitor, its capacitance increases.
• The electric field intensity in the inner region between the plates of a parallel plate capacitor is given as,

• The potential difference between the plates is given as,

• The capacitance C of the parallel plate capacitor is  given as,

Where A = area of the plates, d = distance between the plates, Q = charge on the plates, σ = surface charge density, and K = dielectric constant

CALCULATION:

• When the dielectric medium is filled in the space between the plates of the parallel plate capacitor, its capacitance increases by K times. Hence, option 1 is correct.

CONCEPT:

• Wave is a disturbance which transfers energy from one place to another.
• The wave which is generated due to vibration between electric field and magnetic field and it does not need any medium to travel is called as electromagnetic wave. It can travel through vacuum.
• Radio waves, Infrared waves, Ultraviolet waves and Microwaves are the examples of the electromagnetic wave.

EXPLANATION:

• The infrared waves present in the light ray coming from the sun is responsible for the heat energy.
• Thus infrared waves are called as waves of heat energy. So option 2 is correct.
• The Radio waves, Ultraviolet waves and Microwaves don’t have the heat energy. So options 1, 3 and 4 are wrong.

CONCEPT:

Capacitive reactance:

• The capacitive reactance is the opposition offered by the capacitor in an AC circuit to the flow of ac current.
• Its SI unit is Ohm(Ω).
• The capacitive reactance is given as,

\(⇒ X_C=\frac{1}{ω C}=\frac{1}{2π fC}\)

Where ω = angular frequency, f = frequency of ac current and C = capacitance

Impedance:

• Impedance is essentially everything that obstructs the flow of electrons within an electrical circuit.
• For a pure capacitor, the capacitive reactance is equal to the impedance.

AC voltage applied to a capacitor:

• When an AC voltage is applied to a capacitor, the current in the circuit is given as,

\(⇒ I=\frac{V}{X_C}\)

• In a pure capacitor circuit, the current is ahead of the voltage by one-fourth of a period.

EXPLANATION:

We know that the capacitive reactance is given as,

\(⇒ X_C=\frac{1}{2π fC}\)

\(⇒ X_C\propto\frac{1}{f}\)     -----(1)

Where f = frequency of ac current and C = capacitance

• By equation 1 it is clear that the capacitive reactance is inversely proportional to the frequency of the AC voltage.
• Therefore if the frequency of the AC voltage is increased in the circuit containing a capacitor, then the capacitive reactance will decrease. Hence option 2 is correct.

CONCEPT:

Faraday's Laws of Electromagnetic Induction:

• First law: Whenever the number of magnetic lines of force (magnetic flux) passing through a circuit changes an emf is produced in the circuit called induced emf. The induced emf persists only as long as there is change or cutting of flux.
• Second law: The induced emf is given by the rate of change of magnetic flux linked with the circuit i.e.

\(ε = -N\frac{d\phi}{dt}\)

• Gauss Law: According to gauss’s law, total electric flux through a closed surface enclosing a charge is 1/ϵ0 times the magnitude of charge enclosed.

\(i.e.~{{\mathbf{\Phi }}_{ele}}=\frac{\left( {{Q}_{in}} \right)}{{{\epsilon }_{0}}}\)

\(i.e.~\oint \vec{E}\cdot d\vec{S}=\frac{{{Q}_{in}}}{{{\epsilon }_{0}}}\)

Where

dΦ = change in magnetic flux,

Φele = electric flux 

dt = change in time,

e = induced emf.

 Qin = charge enclosed the sphere,

ϵ0 = permittivity of space (8.85 × 10-12 C2/Nm2),

dS = surface area

Calculation:

From the above explanation, we can see that,

• Option 2 is Gauss theorem which states that when a closed surface encloses a charge (q), the electric flux through the surface is \(\frac{q}{\epsilon_ \circ}.\)
• Electromagnetic induction was discovered by Faraday. The results of his experiments are called Faraday's laws of electromagnetic induction and are as follows:

1. When the magnetic field associated with a circuit is changed, induced e.m.f. is produced in the circuit.
2. This e.m.f lasts as long as the change in the magnetic field continues.
3. The rate of change of the magnetic field and the magnitude of the induced e.m.f. are directly proportional to each other.

Hence option 1, 3 and 4 are correct 

CONCEPT:

Electric flux:

• It is defined as the number of electric field lines associated with an area element.
• Electric flux is a scalar quantity.
• The SI unit of the electric flux is N-m2/C.
• If the electric field is E and the area is A then the electric flux associated with the area is given as,

\(⇒ ϕ = \vec{E}.\vec{A}\)

⇒ ϕ = EAcosθ

Where θ = angle between the surface and the electric field

EXPLANATION:

• If the electric field is E and the area is A then the electric flux associated with the area is given as,

\(⇒ ϕ = \vec{E}.\vec{A}\)     -----(1)

Where θ = angle between the surface and the electric field

• By equation 1 it is clear that the electric flux is a dot product of electric field intensity and area.
• Hence, option 3 is correct.

CONCEPT:

Gauss's law:

• According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is \(\frac{1}{ϵ_o}\) the charge enclosed by the closed surface.

\(⇒ ϕ=\frac{Q}{ϵ_o}\)

Where ϕ = electric flux linked with a closed surface, Q = total charge enclosed in the surface, and ϵo = permittivity​

• Overall the Electric Field due to the hollow conducting sphere is given as 

⇒ E = 0, ( r < R )

\(⇒ E = \frac{q}{4\pi \epsilon _0 R^2}\) ( r = R)

\(⇒ E = \frac{q}{4\pi \epsilon _0 r^2}\) (r > R)

where r is the distance of the point from the center of the sphere, and R is the radius of the sphere.

\(\Rightarrow {{E}}=\frac{1}{4\pi {{\epsilon }_{0}}}\times \frac{q}{{{R}^{2}}}⇒ {E}\propto \frac{1}{{{R}^{2}}}\) (Electric field outside)

• Therefore, graphically it can be represented as

EXPLANATION:

For r < R (Inside hollow sphere)

By applying gauss law

\(\Rightarrow \oint {{E}_{in}}.dS~=\frac{{{q}_{en}}}{{{\epsilon }_{0}}}=0\)

• This is because the charge will be distributed across the surface but we are measuring the Electric field inside the hollow surface hence net charge enclosed will be zero i.e., qen = 0. Therefore, inside a hollow conducting sphere electric field. Hence option 1 is correct.

Nicol prism:

1). Nicol prism is used for producing a polarised beam of light from an un-polarised beam.

2). It is based on the principle of action which involves refraction as it passes into the lower half of the prism. It leaves the prism as polarised light after undergoing another refraction as it exits the far right side of the prism. Thus its action is based on double refraction.

Double refraction is an optical property in which a single ray of unpolarized light entering an anisotropic medium is split into two rays, each traveling in a different direction.

CONCEPT:

Power:

• The rate of work done by an electric current is called power. It is denoted by P. The SI unit of power is the watt (W).
• Power dissipation is given by:

⇒ Power (P) = V I = V2/R = I2 R

Where V is the potential difference across resistance, I is current flowing and R is resistance.

• Heat dissipation/work done/Electrical energy is given by:

⇒ Heat dissipated (H) = Power (P) × Time (t)

CALCULATION:

Given that: Power (P) = 400 W, Per day working time = 8 hrs, Total days = 30 days, and Cost per unit = 2.5 per kWh

⇒ Total time (t) = 8 × 30 = 240 hrs

• The total energy consumed by the refrigerator in 30 days would be:

⇒ Energy (H) = P × t = 400 × 240 hrs = 96000 Wh = 96 kWh

⇒ The cost of energy to operate the refrigerator for 30 days = total energy × cost per unit 

⇒ ​The cost of energy to operate the refrigerator for 30 days = 96 kW h × Rs 2.50 per kW h = Rs 240.00.

• Hence option 4 is correct.

CONCEPT:​

• Resistances in series combination: When two or more resistances are connected one after another such that the same current flows through them are called as resistances in series.

The net resistance/equivalent resistance (R) of resistances in series is given by:

Equivalent resistance, R = R1 + R2

• Resistances in parallel combination: When the terminals of two or more resistances are connected at the same two points and the potential difference across them is equal is called resistances in parallel.

The net resistance/equivalent resistance(R) of resistances in parallel is given by:

\(\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

• Ohm’s law: At constant temperature and other physical quantities, the potential difference across a current-carrying wire is directly proportional to the current flowing through it.

V= R I

Where V is the potential difference, R is resistance and I is current.

CALCULATION:

Given that: 

Here 15 Ω and 30 Ω are in parallel combination.

So net resistance (R) will be:

\(\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

\(\frac{1}{R} = \frac{1}{{{15}}} + \frac{1}{{{30}}} = 3/30 = 1/10\)

So R = 10 Ω 

Now 10 Ω and 5 Ω are in series combination,

so equivalent resistance (R') = 10 Ω + 5 Ω = 15 Ω 

Now use ohm's law:

Electric current (I) = V/R' = 4.5/15 = 0.3 A

So reading of ammeter (I) = 0.3 A

Hence option 1 is correct.

CONCEPT:

• Binding Energy of the nucleus: The energy that holds a nucleus together. or the energy required to disassemble a nucleus into its constituent protons and neutrons completely is called binding energy.
  • or the energy that would be liberated by combining individual protons and neutrons into a single nucleus is called binding energy.
• The higher the binding energy of the nucleus the more stable the atomic nucleus will be.
• The weak nuclear binding force is responsible for radioactive decay.

EXPLANATION:

• The particles present in the nucleus (proton and neutron) are called nucleons.
  • The sum of the individual masses of various particles in the nucleus must be equal to the nuclear mass, but this is not happening.
  • The nuclear mass is somewhat less than the sum of the individual masses of various nuclear particles.
  • The difference between the actual nuclear mass and the expected nuclear mass is called the mass defect.
  • The energy equivalent to mass defect is responsible for holding the nucleons together and is called the binding energy of the nucleus.
• So binding energy is the loss of energy from the nucleus during its formation.
• Hence, option 2 is correct.

CONCEPT:

• Young's Double Slit Experiment (YDSE): Monochromatic light (single wavelength) falls on two narrow slits S1 and S2 which are very close together act as two coherent sources when waves coming from two coherent sources, (S1, S2) superimposes on each other, an interference pattern is obtained on the screen.
  • In Young's Double Slit Experiment alternate bright and dark bands obtained on the screen. These bands are called Fringes.

• Fringe width (β): The separation between any two consecutive bright or dark fringes is called fringe width.
  • In Young's Double Slit Experiment all fringes are of equal width.

\( \beta = \frac{{\lambda \;D}}{d}\)

Where d = Distance between slits D = Distance between slits and screen, λ = Wavelength of monochromatic light emitted from the source

EXPLANATION:

Given - D₁ = 2D and d₁ = d/2

• In Young's Double Slit Experiment, the fringe width is written as

\(\Rightarrow \beta = \frac{{\lambda \;D}}{d}\)       ------ (1)

• When the distance between slit and screen is doubled, while the separation of the slit is halved, then fringe width will be

\(\Rightarrow \beta_1 = \frac{{\lambda \;D_1}}{d_1}=\frac{{\lambda \;2D}}{\frac{d}{2}}\)

\(\Rightarrow \beta_1 = \frac{{4\lambda \;D}}{d}=4\beta \)                 \([\because \beta = \frac{{\lambda \;D}}{d}]\)

CONCEPT:

Electric Field Intensity:

• The electric field intensity at any point is the strength of the electric field at the point.
• It is defined as the force experienced by the unit positive charge placed at that point.

\(\vec E = \frac{{\vec F}}{{{q_o}}}\)

Where F = force and qo = small test charge​

Electric potential (V):

• The potential difference between two points in an electric field may be defined as the amount of work done in moving a unit positive charge from one point to the other against the electrostatic force i.e.,

Electric dipole:

• When two equal and opposite charges are separated by a small distance then this combination of charges is called an electric dipole.
• The strength of an electric dipole is measured by a quantity known as a dipole moment i.e.

\(\vec P = q \times \overrightarrow {2a} \)

Where q = charge and 2a = distance between two charged particles

EXPLANATION:

• The electric field intensity at any point due to a short electric dipole is 

​

\(\Rightarrow {|\vec E|}=\frac{p\sqrt{3\,cos^2\, \theta+1}}{4πϵ_or^3}\)

Here p, π, and ϵo are constant, therefore

\(\Rightarrow E\propto \frac{1}{r^3}\)

• The electric potential due to point of dipole given by 

\( \Rightarrow V = \frac{{p{\rm{cos}}\theta }}{{4\pi {\varepsilon _0}{r^2}}}\)

Here p, π, and ϵo are constant, therefore

\(\Rightarrow V\propto \frac{1}{r^2}\)

Electric field: An electric field is a physical region that surrounds each electric charge and exerts a force on all other charges in the region, either by attraction or repulsion.

Electric fields originate from electric charges, or from time-varying magnetic fields.

​Magnetic field: A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

Important Conclusions:

• A charge that is moving in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.
• Moving charges will generate a magnetic field.
• Electric fields originate from electric charges, or from time-varying magnetic fields.
• Moving charges will generate a magnetic field.
• So a moving electrical charge produces an electric as well as a magnetic field. Hence, option 1 is correct.

If the circular loop is considered as a magnetic dipole, then the dipole moment is the product of current and area. But the circular loop has multiple number of turns.

Therefore, the magnitude of dipole moment = area × current × number of turns.

Substituting the values we get,

Magnitude of dipole moment = πr² × I × N

= 3.14 × 0.05² × 10 × 6

= 0.471 Am²

The dipole moment is in perpendicular direction with the plane of loop.

CONCEPT:

• Nuclear Decay: Nuclear decay is the process by which an unstable isotope of a particular element spontaneously transforms into a new element by the emission of ionizing radiation. Example: Decay of C-14 to C-12.
• In the fission reaction α, β, and γ rays are released.

• α-decay: when an α particle is leaving the parent atom
  • It reduces the atomic number (Z) by 2.
  • It reduces the atomic mass (A) by 4.
• β-decay: when a β particle is leaving the parent atom
  • It increases the atomic number (Z) by 1.
  • It does not change in atomic mass.
• γ - Decay: it neither changes the atomic number nor atomic mass.

​EXPLANATION:

• A beta particle is emitted by radioactive nucleus during Beta decay. 

Two kinds of Beta decay:

• Beta minus decay: Neutron is converted into a proton and this process results into an electron and an electron anti-neutrino.

\(n \to p + {\beta ^ - }\)

• Beta plus decay: A proton is converted to a neutron and this process results into a positron and an electron neutrino.

\(p\to n+ {\beta ^ +}\)

The correct answer is Convex Mirrors.

Key Points

• Convex mirror:
  • These are the spherical mirror that is curved outward and the image obtained is virtual, diminished and erect for a real object.
  • Commonly used as rear-view (wing) mirrors in vehicles.
  • Convex mirrors are used in automobiles because they give an upright (not inverted), though diminished (smaller) image and provide a wider field of view as they are curved outwards. However, they don’t provide correct distant perception.
  • Hence, the outside rear-view mirror of modern automobiles is marked with the warning objects in the mirror are closer than they appear. Hence, Option 4 is correct.
  • These mirrors are fitted on the sides of the vehicle, enabling the driver to see traffic behind him/her to facilitate safe driving.
  • It enables the driver to view a much larger area than would be possible with a plane mirror.
  • In big showrooms and departmental stores, convex mirrors are used to have a view of the customers entering in as well as going out.

Additional Information

• Concave mirrors:
  • These are the spherical mirror that is curved inward and the image obtained from these mirrors depend on the placement of the object.
  •  Used in torches, searchlights, and vehicles headlights to get powerful parallel beams of light.
  • As shaving mirrors to see a larger image of the face.
  • The dentists use concave mirrors to see large images of the teeth of patients.
  • Large concave mirrors are used to concentrate sunlight to produce heat in solar furnaces.

• Plane mirror:
  • The images formed from a plane mirror are the reflected images in their normal proportions but reversed from left to right. These are the most widely used mirrors.

CONCEPT:

• Bar magnet in a magnetic field: When a bar magnet is left free in a uniform magnetic field if it aligns itself in the directional field.

EXPLANATION:

Given - θ₁ = 0° and θ₂ = 90°  

• The potential energy of a magnetic dipole in a magnetic field is the energy possessed by the dipole due to its particular position in the field.
• When a magnetic dipole of moment  is held at an angle θ with the direction of a uniform magnetic field  , the magnitude of the torque acting on the dipole is
  τ = M B sin θ       ---- (i)

Total work done in rotating the dipole from θ₁ to θ₂ is

\(W = \mathop \smallint \nolimits_{{\theta _1}}^{{\theta _2}} {\rm{MBsin\theta \;d\theta \;}} = MB\left[ { - cos\theta } \right]_{{\theta _1}}^{{\theta _2}} = - MB\left[ {cos{\theta _2} - cos{\theta _1}} \right]\)

W = - MH (cos 90° – cos 0°)

W = - MH (0  – 1)

W = MH

CONCEPT:

Total Internal Reflection (TIR):

• When a ray of light goes from denser to the rarer medium it bends away from the normal and as the angle of incidence in denser medium increases, the angle of refraction in the rarer medium also increases and at a certain angle, angle of refraction becomes 90°, this angle of incidence is called critical angle (C).
• When Angle of incidence exceeds the critical angle than light ray comes back into the same medium after reflection from the interface. This phenomenon is called Total internal reflection (TIR).

Conditions for TIR: 

1. The ray must travel from denser medium to rarer medium.
2. The angle of incidence i must be greater than the critical angle C.

​EXPLANATION:

• From above it is clear that the ray must travel from denser medium to rarer medium and the angle of incidence i must be greater than the critical angle C. Therefore option 2 is correct.

CONCEPT:

• A rectangular coil always acts as a magnetic dipole of dipole moment M.
• The dipole moment of a rectangular coil is given by:
• The dipole moment of the coil (M) = N I A
  Where N is number of turns, I is current in the coil and A is the area of the rectangular coil

• The torque of the magnetic field on any magnetic dipole moment is given by;

Torque (T) = M × B = MB Sinθ
Where M × B = cross product of Area vector A and magnetic field vector B and θ = the angle between area vector A and magnetic field B.
EXPLANATION:

The dipole moment of the coil (M) = N I A

Torque on rectangular coil (T) = M × B = MB Sinθ = N I A B Sinθ

• According to the above formula of the torque on a rectangular coil in a magnetic field, torque will be large when the number of turns is large. So option 1 is correct and option 2 is wrong.
• If the magnetic field is perpendicular to the plane of coil then it will be along the area vector. So angle between area vector and the magnetic field will be zero and finally the torque will be zero. So option 3 is wrong.
• If the area will be small then torques will also be small. So option 4 is wrong.

The correct answer is option 2) i.e. Skywave propagation

CONCEPT:

• Propagation of electromagnetic waves: An electromagnetic wave is transmitted between two points by using an antenna in the transmitter which radiates these waves through a medium, and is made available at another antenna at the receiving end. This is the basic principle of communication systems.
• The propagation is dependent on different layers of Earth and can be classified as follows.

EXPLANATION:

• The given signal has a frequency of 20 MHz.
• Hence, skywave propagation is the most suitable mode of transmission.

CONCEPT:

• The capacitance of a capacitor (C): The capacitance of a conductor is the ratio of charge (Q) to it by a rise in its potential (V), i.e.

C = Q/V

For a Parallel Plate Capacitor:

• A parallel plate capacitor consists of two large plane parallel conducting plates of area A and separated by a small distance d.

• Mathematical expression for the capacitance of the parallel plate capacitor is given by:

\(\Rightarrow C = \frac{{{\epsilon_o}A}}{d}\)

Where C = capacitance, A = area of the two plates, ε = dielectric constant (simplified!), d = separation between the plates.

• The unit of capacitance is the farad, (symbol F ).

EXPLANATION:

• From the above discussion, it is clear that the capacitance of the capacitor is inversely proportional to the distance between the parallel plate capacitor.
• Hence, if the distance between the parallel plate capacitor is increased then the capacitance of the capacitor will decrease. Therefore option 3 is correct.

CONCEPT:

• Conductor: An object in which charge can flow in one or more directions.
  • Materials made of metal are common electrical conductors.

EXPLANATION:

• In the case of the conductor, the valance band and conduction band overlap with each other. So now there is an excess of electrons in the conduction band of it.
• So, Resistance increases with an increase in temperature for conductors and vice versa.

• Hence the correct answer is option 4.

Additional Information

• Resistance decreases with an increase in temperature for semiconductors and vice versa.

CONCEPT:

• When a p-type semiconductor crystal is brought into close contact with an n-type semiconductor crystal, the resulting arrangement is called p-n junction diode.

EXPLANATION:

Forward Bias:

• When the negative terminal of the battery is connected to the N – side and the positive terminal to P –side, then the connection is called forward bias.
• In forward biasing, the applied voltage V of the battery mostly drops across the depletion region and the voltage drops across the p-side and n-side of the p-n junction is negligibly small.
• It is because the resistance of the depletion region is very high as it has no free charge carriers.
• A p - n junction diode allows electric current in one direction and blocks electric current in another direction.
• It allows electric current when it is forward biased and blocks electric current when it is reverse biased.
• An actual diode offers a very small resistance (ideally zero but actually very low) when forward biased and is called forward resistance. Therefore option 3 is correct.

• In forward biasing the forward voltage opposes the potential barrier Vb. As a result of it, the potential barrier height is reduced and the width of the depletion layer decreases.
• As forward voltage is increased, at a particular value the depletion region becomes very much narrow such that a large number of majority charge carriers can cross the junction.

CONCEPT:

• Cyclotron: A cyclotron is an apparatus in which atomic and subatomic charged particles are accelerated by an alternating electric field while following a spiral or circular path in a magnetic field.
• A cyclotron is used to accelerate, positively charged particles but negatively charged particles and neutrally charged particles   (e.g neutron) cannot be accelerated in the cyclotron.

EXPLANATION:

• Cyclotron can accelerate positively charged particles through the electric field only. 
• The working of the cyclotron is based on the fact that a positively charged particle can be accelerated to sufficiently high energy with the help of similar values of the oscillating electric field by making it to cross the same electric field time and again with the use of a strong magnetic field.

So option 1 is correct.

CONCEPT :

• Power is the rate of doing work in an electrical circuit, the various equation for power calculation is given by

\(⇒ P = VI=I^2R=\frac{V^2}{R}\)

Where V = Potential difference , R = Resistance, I= Current

CALCULATION :

Given - P = 20 W,  I = 4Amp, V = 220 V, R = ?

• The resistance of the motor can be written as 

\(⇒ R = \frac{P}{I^{2}}\)

\(⇒ R = \frac{20}{4^{2}} = 1.25 \Omega\)

• The applied EMF can be calculated as

⇒ V = E - I R

⇒ E = V + IR = 220 + 4 × 1.25 = 225V

• Hence, option 2 is the answer